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Polish United Workers' Party

The Polish United Workers' Party (Polska Zjednoczona Partia Robotnicza, ), commonly abbreviated to PZPR, was the communist party which ruled the Polish People's Republic as a one-party state from 1948 to 1989. The PZPR had led two other legally permitted subordinate minor parties together as the Front of National Unity and later Patriotic Movement for National Rebirth. Ideologically, it was based on the theories of Marxism-Leninism, with a strong emphasis on left-wing nationalism. The Polish United Workers' Party had total control over public institutions in the country as well as the Polish People's Army, the UB and SB security agencies, the Citizens' Militia (MO) police force and the media.

The falsified 1947 Polish legislative election granted the Communist Polish Workers' Party (PPR) complete political authority in post-war Poland. The PZPR was founded forthwith in December 1948 through the unification of the PPR and the Polish Socialist Party (PPS). From 1952 onward, the position of "First Secretary" of the Polish United Workers' Party was de facto equivalent to Poland's head of state. Throughout its existence, the PZPR maintained close ties with ideologically-similar parties of the Eastern Bloc, most notably the Socialist Unity Party of Germany, Communist Party of Czechoslovakia and the Communist Party of the Soviet Union. Between 1948 and 1954, nearly 1.5 million individuals registered as Polish United Workers' Party members, and membership rose to 3 million by 1980.

The party's primary objective was to impose socialist agenda unto Polish society. The communist government sought to nationalize all institutions. Some concepts imported from abroad, such as large-scale collective farming and secularization, failed in their early stages. The PZPR was considered more liberal and pro-Western than its counterparts in East Germany or the Soviet Union, and was more averse to radical politics. Although propaganda was utilized in major media outlets like Trybuna Ludu (lit. 'People's Tribune') and televised Dziennik ('Journal'), censorship became ineffective by the mid-1980s and was gradually abolished. On the other hand, the Polish United Worker's Party was responsible for the brutal pacification of civil resistance and protesters in the Poznań protests of 1956, the 1970 Polish protests and throughout martial law between 1981 and 1983. The PZPR also initiated a bitter anti-Semitic campaign during the 1968 Polish political crisis, which forced the remainder of Poland's Jews to emigrate.

Amidst the ongoing political and economic crises, the Solidarity movement emerged as a major anti-bureaucratic social movement that pursued social change. With communist rule being relaxed in neighbouring countries, the PZPR systematically lost support and was forced to negotiate with the opposition and adhere to the Polish Round Table Agreement, which permitted free democratic elections. The elections on 4 June 1989 proved victorious for Solidarity, thus bringing 40-year communist rule in Poland to an end. The Polish United Workers' Party was dissolved in January 1990.

Programme and goals
Until 1989, the PZPR held dictatorial powers (the amendment to the constitution of 1976 mentioned "a leading national force") and controlled an unwieldy bureaucracy, the military, the secret police, and the economy. Its main goal was to create a Communist society and help to propagate Communism all over the world. On paper, the party was organised on the basis of democratic centralism, which assumed a democratic appointment of authorities, making decisions, and managing its activity. These authorities decided about the policy and composition of the main organs; although, according to the statute, it was a responsibility of the members of the congress, which was held every five or six years. Between sessions, the regional, county, district and work committees held party conferences. The smallest organizational unit of the PZPR was the Fundamental Party Organization (FPO), which functioned in workplaces, schools, cultural institutions, etc.

The main part in the PZPR was played by professional politicians, or the so-called "party's hardcore", formed by people who were recommended to manage the main state institutions, social organizations, and trade unions. The crowning time of the PZPR development (the end of the 1970s) consisted of over 3.5 million members. The Political Office of the Central Committee, Secretariat and regional committees appointed the key posts within the party and in all organizations having ‘state’ in its name – from central offices to even small state and cooperative companies. It was called the nomenklatura system of state and economy management. In certain areas of the economy, e.g., in agriculture, the nomenklatura system was controlled with the approval of the PZPR and by its allied parties, the United People's Party (agriculture and food production), and the Democratic Party (trade community, small enterprise, some cooperatives). After martial law began, the Patriotic Movement for National Rebirth was founded to organize these and other parties.

Establishment and Sovietization period
The Polish United Workers' Party was established at the unification congress of the Communist Polish Workers' Party and the Polish Socialist Party during meetings held at the main building of the Warsaw University of Technology from 15 to 21 December 1948. The unification was possible because the PPS had effectively been taken over by pro-Communist fellow travelers, and the activists who opposed unification had been forced out of the party. Similarly, the members of the PPR who were accused of "rightist–nationalist deviation" (odchylenie prawicowo-nacjonalistyczne) were expelled. Thus, the merger was actually an absorption of the PPS by the PPR, resulting in what was a renamed and enlarged PPR for all intents and purposes.

"Rightist-nationalist deviation" was a political propaganda term used by the Polish Stalinists against prominent activists, such as Władysław Gomułka and Marian Spychalski who opposed Soviet involvement in the Polish internal affairs, as well as internationalism displayed by the creation of the Cominform and the subsequent merger that created the PZPR. It is believed that it was Joseph Stalin who put pressure on Bolesław Bierut and Jakub Berman to remove Gomułka and Spychalski as well as their followers from power in 1948. It is estimated that over 25% of socialists were removed from power or expelled from political life.

Bolesław Bierut, an NKVD agent and a hardline Stalinist, served as first Secretary General of the ruling PZPR from 1948 to 1956, playing a leading role in imposing communism and the installation of its repressive regime. He had served as President since 1944 (though on a provisional basis until 1947). After a new constitution abolished the presidency, Bierut took over as Prime Minister, a post he held until 1954. He remained party leader until his death in 1956.

Bierut oversaw the trials of many Polish wartime military leaders, such as General Stanisław Tatar and Brig. General Emil August Fieldorf, as well as 40 members of the Wolność i Niezawisłość (Freedom and Independence) organisation, various Church officials and many other opponents of the new regime including Witold Pilecki, condemned to death during secret trials. Bierut signed many of those death sentences.

Bierut's mysterious death in Moscow in 1956 (shortly after attending the 20th Congress of the Communist Party of the Soviet Union) gave rise to much speculation about poisoning or a suicide, and symbolically marked the end of Stalinism era in Poland.

Gomułka's autarchic communism

In 1956, shortly after the 20th Congress of the Communist Party of the Soviet Union, the PZPR leadership split into two factions, dubbed Natolinians and Puławians. The Natolin faction – named after the place where its meetings took place, in a government villa in Natolin – were against the post-Stalinist liberalization programs (Gomułka thaw). The most well known members included Franciszek Jóźwiak, Wiktor Kłosiewicz, Zenon Nowak, Aleksander Zawadzki, Władysław Dworakowski, Hilary Chełchowski.

The Puławian faction – the name comes from the Puławska Street in Warsaw, on which many of the members lived – sought great liberalization of socialism in Poland. After the events of Poznań June, they successfully backed the candidature of Władysław Gomułka for First Secretary of party, thus imposing a major setback upon Natolinians. Among the most prominent members were Roman Zambrowski and Leon Kasman. Both factions disappeared towards the end of the 1950s.

Initially very popular for his reforms and seeking a "Polish way to socialism", and beginning an era known as Gomułka's thaw, he came under Soviet pressure. In the 1960s he supported persecution of the Roman Catholic Church and intellectuals (notably Leszek Kołakowski who was forced into exile). He participated in the Warsaw Pact intervention in Czechoslovakia in 1968. At that time he was also responsible for persecuting students as well as toughening censorship of the media. In 1968, he incited an anti-Zionist propaganda campaign, as a result of Soviet bloc opposition to the Six-Day War.

In December 1970, a bloody clash with shipyard workers in which several dozen workers were fatally shot forced his resignation (officially for health reasons; he had in fact suffered a stroke). A dynamic younger man, Edward Gierek, took over the Party leadership and tensions eased.

Gierek's economic opening
In the late 1960s, Edward Gierek had created a personal power base and become the recognized leader of the young technocrat faction of the party. When rioting over economic conditions broke out in late 1970, Gierek replaced Gomułka as party first secretary. Gierek promised economic reform and instituted a program to modernize industry and increase the availability of consumer goods, doing so mostly through foreign loans. His good relations with Western politicians, especially France's Valéry Giscard d'Estaing and West Germany's Helmut Schmidt, were a catalyst for his receiving western aid and loans.

The standard of living improved in Poland in the early 1970s, the economy, however, began to falter during the 1973 oil crisis, and by 1976 price hikes became necessary. New protests broke out in June 1976, and although they were forcibly suppressed, the planned price increases were suspended. High foreign debts, food shortages, and an outmoded industrial base compelled a new round of economic reforms in 1980. Once again, price increases set off protests across the country, especially in the Gdańsk and Szczecin shipyards. Gierek was forced to grant legal status to Solidarity and to concede the right to strike. (Gdańsk Agreement).

Shortly thereafter, in early September 1980, Gierek was replaced by Stanisław Kania as General Secretary of the party by the Central Committee, amidst much social and economic unrest. Kania admitted that the party had made many economic mistakes, and advocated working with Catholic and trade unionist opposition groups. He met with Solidarity leader Lech Wałęsa, and other critics of the party. Though Kania agreed with his predecessors that the Communist Party must maintain control of Poland, he never assured the Soviets that Poland would not pursue actions independent of the Soviet Union. On 18 October 1981, the Central Committee of the Party withdrew confidence in him, and Kania was replaced by Prime Minister (and Minister of Defence) Gen. Wojciech Jaruzelski.

Jaruzelski's autocratic rule
On 11 February 1981, Jaruzelski was elected Prime Minister of Poland and became the first secretary of the Polish United Workers' Party on 18 October 1981. Before initiating the plan of suppressing Solidarity, he presented it to Soviet Premier Nikolai Tikhonov. On 13 December 1981, Jaruzelski imposed martial law in Poland.

In 1982, Jaruzelski revitalized the Front of National Unity, the organization the Communists used to manage their satellite parties, as the Patriotic Movement for National Rebirth.

In 1985, Jaruzelski resigned as prime minister and defence minister and became chairman of the Polish Council of State, a post equivalent to that of president or a dictator, with his power centered on and firmly entrenched in his coterie of "LWP" generals and lower rank officers of the Polish People's Army.

Breakdown of autocracy
The attempt to impose a naked military dictatorship notwithstanding, the policies of Mikhail Gorbachev stimulated political reform in Poland. By the close of the tenth plenary session in December 1988, the Polish United Workers Party was forced, after strikes, to approach leaders of Solidarity for talks.

From 6 February to 15 April 1989, negotiations were held between 13 working groups during 94 sessions of the roundtable talks.

These negotiations resulted in an agreement that stated that a great degree of political power would be given to a newly created bicameral legislature. It also created a new post of president to act as head of state and chief executive. Solidarity was also declared a legal organization. During the following Polish elections the Communists won 65 percent of the seats in the Sejm, though the seats won were guaranteed and the Communists were unable to gain a majority, while 99 out of the 100 seats in the Senate — all freely contested — were won by Solidarity-backed candidates. Jaruzelski won the presidential ballot by one vote.

Jaruzelski was unsuccessful in convincing Wałęsa to include Solidarity in a "grand coalition" with the Communists and resigned his position of general secretary of the Polish United Workers Party. The PZPR' two allied parties broke their long-standing alliance, forcing Jaruzelski to appoint Solidarity's Tadeusz Mazowiecki as the country's first non-communist prime minister since 1948. Jaruzelski resigned as Poland's President in 1990, being succeeded by Wałęsa in December.

Dissolution of the PZPR

Starting from January 1990, the collapse of the PZPR became inevitable. All over the country, public occupations of the party buildings started in order to prevent stealing the party's possessions and destroying or taking the archives. On 29 January 1990, XI Congress was held, which was supposed to recreate the party. Finally, the PZPR dissolved, and some of its members decided to establish two new social-democratic parties. They got over $1 million from the Communist Party of the Soviet Union known as the Moscow loan. Of this, $300,000 was spent to set up Trybuna, a left-wing newspaper, $200,000 on severance pay for employees of PUWP, $500,000 given back to the Russians, and $200,000 circulated to pay off the loan in installments.

The former activists of the PZPR established the Social Democracy of the Republic of Poland (in Polish: Socjaldemokracja Rzeczpospolitej Polskiej, SdRP), of which the main organizers were Leszek Miller and Mieczysław Rakowski. The SdRP was supposed (among other things) to take over all rights and duties of the PZPR, and help to divide out the property. Up to the end of the 1980s, it had considerable incomes mainly from managed properties and from the RSW company ‘Press- Book-Traffic’, which in turn had special tax concessions. During this period, the income from membership fees constituted only 30% of the PZPR's revenues. After the dissolution of the Polish United Workers' Party and the establishment of the SdRP, the rest of the activists formed the Social Democratic Union of the Republic of Poland (USdRP), which changed its name to the Polish Social Democratic Union, and The 8th July Movement. The Moscow loan caused controversy in Polish politics and occasioned a year-long prosecution effort. In the end nobody was sentenced.

At the end of 1990, there was an intense debate in the Sejm on the takeover of the wealth that belonged to the former PZPR. Over 3000 buildings and premises were included in the wealth and almost half of it was used without legal basis. Supporters of the acquisition argued that the wealth was built on the basis of plunder and the Treasury grant collected by the whole society. Opponents of SdRP claimed that the wealth was created from membership fees; therefore, they demanded wealth inheritance for SdPR which at that time administered the wealth. Personal property and the accounts of the former PZPR were not subject to control of a parliamentary committee.

On 9 November 1990, the Sejm passed "The resolution about the acquisition of the wealth that belonged to the former PZPR". This resolution was supposed to result in a final takeover of the PZPR real estate by the Treasury. As a result, only a part of the real estate was taken over mainly for a local government by 1992, whereas a legal dispute over the other party carried on till 2000. Personal property and finances of the former PZPR practically disappeared. According to the declaration of SdRP Members of Parliament, 90–95% of the party's wealth was allocated for gratuity or was donated for social assistance.

Structure
The highest statutory authority of the Voivodeship party organization was the voivodeship conference, and in the period between conferences – the PZPR voivodeship committee. To drive current party work, the provincial committee chose the executive. Voivodeship conferences convened a provincial committee in consultation with the Central Committee of PZPR – formally at least once in year. Plenary meetings of the Voivodeship committee were to be convened at least every two months and executive meetings – once a week.

In practice, the frequency of holding provincial conferences and plenary meetings KW deviated from the statutory standards were held less often. Dates and basic Topics of session of Voivodeship party conferences and plenary sessions of Voivodeship Committee PZPR in the provinces of Poland were generally correlated with dates and topics of plenary sessions Central Committee of the PZPR. They were devoted mainly to "transferring" resolutions and decisions of the Central Committee to the provincial party organization. The provincial committee had no freedom in shaping the original, its own meeting plan. The initiative could be demonstrated – in accordance with the principle of democratic centralism – only in the implementation of resolutions and orders of instances supreme.

The dependence of the Voivodeship party organization and its authorities was also determined by that its activity was financed almost entirely from a subsidy received from the Central Committee of PZPR. Membership fees constituted no more than 10% of revenues. The activities of the Voivodeship Committee between PZPR Voivodeship conferences were formally controlled by the Audit Committee (elected during these conferences). Initially only examined the budget implementation and accounting of PZPR Voivodeship Committee. In the following years, the scope of its activities was expanded, including control over the management of party membership cards, security OF confidential documents, how to deal with complaints and complaints addressed to the party. The number of inspections carried out grew systematically, and the work of committees accepted more planned and formalized character.

Building
The Central Committee had its seat in the Party's House, a building erected by obligatory subscription from 1948 to 1952 and colloquially called White House or the House of Sheep. Since 1991 the Bank-Financial Center "New World" is located in this building. From 1991–2000 the Warsaw Stock Exchange also had its seat there.

Party leaders
By the year 1954 the head of the party was the Chair of Central Committee:

Presidents

 * Wojciech Jaruzelski
 * Aleksander Kwaśniewski

Prime ministers

 * Józef Oleksy
 * Włodzimierz Cimoszewicz
 * Leszek Miller
 * Marek Belka

European Commissioners

 * Danuta Hübner

title: Polish United Workers' Party


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Peritoneum

The peritoneum is the serous membrane forming the lining of the abdominal cavity or coelom in amniotes and some invertebrates, such as annelids. It covers most of the intra-abdominal (or coelomic) organs, and is composed of a layer of mesothelium supported by a thin layer of connective tissue. This peritoneal lining of the cavity supports many of the abdominal organs and serves as a conduit for their blood vessels, lymphatic vessels, and nerves.

The abdominal cavity (the space bounded by the vertebrae, abdominal muscles, diaphragm, and pelvic floor) is different from the intraperitoneal space (located within the abdominal cavity but wrapped in peritoneum). The structures within the intraperitoneal space are called "intraperitoneal" (e.g., the stomach and intestines), the structures in the abdominal cavity that are located behind the intraperitoneal space are called "retroperitoneal" (e.g., the kidneys), and those structures below the intraperitoneal space are called "subperitoneal" or "infraperitoneal" (e.g., the bladder).

Layers
The peritoneum is one continuous sheet, forming two layers and a potential space between them: the peritoneal cavity.

The outer layer, the parietal peritoneum, is attached to the abdominal wall and the pelvic walls. The tunica vaginalis, the serous membrane covering the male testis, is derived from the vaginal process, an outpouching of the parietal peritoneum.

The inner layer, the visceral peritoneum, is wrapped around the visceral organs, located inside the intraperitoneal space for protection. It is thinner than the parietal peritoneum. The mesentery is a double layer of visceral peritoneum that attaches to the gastrointestinal tract. There are often blood vessels, nerves, and other structures between these layers. The space between these two layers is technically outside of the peritoneal sac, and thus not in the peritoneal cavity.

The potential space between these two layers is the peritoneal cavity, filled with a small amount (about 50 mL) of slippery serous fluid that allows the two layers to slide freely over each other.

The right paracolic gutter is continuous with the right and left subhepatic spaces. The epiploic foramen allows communication between the greater sac and the lesser sac. The peritoneal space in males is closed, while the peritoneal space in females is continuous with the extraperitoneal pelvis through openings of the fallopian tubes, the uterus, and the vagina.

Subdivisions
Peritoneal folds are omentums, mesenteries and ligaments; they connect organs to each other or to the abdominal wall. There are two main regions of the peritoneal cavity, connected by the omental foramen.
 * The greater sac, represented in red in the diagrams above.
 * The lesser sac, represented in blue. The lesser sac is divided into two "omenta":
 * The lesser omentum (or hepatogastric) is attached to the lesser curvature of the stomach and the liver.
 * The greater omentum (or gastrocolic) hangs from the greater curvature of the stomach and loops down in front of the intestines before curving back upwards to attach to the transverse colon. In effect it is draped in front of the intestines like an apron and may serve as an insulating or protective layer.

The mesentery is the part of the peritoneum through which most abdominal organs are attached to the abdominal wall and supplied with blood and lymph vessels and nerves.

Other ligaments and folds
In addition, in the pelvic cavity there are several structures that are usually named not for the peritoneum, but for the areas defined by the peritoneal folds:

Classification of abdominal structures
The structures in the abdomen are classified as intraperitoneal, mesoperitoneal, retroperitoneal or infraperitoneal depending on whether they are covered with visceral peritoneum and whether they are attached by mesenteries (mensentery, mesocolon).

Structures that are intraperitoneal are generally mobile, while those that are retroperitoneal are relatively fixed in their location.

Some structures, such as the kidneys, are "primarily retroperitoneal", while others such as the majority of the duodenum, are "secondarily retroperitoneal", meaning that structure developed intraperitoneally but lost its mesentery and thus became retroperitoneal.

Development
The peritoneum develops ultimately from the mesoderm of the trilaminar embryo. As the mesoderm differentiates, one region known as the lateral plate mesoderm splits to form two layers separated by an intraembryonic coelom. These two layers develop later into the visceral and parietal layers found in all serous cavities, including the peritoneum.

As an embryo develops, the various abdominal organs grow into the abdominal cavity from structures in the abdominal wall. In this process they become enveloped in a layer of peritoneum. The growing organs "take their blood vessels with them" from the abdominal wall, and these blood vessels become covered by peritoneum, forming a mesentery.

Peritoneal folds develop from the ventral and dorsal mesentery of the embryo.

Imaging assessment
CT scan is a fast (15 seconds) and efficient way in visualising the peritoneal spaces. Although ultrasound is good at visualizing peritoneal collections and ascites, without ionising radiation, it does not provide a good overall assessment of all the peritoneal cavities. MRI scan is also increasingly used to visualise peritoneal diseases, but requires long scan time (30 to 45 minutes) and prone to motion artifacts due to respiration and peristalsis and chemical shift artifacts at the bowel-mesentery interface. Those with peritoneal carcinomatosis, acute pancreatitis, and intraabdominal sepsis may not tolerate prolonged MRI scan.

Peritoneal dialysis
In one form of dialysis, called peritoneal dialysis, a glucose solution is sent through a tube into the peritoneal cavity. The fluid is left there for a prescribed amount of time to absorb waste products, and then removed through the tube. The reason for this effect is the high number of arteries and veins in the peritoneal cavity. Through the mechanism of diffusion, waste products are removed from the blood.

Peritonitis
Peritonitis is the inflammation of the peritoneum. It is more commonly associated to infection from a punctured organ of the abdominal cavity. It can also be provoked by the presence of fluids that produce chemical irritation, such as gastric acid or pancreatic juice. Peritonitis causes fever, tenderness, and pain in the abdominal area, which can be localized or diffuse. The treatment involves rehydration, administration of antibiotics, and surgical correction of the underlying cause. Mortality is higher in the elderly and if present for a prolonged time.

Primary peritoneal carcinoma
Primary peritoneal cancer is a cancer of the cells lining the peritoneum.

Etymology
"Peritoneum" is derived from περιτόναιον via Latin. In Greek, περί means "around", while τείνω means "to stretch"; thus, "peritoneum" means "stretched over".

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Peritoneum


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Pistachio

The pistachio (Pistacia vera), a member of the cashew family, is a small tree originating in Persia. The tree produces seeds that are widely consumed as food. The word can be countable or uncountable, meaning its plural is with or without an 's'.

In 2022, world production of pistachios was one million tonnes, with the United States, Iran, and Turkey combined accounting for 88% of the total.

Description
The tree grows up to 10 m tall. It has deciduous, pinnate leaves 10–20 cm long. The plants are dioecious, with separate male and female trees. The flowers are apetalous and unisexual and borne in panicles.

The fruit is a drupe, containing an elongated seed, which is the edible portion. The seed, commonly thought of as a nut, is a culinary nut, not a botanical nut. The fruit has a hard, cream-colored exterior shell. The seed has a mauve-colored skin and light green flesh, with a distinctive flavor. When the fruit ripens, the shell changes from green to an autumnal yellow/red and abruptly splits partly open. This is known as dehiscence, and happens with an audible pop. The splitting open is a trait that has been selected by humans. Commercial cultivars vary in how consistently they split open.

Each mature pistachio tree averages around 50 kg of seeds, or around 50,000, every two years.

Etymology
Pistachio is from late Middle English pistace, from Old French, superseded in the 16th century by forms from Italian pistacchio, via Latin from Greek πιστάκιον pistákion, and from Middle Persian pistakē.

Distribution and habitat
The pistachio tree is native to regions of Central Asia, including present-day Iran and Afghanistan. .Pistachio is a desert plant and is highly tolerant of saline soil. It has been reported to grow well when irrigated with water having 3,000–4,000 ppm of soluble salts. Pistachio trees are fairly hardy in the right conditions and can survive temperatures ranging between -10 C in winter and 118 F in summer. They need a sunny position and well-drained soil. Pistachio trees do poorly in conditions of high humidity and are susceptible to root rot in winter if they get too much water and the soil is not sufficiently free-draining. Long, hot summers are required for proper ripening of the fruit.

Cultivation
The pistachio tree may live up to 300 years. The trees are planted in orchards, and take around 7 to 10 years to reach significant production. Production is alternate-bearing or biennial-bearing, meaning the harvest is heavier in alternate years. Peak production is reached around 20 years. Trees are usually pruned to size to make the harvest easier. One male tree produces enough pollen for 8 to 12 drupe-bearing females. Harvesting in the United States and in Greece is often accomplished using equipment to shake the drupes off the tree. After hulling and drying, pistachios are sorted according to open-mouth and closed-mouth shells, then roasted or processed by special machines to produce pistachio kernels.

History
Archaeological evidence shows that pistachio seeds were a common food as early as 6750 BCE. The earliest evidence of pistachio consumption goes back to the Bronze Age Central Asia and comes from Djarkutan, modern Uzbekistan.

Pistachio trees were introduced from Asia to Europe in the first century AD by the Romans. They are cultivated across Southern Europe and North Africa.

Theophrastus described it as a terebinth-like tree with almond-like nuts from Bactria.

It appears in Dioscorides' writings as pistákia (πιστάκια), recognizable as P. vera by its comparison to pine nuts.

Pliny the Elder wrote in his Natural History that pistacia, "well known among us", was one of the trees unique to Syria, and that the seed was introduced into Italy by the Roman proconsul in Syria, Lucius Vitellius the Elder (in office in 35 AD), and into Hispania at the same time by Flaccus Pompeius.

The early sixth-century manuscript De observatione ciborum (On the Observance of Foods) by Anthimus implies that pistacia remained well known in Europe in late antiquity.

An article on pistachio tree cultivation was brought down in Ibn al-'Awwam's 12th-century agricultural work, Book on Agriculture.

Archaeologists have found evidence from excavations at Jarmo in northeastern Iraq for the consumption of Atlantic pistachio.

The Hanging Gardens of Babylon were said to have contained pistachio trees during the reign of King Marduk-apla-iddina II about 700 BCE.

In the 19th century, the pistachio was cultivated commercially in parts of the English-speaking world, such as Australia and in the US in New Mexico and California, where it was introduced in 1854 as a garden tree.

In 1904 and 1905, David Fairchild of the United States Department of Agriculture introduced hardier cultivars to California collected from China, but it was not promoted as a commercial crop until 1929. Walter T. Swingle's pistachios from Syria had already fruited well at Niles, California, by 1917.

In 1969 and 1971, changes to the tax code in the United States eliminated tax shelters for almonds and citrus fruits. That encouraged California farmers to plant pistachio trees, because they were still eligible for such tax breaks. In 1972, the Shah of Iran began a school breakfast program that included packets of pistachios. This resulted in a decline of pistachio exports from Iran, resulting in increased prices in other countries and additional incentives to plant pistachio trees in California. The first commercial pistachio harvest in California took place in 1976. The Shah was forced into exile in January, 1979 during the Iranian Revolution, resulting in an end to trade between the United States and Iran, providing additional incentives for American farmers to plant dramatically more pistachio trees.

By 2008, U.S. pistachio production rivaled that of Iran. Drought and unusually cold weather in Iran led to severe declines in production there, while U.S. production was increasing. At that time, pistachios were Iran's #2 export product, after the oil and gas sector.

By 2020, there were 150,000 pistachio farmers in Iran, approximately 70% of whom were small-scale producers using inefficient manual picking and processing techniques. There were 950 far larger U.S. producers, using highly efficient mechanized production techniques. Between them, the U.S. and Iran control 70% of the world export market, with the U.S. in the lead. Worldwide demand exceeds production, so both countries have the ability to sell their production to various export markets.

In 2021, Fresno County, California accounted for about 40% of U.S. pistachio production, with a value of $722 million.

Diseases and environment
Pistachio trees are vulnerable to numerous diseases and infestation by insects such as Leptoglossus clypealis in North America. Among these is infection by the fungus Botryosphaeria, which causes panicle and shoot blight (symptoms include death of the flowers and young shoots), and can damage entire pistachio orchards. In 2004, the rapidly growing pistachio industry in California was threatened by panicle and shoot blight first discovered in 1984. In 2011, anthracnose fungus caused a sudden 50% loss in the Australian pistachio harvest. Several years of severe drought in Iran around 2008 to 2015 caused significant declines in production.

Production
In 2022, world production of pistachios was one million tonnes, with the United States, Iran, and Turkey together accounting for 88% of the total (table).

Toxicity
As with other tree seeds, aflatoxin is found in poorly harvested or processed pistachios. Aflatoxins are potent carcinogenic chemicals produced by molds such as Aspergillus flavus and A. parasiticus. The mold contamination may occur from soil or poor storage, and be spread by pests. High levels of mold growth typically appear as gray to black filament-like growth. Eating mold-infected and aflatoxin-contaminated pistachios is unsafe. Aflatoxin contamination is a frequent risk, particularly in warmer and humid environments. Food contaminated with aflatoxins has been found as the cause of frequent outbreaks of acute illnesses in parts of the world. In some cases, such as in Kenya, this has led to several deaths.

Pistachio shells typically split naturally prior to harvest, with a hull covering the intact seeds. The hull protects the kernel from invasion by molds and insects, but this hull protection can be damaged in the orchard by poor orchard management practices, by birds, or after harvest, which makes exposure to contamination much easier. Some pistachios undergo so-called "early split", wherein both the hull and the shell split. Damage or early splits can lead to aflatoxin contamination. In some cases, a harvest may be treated to keep contamination below strict food safety thresholds; in other cases, an entire batch of pistachios must be destroyed because of aflatoxin contamination.

Like other members of the family Anacardiaceae (which includes poison ivy, sumac, mango, and cashew), pistachios contain urushiol, an irritant that can cause allergic reactions. Large quantities of pistachios are self-heating in the presence of moisture due to their high oil content in addition to naturally occurring lipases, and can spontaneously combust if stored with a combustible fabric such as jute.

Uses
The kernels are often eaten whole, either fresh or roasted and salted, and are also used in pistachio ice cream, traditional Persian ice cream, kulfi, spumoni, pistachio butter, pistachio paste, and confections such as baklava, pistachio chocolate, pistachio halva, pistachio lokum or biscotti, and cold cuts such as mortadella. Americans make pistachio salad, which includes fresh pistachios or pistachio pudding, whipped cream, and canned fruit. Indian cooking uses pounded pistachios with grilled meats, and in pilao rice dishes.

The shell of the pistachio is naturally a beige color, but it may be dyed red or green in commercial pistachios. Originally, dye was applied to hide stains on the shells caused when the nuts were picked by hand. In the 21st century, most pistachios are harvested by machine and the shells remain unstained.

Nutrition
Raw pistachios are 4% water, 45% fat, 28% carbohydrates, and 20% protein (table). In a 100-gram reference amount, pistachios provide 2351 kJ of food energy and are a rich source (20% or more of the Daily Value or DV) of protein, dietary fiber, several dietary minerals, and the B vitamins thiamin (73% DV) and vitamin B6 (100% DV) (table). Pistachios are a moderate source (10–19% DV) of riboflavin, vitamin B5, folate, vitamin E, and vitamin K (table).

The fat profile of raw pistachios consists mainly of monounsaturated fats and polyunsaturated fats, with a small amount of saturated fats (table). Saturated fatty acids include palmitic acid (10% of total) and stearic acid (2%) (table). Oleic acid is the most common monounsaturated fatty acid (52% of total fat). and linoleic acid, a polyunsaturated fatty acid, is 30% of total fat. Relative to other tree nuts, pistachios have a lower amount of fat and food energy, but higher amounts of potassium, vitamin K, γ-tocopherol, and certain phytochemicals such as carotenoids, and phytosterols.

Research and health effects
In July 2003, the United States Food and Drug Administration approved the first qualified health claim specific to consumption of seeds (including pistachios) to lower the risk of heart disease: "Scientific evidence suggests but does not prove that eating 1.5 oz per day of most nuts, such as pistachios, as part of a diet low in saturated fat and cholesterol may reduce the risk of heart disease". Although a typical serving of pistachios supplies substantial food energy (nutrition table), their consumption in normal amounts is not associated with weight gain or obesity.

One review found that pistachio consumption lowered blood pressure in persons without diabetes mellitus. A 2021 review found that pistachio consumption for three months or less significantly reduced triglyceride levels.

title: Pistachio


text:
Dorididae

Sea lemon is a loosely applied common name for a group of medium-sized to large shell-less colorful sea slugs or nudibranchs, specifically dorid nudibranchs in the taxonomic family Dorididae and other closely related families. These are marine gastropod mollusks.

The Monterey sea lemon is Doris montereyensis and the mottled pale sea lemon is Diaulula lentiginosa.

The common name sea lemon probably comes from these animal's visual similarity to a lemon based on such qualities as the roughened skin, the oval form when seen from above, and the common but not inevitable orange to pale yellow coloration.

Description
These dorid nudibranchs can be large (up to 20 cm), rather flattened, and oval in shape when seen from above. They have two hornlike projections (rhinophores) on the head, and a rosette-like tuft of gills on the back of the animal. The mantle is sometimes sprinkled with black dots, and it is covered in small bumps, which are called tubercles.

Life habits
Sea lemons feed on sponges. They lay ribbons of white or yellow eggs. Taxonomically the Dorididae is a family of several genera, the dorids named after the mythological ancient Greek sea nymph Doris. (See Ovidius, Metamorphoses 2.6)

Genera
Genera within the family Dorididae include:
 * Aphelodoris (Bergh, 1879)
 * Artachaea Bergh, 1881
 * Conualevia Collier & Farmer, 1964
 * Doriopsis Bergh, 1889
 * Doris Linnaeus, 1758
 * Goslineria Valdés, 2001
 * Homoiodoris Bergh, 1881 : nomen dubium
 * Pharodoris Valdés, 2001
 * Genera brought into synonymy:
 * Anisodoris Bergh, 1898 : synonym of Diaulula (Discodorididae)
 * Anoplodoris P. Fischer, 1883 : synonym of Doris Linnaeus, 1758
 * Aporodoris Ihering, 1886 [synonym of Taringa in family Discodorididae]
 * Archidoris Bergh, 1878 : synonym of Doris Linnaeus, 1758
 * Austrodoris Odhner, 1926 : synonym of Doris Linnaeus, 1758
 * Ctenodoris Eliot, 1907 : synonym of Doris Linnaeus, 1758
 * Doridigitata d'Orbigny, 1839: synonym of Doris Linnaeus, 1758
 * Doriopsis Bergh, 1889 : synonym of Doris Linnaeus, 1758
 * Doriorbis Kay & Young, 1969 : synonym of Doris Linnaeus, 1758
 * Etidoris Ihering, 1886 a: synonym of Thordisa Bergh, 1877
 * Guyonia Risbec, 1928 : synonym of Doris Linnaeus, 1758
 * Neodoris Baba, 1938 : synonym of Doris Linnaeus, 1758
 * Praegliscita Burn, 1957 : synonym of Doris Linnaeus, 1758
 * Rhabdochila P. Fischer, 1883 : synonym of Rostanga Bergh, 1879
 * Siraius Marcus, 1955 : synonym of Doris Linnaeus, 1758
 * Staurodoris Bergh, 1878 : synonym of Doris Linnaeus, 1758

title: Dorididae


text:
Talk:WYSIAYG

Merge with WYSIWYG
I don't think this is worth a seperate encyclopedia article - instead, it should be integrated into the WYSIWYG article. --Robert Merkel
 * Although WYSIAYG as an abbreviation is obviously WYSIWYG inspired, it is an entirely different concept. It doesn't belong on the WYSIWYG page. This article needs an extensive rewrite, but not a merge or delete. Shinobu 21:53, 22 Apr 2005 (UTC)

POV
Ties WYSIAYG to WYSIWYG. However any relation between the two terms is only historical, if existant at all. Whether something alows for WYSIWYG is an interface question. WYSIAYG is an entirely different issue about reusable styles and structured documents.

For example Word and Tex can both be used to create documents that do and don't suffer from WYSIAYG. One is WYSIWYG, the other is not.

In HTML CSS can be used, or not. Whether a WYSIWYG editor is used doesn't matter (in sofar as it supports CSS that is). If you don't use CSS, the document will suffer from WYSIAYG, whether you used a WYSIWYG editor or not. Shinobu 18:29, 22 Apr 2005 (UTC)

Meaning
I've encountered two meanings of this word: Note that these issues are not the same. Shinobu 09:56, 28 May 2005 (UTC)
 * The issue described in the WYSIWYG article.
 * The issue described in this article.

title: Talk:WYSIAYG


text:
Thomas Joannes Stieltjes

Thomas Joannes Stieltjes (, 29 December 1856 – 31 December 1894) was a Dutch mathematician. He was a pioneer in the field of moment problems and contributed to the study of continued fractions. The Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral.

Biography
Stieltjes was born in Zwolle on 29 December 1856. His father (who had the same first names) was a civil engineer and politician. Stieltjes Sr. was responsible for the construction of various harbours around Rotterdam, and also seated in the Dutch parliament. Stieltjes Jr. went to university at the Polytechnical School in Delft in 1873. Instead of attending lectures, he spent his student years reading the works of Gauss and Jacobi — the consequence of this being he failed his examinations. There were two further failures (in 1875 and 1876), and his father despaired. His father was friends with H. G. van de Sande Bakhuyzen (who was the director of Leiden University), and Stieltjes Jr. was able to get a job as an assistant at Leiden Observatory.

Soon afterwards, Stieltjes began a correspondence with Charles Hermite which lasted for the rest of his life. He originally wrote to Hermite concerning celestial mechanics, but the subject quickly turned to mathematics and he began to devote his spare time to mathematical research.

The director of Leiden Observatory, van de Sande-Bakhuyzen, responded quickly to Stieltjes' request on 1 January 1883 to stop his observational work to allow him to work more on mathematical topics. In 1883, he also married Elizabeth Intveld in May. She also encouraged him to move from astronomy to mathematics. And in September, Stieltjes was asked to substitute at University of Delft for F.J. van den Berg. From then until December of that year, he lectured on analytical geometry and on descriptive geometry. He resigned his post at the observatory at the end of that year.

In 1884, Stieltjes applied for a chair in Groningen. He was initially accepted, but in the end turned down by the Department of Education, since he lacked the required diplomas. In 1884, Hermite and professor David Bierens de Haan arranged for an honorary doctorate to be granted to Stieltjes by Leiden University, enabling him to become a professor. In 1885, he was appointed as member of the Royal Dutch Academy of Sciences (Koninklijke Nederlandse Akademie van Wetenschappen, KNAW), and the next year he became a foreign member. In 1889, he was appointed professor of differential and integral calculus at Toulouse University.

Stieltjes died on 31 December 1894 in Toulouse, France. He was buried in on 2 January 1895.

Research
Stieltjes worked on almost all branches of analysis, continued fractions and number theory. For his work, he is sometimes referred to as "the father of the analytic theory of continued fractions".

His work is also seen as important as a first step towards the theory of Hilbert spaces. Other important contributions to mathematics that he made involved discontinuous functions and divergent series, differential equations, interpolation, the gamma function and elliptic functions. He became known internationally because of the Riemann–Stieltjes integral.

Awards
Stieltjes' work on continued fractions earned him the Ormoy Prize of the Académie des Sciences.

title: Thomas Joannes Stieltjes


text:
Vladimir of Novgorod

Vladimir Yaroslavich (Владимир Ярославич; Valdamarr Jarizleifsson; 1020 – October 4, 1052) was Prince of Novgorod from 1036 until his death in 1052. He was the eldest son of Yaroslav I the Wise by Ingegerd Olofsdotter, a daughter of Olof Skötkonung, the king of Sweden.

Life
In state affairs, he was assisted by the voivode Vyshata and the bishop Luka Zhidiata. In 1042, Vladimir may have been in conflict with Finns, according to some interpretations even making a military campaign in Finland. In the next year he led the Russian armies against the Byzantine emperor Constantine IX. He predeceased his father by two years and was buried by him in St Sophia Cathedral he had built in Novgorod. His sarcophagus is in a niche on the south side of the main body of the cathedral overlooking the Martirievskii Porch. He is depicted in an early twentieth-century fresco above the sarcophagus and on a new effigial icon on top of the sarcophagus. The details of his death is unknown, however his son Rostislav and his descendants were in unfriendly relationship with the descendants of the Yaroslaviches triumvirate (Iziaslav, Sviatoslav, and Vsevolod). Three of Vladimir's younger brothers Izyaslav I, Svyatoslav II and Vsevolod I all reigned in Kiev, while other two (Igor and Vyacheslav) died in their early twenties after which their lands were split between the Yaroslaviches triumvirate. Coincidentally, the Vyshata of Novgorod pledged his support to Rostislav in the struggle against the triumvirate.

Vladimir's only son, Rostislav Vladimirovich, was a landless prince who usurped power in Tmutarakan. His descendants were dispossessed by their uncles and were proclaimed as izgoi (outcast), but gradually managed to establish themselves in Halychyna, ruling the land until 1199, when their line became extinct. In order to downplay their claims to Kiev, the records of Vladimir's military campaigns seem to have been obliterated from Kievan chronicles. As a result, medieval historians often confuse him with two more famous namesakes — Vladimir the Great and Vladimir Monomakh. The name of Vladimir's consort is uncertain either. According to Nikolai Baumgarten, Vladimir was married to the daughter of count Leopold of Staden, Ode. Others (Aleksandr Nazarenko) disregard that assumption or claim a different person.

Vladimir's memory was better preserved in foreign sources. In Norse sagas he frequently figures as Valdemar Holti (that is, "the Nimble"). George Cedrenus noticed Vladimir's arrogance in dealing with the Byzantines.

title: Vladimir of Novgorod


text:
Johann Mühlegg

Johann Mühlegg (born 8 November 1970 in Ostallgäu, Germany) is a former top level cross-country skier who competed in international competitions first representing Germany and then Spain, after becoming a Spanish citizen in 1999. He was excluded and disqualified from the 2002 Winter Olympics in Salt Lake City for doping.

Early career
Mühlegg participated for Germany in the 1992, 1994 and 1998 Winter Olympics, even though he began having trouble with Germany's ski federation in 1993. From the beginning, Mühlegg singled himself out, at one point accusing German head coach Georg Zipfel for "damaging him spiritually" (the so-called Spiritistenaffäre). He was thrown off the team in 1995, but was reinstated later. But from that moment on, the ever eccentric Mühlegg insisted on taking a flask of holy water with him at all times, and trusting only his Portuguese cleaning woman/chaperone Justina Agostinho. In the end, Mühlegg was branded as a team cancer and was thrown out.

Competing for Spain
After being ejected from the national team after the 1998 Nagano Games, his good relations with members of the Spanish cross-country skiing team, in particular Juan Jesús Gutiérrez Cuevas and Haritz Zunzunegui, opened the door for Mühlegg to obtain Spanish citizenship.

In late 1999, competing for Spain, he won a World Cup race for the first time. At the 2001 FIS Nordic World Ski Championships in Lahti, he won two medals with a silver in the 10 km + 10 km combined pursuit (stepping up when the original medalist Jari Isometsä was disqualified for hemohes use), and a gold in the 50 km freestyle race. These are the only medals ever that Spain has won at the FIS Nordic World Ski Championships.

In the 2002 Winter Olympics in Salt Lake City, Mühlegg won gold medals in the 30 km freestyle and the 10 km + 10 km pursuit races, the successes gaining him congratulations from King Juan Carlos I of Spain.

Mühlegg finished first in the 50 km classical race held on the final Saturday of the Salt Lake City Winter Olympic Games on 23 February 2002 but was disqualified from that race and was expelled from the Games the next day, after testing positive for darbepoetin (a medicine which boosts red blood cell count; the substance was not banned at the time since it had only recently been developed).

Doping controversy
Following the darbepoetin scandal, the International Olympic Committee (IOC) initially let Mühlegg keep his gold medals from the first two races. But in December 2003 a ruling by the Court of Arbitration for Sport (CAS) found that these medals should also be withdrawn. The CAS remitted this case as well as similar ones involving Olga Danilova and Larisa Lazutina (both from Russia) to the IOC Executive Board, which confirmed the rulings in February 2004.

Cross-country skiing results
All results are sourced from the International Ski Federation (FIS).

World Championships

 * 2 medals – (1 gold, 1 silver)

Individual podiums

 * 7 victories
 * 12 podiums

title: Johann Mühlegg


text:
Darbepoetin alfa

Darbepoetin alfa (INN) is a re-engineered form of erythropoietin containing 5 amino acid changes (N30, T32, V87, N88, T90) resulting in the creation of 2 new sites for N-linked carbohydrate addition. It has a 3-fold longer serum half-life compared to epoetin alpha and epoetin beta. It stimulates erythropoiesis (increases red blood cell levels) by the same mechanism as rHuEpo (binding and activating the Epo receptor) and is used to treat anemia, commonly associated with chronic kidney failure and cancer chemotherapy. Darbepoetin is marketed by Amgen under the trade name Aranesp.

The medication was approved in September 2001, by the US Food and Drug Administration for treatment of anemia in patients with chronic kidney failure by intravenous or subcutaneous injection. In June 2001, it had been approved by the European Medicines Agency for this indication as well as the treatment of anemia in cancer patients undergoing chemotherapy.

Dr. Reddy's Laboratories launched darbepoetin alfa in India under the brand name Cresp in August 2010. This is the world's first follow-on biologic of darbepoetin alfa.

Darbepoetin is produced by recombinant DNA technology in modified Chinese hamster ovary cells. It differs from endogenous erythropoietin (EPO) by containing two more N-linked oligosaccharide chains. It is an erythropoiesis-stimulating 165-amino acid protein.

It is on the World Health Organization's List of Essential Medicines.

Contraindications
Use of darbepoetin alfa is contraindicated in patients with hypersensitivity to the drug, pre-existing uncontrolled hypertension, and pure red cell aplasia.

Adverse effects
Darbepoetin alfa has black box warnings in the United States for increased risk of death, myocardial infarction, stroke, venous thromboembolism, thrombosis of vascular access, and tumor progression or recurrence. To avoid side effects, it is recommended for patients with chronic kidney failure or cancer to use the lowest possible dose needed to avoid red blood cell (RBC) transfusions.

In addition to those listed in the black box warning, use of darbepoetin alfa also increases the risk of cardiovascular problems, including cardiac arrest, arrhythmia, hypertension and congestive heart failure, and edema. A recent study has extended these findings to treatment of patients exhibiting cancer-related anemia (distinct from anemia resulting from chemotherapy). Other reported adverse reactions include increased risk of seizure, hypotension, and chest pain.

Pregnancy and lactation
Darbepoetin alfa is not assigned a pregnancy category in the United States.

It is not known if darbepoetin alfa is excreted in breast milk.

Mechanism of action
Darbepoetin alfa binds to the erythropoietin receptor on erythroid progenitor cells, stimulating RBC production and differentiation.

Safety advisories in anemic cancer patients
Amgen sent a "dear stockholders" letter in January 2007, that highlighted results from a recent anemia of cancer trial, and warned doctors to consider use in that off-label indication with caution.

Amgen advised the U.S. Food and Drug Administration (FDA) as to the results of the DAHANCA 10 clinical trial. The DAHANCA 10 data monitoring committee found that 3-year loco-regional control in subjects treated with Aranesp was significantly worse than for those not receiving Aranesp (p=0.01).

In response to these advisories, the FDA released a Public Health Advisory on 9 March 2007, and a clinical alert for doctors on 16 February 2007, about the use of erythropoeisis-stimulating agents (ESAs) such as epoetin alfa (marketed as Epogen) and darbepoetin alfa. The advisory recommended caution in using these agents in cancer patients receiving chemotherapy or off chemotherapy, and indicated a lack of clinical evidence to support improvements in quality of life or transfusion requirements in these settings.

According to the 2010 update to clinical practice guidelines from the American Society of Clinical Oncology (ASCO) and the American Society of Hematology (ASH), use of ESAs such as darbepoetin alfa in cancer patients is appropriate when following stipulations outlined in FDA-approved labeling.

Society and culture
Like EPO, darbepoetin alfa has the potential to be abused by athletes seeking a competitive advantage. Its use during the 2002 Winter Olympic Games to improve performance led to the disqualification of cross-country skiers Larisa Lazutina of Russia, Olga Danilova of Russia and Johann Mühlegg of Spain from their final races.

Economics
Epogen and Aranesp had more than $6 billion in combined sales in 2006. Procrit sales were about $3.2 billion in 2006.

title: Darbepoetin alfa


text:
Talk:Möbius function

Move from "Moebius arithmetical function"
The following was moved here from the "Moebius arithmetical function" article:

In number theory is very important another sum, defined by:

 * M(n) = ∑ μ(n).

This function is closely linked with the positions of zeroes of Euler - Riemann ζ- function. The connection between M(n) and Riemann conjecture was known to Thomas Joannes Stieltjes.

How exactly is M(n) defined. What is the connection with the Riemann conjecture? AxelBoldt

Whuh Axel. Let me first say you didn't just put M(n) from the Möbius arithmetical function article - but you changed the whole article I had started, (I guess so). Who's gonna say that now a new article is more comprehensible? I think now μ(n) is not 'well defined'. It is hard to calculate its values by hand, isn't it? And perhaps it is just because someone does not fully understand the meaning of such great and innovative function. The best definition is I suppose that one I gave - but you just put it away. In TeX notation it would be even better, but no-one can say which way is the best for writting formulas in Wikipedia. I am new herein but I have at least about 15 years experience in math related stuff and futhermore I am not educated mathematician (but in the end it doesn't matter for the Feynman's curious character, right...)

Yes, μ(n) is multiplicative function, but basically it is arithmetic function - the most important one, as it is written in appropriate article. Let us not split hairs for everything here - because such free encyclopedia won't be very good after all. I know it mainly from number theory, combinatorics and in connection with the famous Riemann hypothesis. For L-systems see bellow...

Do you know the answer about definition range of M(n), or are you just asking so? I can't answer this without any preparations. And in the other side I don't quite understand your claim. I guess it is defined for every natural n if this answers the question? The first values for M(n) are:

 * M(1) = 1
 * M(2) = 0
 * M(3) = -1
 * M(4) = -1
 * M(5) = -2
 * M(6) = -1
 * M(7) = -2
 * M(8) = -2
 * M(9) = -2
 * M(10) = -1
 * M(11) = -2
 * M(12) = -2
 * M(13) = -3
 * M(14) = -2
 * M(15) = -1
 * M(16) = -1
 * M(17) = -2
 * M(18 = -2
 * M(19) = -3
 * M(20) = -3

Is M(n) for every further number negative? No! M(39) = 0 is the first counterexample to this foolish assumption. And here's a few more values to consider:

 * M(92) = -1
 * M(93) = 0
 * M(94) = 1
 * M(95) = 2
 * M(96) = 2
 * M(97) = 1
 * M(98) = 1
 * M(99) = 1
 * M(100) = 1

Stieltjes wrote to his collaborator Hermite 1885 that he succeeded to proove for every "big" n:

 * | M(n) | < √ n.

Stieltjes never published his 'proof' because he probably found an error. It is valid onward, if for every constant number a with every natural n:

 * | M(n) | < √ n,

then Riemann conjecture is correct. Stieltjes announcement had pulled the trigger of researching wave about function M(n). Franz Mertens 1897 published 50 pages long table with values for M(n) for number up to 10000. With this table he thought that Stieltjes inequality is very likely. We say today this is Mertens conjecture. Another one was R. D. Sterneck who from 1897 to 1913 in several articles published tables for function M(n) for selected values n up to 5,000,000. On this basis he conjectured that for every numbers n larger than 200 is:

 * | M(n) | < (1/2) √ n

W. B. Jurkat found this conjecture is wrong not until 1960. The least n for above inequality is wrong is n = 7725038629, because | M(n) | = 43947 > 43946.0994544. Futher researches was made by H. Cohen and Dress in 1979. They showed that for every first 7,800,000,000 numbers is valid:

 * | M(n) | < (3/5) √ n

Many computer calculations indicated the truthfulness of Mertens conjecture and some theoretical results showed just the opposite. The final judgement was said 1983 by H. te Riele and A. Odlyzko. Their 8 years joint research work via electronic mail finally canceled Mertens conjecture. Te Riele mainly worked in Mathematical Centre in Amsterdam and Odlyzko in Bell Laboratories in New Jersey.

How te Riele and Odlyzko disproved Mertens conjecture? Not in the way that they found certain number n, for which is | M(n) | > √ n. We don't know today the number and it is very likely that it must be larger than 10 30. They used a method, which was discovered by A. E. Ingham 1942. Ingham defined a function h(x), which has this property: Let x be arbitrary real number and let A < h(x) - at that time there is a natural number n, that | M(n) | > A √ n. Let us say if we know a number x for which is | M(n) | > 1.06 √ n. This would disprove Mertens conjecture. But Ingham method does not tell us for which number n this inequality is valid.

Te Riele and Odlyzko decided to find a number x for which is h(x) > 1. It soon showed up this is very hard exploit. Calculating of function h(x) is very difficult and it needs supercomputers. Among other we need for calculating of h(x) many exact values of zeroes of Euler - Riemann ζ- function. And this is not the only trouble. Even if we have a computer programme for h(x), we should find a certain real number x, for which is h(x) > 1. And this is much harder because function h(x) is almost much smaller than 1. The search for suitable number x is very similar to searching of the needle in a hay. They have realised this fact and till 1979 they have found the largest value for x 0.86. They had abandoned their own research.

The further research on this field came with an algorithm from A. K. Lenstra, H. W. Lenstra junior and L. Lovasz. First application of this algorithm on Mertens conjecture enabled successful search of number x.

The first task was to calculate (non-trivial ??) zeroes of Euler - Riemann ζ-function. It was made by computer CDC CYBER 750 at Amsterdam's Mathematical Centre. Within 40 hours the computer defined with accuracy of 100 decimals 2000 zeroes. The other task - calculation of h(x) was running on CRAY-1 at Bell's Laboratories 10 hours with newest Lenstra's algorithms. They had found a number x for which is h(x) = 1.061 545 and thus disproved Mertens conjecture. This so wanted number x is:

- 14 045 289 680 592 998 046 790 361 630 399 781 127 400 591  999 789 738 039 965 960 762.521 505.

Te Riele and Odlyzko publicated their work in an article Disproff of the Mertens conjecture in Journal für die reine und angewandte Mathematik, Vol. 357 (1985), p. 138-160. We know today that

 * | M(n) | < 1.06 √ n

is not correct for every natural n. We can ask if there is a certain constant A that it would be:

 * | M(n) | < A √ n.

Te Riele and Odlyzko think this constant does not exist. We can not disprove above inequality as an example for A = 2 with today's algorithms and computers.

What does it mean for Riemann conjecture a fact that Mertens conjecture is wrong? Strictly speaking - nothing. From regularity of Mertens conjecture follows the regularity of Riemann conjecture. But in the other side irregularity of Mertens conjecture does not say anything about Riemann conjecture. Mertens and Riemann conjectures are not equivalent.

Riemann conjecture is equivalent to weak form of Mertens conjecture, that for every real r, which is larger than 1/2 exists a real constant A that for every natural n is valid:

 * | M(n) | < A n r.

And this is something very different than Mertens conjecture. Okay, this is this all about connection of M(n) with Riemann conjecture in breaf. Mainly this comes from Keith Devlin's book from 1988 Mathematics: The New Golden Age in Slovene translation from 1993. Since then had passed 13 years and I believe that mathematicians and computer scientists didn'l sleep a lot... We should see herewith another similar result with Bieberbach conjecture and some more.

I deeply hope I gave you some clue on what you meant. And futhermore more I would like to writte that I would be very glad if my contributions would be accepted as they are mentioned to be, otherwise I won't participate in a run for a "giant question" and a "final quest of everything". And I won't put this lines to very nice article on Riemann hypothesis until I am not shure that someone won't totaly change them. It is very relative what one article should contain. I say it is better if contains more rather than less... I believe Axel, you won't go to my article on leading Slovene mathematician Josip Plemelj and throw everything out just because. You should consider other opinions and knowledge too.

I have another questions for everyone who can help on. Let us say that Möbius (multiplicative arithmetical) function μ(n) behave like an L-system. What is then its Besicovitch - Hausdorff dimension? Let someone say too, if this question is senseless? I should say I had written years ago a computer graphical programme that draws Möbius function μ(n) as a L-system and I still have no clue what is its BH dimension. Remember that every modern computer shows that Hilberts curve fills a square already with n ~ 10. And in the other side I haven't found what is BH dimension for Mandelbrodt set either? Does anyone know how should such questions be solved? I also had made a programme which extends Ulam sheet with Möbius function μ(n).

Axel, how we should name such curve: a Möbius curve or Möbius function in plane? I look forward for every little suggestion.

In and out over the hills... XJamRastafire

Sorry, but I didn't mean to remove your definition and replace it with a "better" one. I simply noticed that there was already a Moebius function article there and I redirected your new one over there and got rid of what I thought was duplication.

Feel free to add your definition back in. You can still get it if you go to http://wikipedia.com/wiki/M%F6bius_arithmetical_function&action=history

Regarding M(n): I was asking about the definition.
 * M(n) = ∑ μ(n)

I think you probably mean
 * M(n) = ∑1≤k≤n μ(k)?

The above story of the Mertens conjecture is really interesting. It deserves its own article, and links from Moebius function and Riemann hypothesis.

Cheers, AxelBoldt

Ahah, Axel I get it. I first thought you just canseled my effort to explain μ(n). I haven't found any articles before I started, so that's why I am apologizeing. I will get used to keep track of all of these meanings. Okay, let that definition of μ(n) in an article stay. Yes, as it is obvious from the first values of k the sum is really

 * M(n) = ∑1≤k≤n μ(k),

defined also for all non negative natural numbers.

I can rewrite the story of Mertens conjecture and put it out. But this takes time and I shall endeavour. Thank you also for correcting my "false" redirections. I think in this way this FE (free encyclopedia) has a great future. You gave me futher stiff doze of mettle.

Best regards. XJamRastafire

I have found Axel that you are changing articles quite a bit. But the questions is - if all your changes are strictly necessary. What is the main difference between words inverse and converse in common English language. Yes, I think you meant the usage of it in the context as in inverse of matrix but, anyway you can use it, as I have, and so and so on. I admit my English is not fully 100 %, so you should too as German Wikipedian, 'richtig'. Please correct sensible or otherwise I won't write a word anymore. Are you some kind of master's tutor here or are your intentions better as it seems. Fist you put away almost all my 'free' stuff, after that you commend my talkings about Mertens conjecture and after that you again put away 1/3 of my 'free' work on Möbius arithmetical function μ(n) and on twin prime conjecture. Perhaps I don't understand quite conceptions of GNU license and fair plays.

Let me say, for example, that if I have fully understandable article on Cantor's great work and I would like to put it in Wikipedia. But first I have to translate it from my native language, which is by chance to be Slovene one. Second this will take me a lot of time. Okay. And then would come one "funny" Axel from Paderborn or so, ha, ha, who shall fix this 'boted' stuff. After all I shall get tired of it in the end. But as Nikolaj Pečenko likes to say: "I could be wrong, too..." A solution of this tricky situation can be in sending e-mails about particular subject. Please, Axes if you shall 'superfix' some more articles, I think it is better for us to hear via e-mail and make things clearer.

To the infinity and back... XJamRastafire [2002.02.25] 1 Monday

I don't think that negotiating an article over email would be easier than negotiating it on Wiki. Of course articles get changed all the time, including the ones I started. If you take a more relaxed attitude, it's a lot easier. If somebody makes an edit that you don't agree with, for instance because you think it was more correct or clearer before, just change it back and discuss on the talk page. Generally, it's not a matter of "which changes are strictly necessary" but "which changes improve the article".

If you intend to write about Cantor's work, you may want to check first the material we already have, to see where your additions fit in: Georg Cantor, Cantors Diagonal argument, Continuum hypothesis, countable, cardinal number. AxelBoldt

I know articles are changing all the time. But in the end with these changes we shall be in a strange situation, let me say, to change Euler arithmetical function and all of his work. What will rest in the far end. I'll be very wanton if I would change one's whole article. My job is just to put dot points on i's and nothing more. If I shall find in some near future, for example, that π is over-super-real number I will all let you know. I'll keep in mind your words on improveing of articles. I think I shall see the whole picture later when I will get used of Wikipedia. There are a lot of GNU based computer programmes all over and that's for shure they must have firm frames to work properly. That's how I imagine good articles in FE (free encyclopedia). Axel check the Paderborn sails if they all work properly :-) XJamRastafire

XJamRastafire, could you please clarify this paragraph:

 * Very simple numbers to above ones are numbers by exactly 5 different primes, but they differ in values of μ(n) for it is μ(4620) = 0, as 4620 = 2 2. 3. 5. 7. 11 (SIDN A051270): 2310,2730,3570,3990,4290,4620,4830,5460,5610,6006,6090,6270,6510,6630,6930,7140,

Exactly what class of numbers are you listing here?

Also, could you add a description of how you generated the pictures? I don't think "colored L-systems" is a very well-known term. AxelBoldt

 * Yes, wellcome to the "strange" and beautiful world of numbers. But let me first say (as I already had somewhere above) that I am not "well educated" mathematician, whatever that really could mean and after all I am not Hilbert to chop Hydra's heads here. I have some knowledge in number theory mainly from the readings of leading Slovene mathematician in this field Jože Grasselli which once lectured at high school in my birth place. And I have done some own 'laic' researches. Gauss was probably up to his 16 years one laic youth from Brunswick and finally he became famous all over with his extraordinary work. It is known too for him that he started to teach himself Russian language when he was already over 60. These 'guys' can teach us all a lot. I am not ashame that I love mathematics very much and such. I put Cantor just for an example. Yes, for shure, I have to check all the material which is available here, than all my material and after that do the rest. We all know how many problems Cantor had outside with his understandings of infinity and such. His case is not alone (let me think - I know some more from head: Johhny (a.k.a Russell Crowe) Forbes Nash, Paul Ehrenfest and perhaps many more with their visionary mind...) I work in a tool factory and I know what a tool (or die) means. If you want to solve one problem you need some tools. In math is the same. And these tools is mathematical language itself, which changes with time too. Beautiful example for this is Hawking which is by Thorne's words the only one in the world who can speculate about topology and thermodynamics of black holes. Deep inside me there are a lot of proper thoughts about math, but I am not able to solve a great deal of problems in an ordinary fashion. It is quite well known that Feynman's (path) integral can not be defined in a strictly mathematical way even today, but physicists use it happily and with great amount. I dream of a day when some more unsolved problems will be revealed to all... We owe a lot to hard work of an army of mathematicians, physicists and astronomers and in recent times to computational scientiest.

 * I dont know what you don't understand with above numbers. Is it their mathematical nature or general one? I don't know either for their class if there is one.
 * μ(4620) = 0, as 4620 = 2 2. 3. 5. 7. 11 (only five different primes but 2 comes twice...)
 * μ(4830) = -1 because 4830 = 2. 3. 5. 7. 23 (just five distict primes and noone comes more than once...)

 * I don't know for the class. I just put it here to distinguish the properties of μ(n). I even hardly believe that there is such tricky classification for this weird numbers. The significance of μ(n) is, as I think, that it tells us so much more about natural numbers (or maybe I am wrong - the time will tell). Perhaps this is another somekind of higher dimension view on the numbers. And strange fact is that Möbius is known to be (theoretical) astronomer. He must have observed very well. He also discovered his famous strip which today may be a child's toy. We can say this was just a lucky chanse but, we should know more about his relations to numbers.

 * Yes, the pictures are all 'mine'. I made all the 'tools' to produce them except for graphical grasping. I haven't found themlike on the net. I just found Ulam's prime spirals or Ulam's square (prime) spirals which in my native language are called Ulam's cloth or sheet (Slovene = Ulamov prt). (I haven't found this term in English either). Check for Ulam's prime spiral at URL I gave in the article:

 * We just draw numbers in a spiral as for example (number 1 lies at the center and than we go in the circle left or right (or clockwise and vice versa):

17 16  15  14  13    ^        18   5   4   3  12    .       19   6   1   2  11    .       20   7   8   9  10    .        21  22  23  24  25 ....

 * Then we check the numbers if they are primes (or they have some different arithmetical or other properties) and sieve or pan them out. What remains is Ulam's cloth:

17             13    ^             5       3        .       19       1   2  11    .            7                .                23         ....

 * I had made a C programme to do the same with μ(n) and colours the definition range {-1, 0, 1} with three different colours. Now (as I call it Ulam - Möbius cloth) with values (colours) for μ(n) looks like:

-1  0   1   1  -1    ^         0  -1   0  -1   0    .       -1   1   1  -1  -1    .        0  -1   0   0   1    .         1   1  -1   0   0 ....

 * I think a pattern which comes from a programme's graphical output is really most beautiful but we won't discuss about its meaning here. Let us just say that primes come mainly on diagonals of such Ulam's spiraled square. This is all just OK mathematically I think.

 * But about Möbius coloured L-systems I am not shure. Typical L-systems (I don't know their definition origin right now, but it is highly implemented in very good computer programme fractint) are a Hilbert's curve or a dragon's curve. My (semi) definition for Möbius L-system (or something else - representation on a plane) is perhaps not quite the same as in L-system, because it is not recursive or selfgenerated over the limit as real L-systems are. We start with particular point, let us say 1 at the center and we do almost the same as in Ulam's prime spiral. We go upwards for a unit (but units does not count in L-systems I guess). We calculate μ(n+1) and we get μ(2)= -1. If it changes the sign, we go left, otherwise we go right -just like a turtle with a supermind would go to catch Zenon on some strange sphere and Universe, ha, ha. Then we calculate and get μ(3)= -1. If μ(n)= 1 we dont change a course of our clever turtle and we turn in the same temporal direction. And if μ(n)= 0, we even don't turn left or right but we let our L-turtie go straight on for a unit. I have to say that my programme does not use turtle graphic or strictly L-system "macro" language but ordinary and "more" understandable Cartesian coordinates. I think it even does not need it, because its main thing is after all just an arithmetical function which is in this case to be our famous and loving μ(n).

 * Now you must tell me if this worths something, mathematically speaking. Hardy spent all of his life doing just the same kind of things as they say, so this is my argument. Please let me know about your opinion. And finally if we "knew" all the values of μ(n) how this strange curve would look like? In my programme monitor x,y coordinates for certain n run out at a certain window and a scaling should be done - this is not implemented in it. And what would be its BH dimension. Perhaps someone knows the answer - I don't. I even don't know if this is the right mathematical question. Help me out. Would this function stretch all over (the flat 2D Universe, R2) or does it have a certain orbit or somekind of atractor to form itself? If we look to another starting point, we get different shape of function - but perhaps there are some selfsimilarities with its edge as in ordinary fractal functions as Mandelbrodt's or others are. But I don't yell for fractal nature of such Möbius function.

 * I hope you understand better now. I must say that an idea to define Möbius function and Ulam prime square spiral are properties of a visionary mind for me. Let me say here too that the similar thing happend when Jožef Stefan defined his Stefan - Boltzmann law of a radiation of a black body where there temperature T strangely came in fourth degree and many physicists thought and perhaps still think he just guessed that. But he was among the first in Europe who fully understand Maxwell's electromagnetic field equations and the whole theory at that time so he must have been a real physicist. --XJamRastafire [2002.02.26] 2 Thuesday (2nd ed.)

Ok, I understand the number spiral and the turtle graphics approach. Maybe we should add an explanation to the main article.

These numbers
 * (SIDN A051270): 2310,2730,3570,3990,4290,4620,4830,5460,5610,6006,6090,6270,6510,6630,6930,7140

are still mysterious to me: where do they come from, how are they defined, what do they have to do with μ? AxelBoldt

 * Edit conflict - hope everything is placed as it should be XJamRastafire [2002.02.26] 2 Thuesday (3rd ed.)

At first glance above Sloane's sequence A0512270 for me does not seem very special. They are well defined with a simple request.

 * How are they defined and why did you add them to the article about the μ function? AxelBoldt

I think it really doesn't matter where they "come from". We can form another sequence which would represent μ(k) of above sequence. First such sequence would be then:

(-1,-1,-1,-1,-1),0,(-1),0,(-1,-1,-1,-1,-1,-1),0,0,(-1,-1,-1,-1),0,0,0,0,(-1,-1,-1,-1),0,(-1,-1),0,(-1,...

or if we write shorter:

5,1,1,1,6,2,4,4,4,1,2,1,... defined for k>0

we get another "strange" sequence which is not in the base o Sloane's On-Line Encyclopedia of Integer Sequences. But we just speculating and we never know where we would come. But what bothers me is for example according to Peter Meyer's home page what are the similarities and connections with such properties found on his page as:
 * Cluster colour of primes, (Similar property would have above sequence 5,1,1,6,... because numbers are somehow grouped with μ(k)),
 * Sequence colour of primes,
 * Palmen's colour of natural numbers and so on

with μ(n)? This is nice challenge for some more computations and thinkings with pure μ(n)... Axel if you think that those pictures don't fit in an article we can throw them away. And in the other side if you think they are just fine, we shall find some time to put futher explanations. Perhaps this free encyclopedia some day becomes a research center for some serious and alternative approaches to natural sciences. After all mathematicians are not the only ones who love mathematics and who knows what really are they thinking. :-) XJamRastafire [2002.02.26] 2 Thuesday (4rd ed.)

Dear friends check this animated *.gif which merges the representations of μ(n)) if we observe it around and along sequenced prime numbers.

Representation of μ(n) on a plane as a coloured L-system around prime points n=1,2,3,5,7,11 cycled.

This picture clearly shows the "strange" behaviour of μ(n). It looks different for every single point but in the other side shows some similarities on the edges. Axel, sorry but I still don't understand your question about Sloane's sequence A0512270! The definition of it states existence of exactly 5 different primes but every one of them can be arbitrary numbered. So everything in these sets is correct for such sequences (I guess so):

 * { 21761. 1417, 3, 11, 21, 1093. 18}
 * { 21761. 1418, 3, 1111, 21, 1133. 18}
 * { 29, 3, 11, 21, 127}
 * { 5, 3, 11, 21, 109}
 * { 5, 7, 11, 21, 131}
 * { 7, 11, 13, 21, 137}

as long there are just and exactly 5 primes. I think this should be enough. Let me know if this is still not good for ya. XJam [2002.02.27] 3 Wednesday (0)

Axel can we agree on one another thing. You use asterix (*) for multiplication of natural numbers. I was tought already at elementary school to use period (.). Generally we use (*) for other kind of multiplications excpecially to multiplicate matrices where there multiplication has some different properties of natural multiplication. We also use multiplication sign (x) in HTML (&times). I use as in:

1. 11. 111. 11111. 1111111. 111111111. ...

you use:

1 * 11 * 111 * 11111 * 1111111 * 111111111 * ...

The best it would be I guess (but it is a little bit longer):

1 · 11 · 111 · 11111 · 1111111 · 111111111 · ...

and in sequences:

11,  3 · 11 · 37, 3 · 11 · 37 · 41 · 271, 3 · 11 · 37 · 41 · 239 · 271 · 4649, 3 · 11 · 37 · 41 · 239 · 271 · 4649 · 21649 · 513239, 3 · 11 · 37 · 41 · 53 ·  79 ·  239 ·   271 ·    4649 · 21649 · 513239 · 265371653 XJam [2002.02.27] 3 Wednesday (1st ed. )

 * I agree that we shouldn't use * for multiplication, since it's just computerese. × and · should display correctly for almost everyone, so there's no reason to avoid them. (And even x and . are better than *.) --Zundark, 2002 Feb 27

 * Ok, I vow to drop *; I'm happy with × and ·, but I don't like x and .: x is often a variable, and . looks too much like the decimal point used in many countries. AxelBoldt

Why was this sentence:
 * In the computational world μ(n) is sometimes designated as M(n), which must not be mistaken with a common mathematical notation of another sum (see below).

thrown out from the article again? This statement clearify the mess with math notation. -- XJam [2002.06.14] 5 Friday (0)

The letter used to denote a function is not fixed; you can use any letter you like as long as you say what function you are talking about. μ is used for lots of functions which aren't the Mobius function, and lots of letters which aren't μ are used for the Mobius function. There are also many functions called M around, not just the one that the article mentions. The name of a function is fixed, not the letter. There are way too few letters around. Especially if you are on a system without greek letters. AxelBoldt, Friday, June 14, 2002

I've moved the relevant info from here to Ulam spiral, and removed the info that doesn't seem related to this page. If anyone wants to resurrect the info here, please add text explaining how it relates to this page.

M(infinity)=-2 ?
I believe I have found $$M({\infin})$$. It should equal -2. Here's why:

I use this well known formula: $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s}$$, and insert s=0. This reduces to $$\frac{1}{\zeta(0)} = \sum_{n=1}^{\infin} \mu(n)$$ and so because $${\zeta(0)}=\frac{-1}{2}$$ and $$\sum_{i=1}^n \mu(i)=M(n)$$ $$M({\infin})=\frac{1}{\frac{-1}{2}}=-2$$. can anybody verify this, can this be used to prove that $$M(n)=O(\sqrt n)$$ and so too the Riemann hypothesis? Ozone 20:12, 24 January 2006 (UTC)

 * You must have meant

 * $$\sum_{k=1}^n \mu(k)=M(n).$$

 * The meaning of the identity

 * $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s}$$

 * seems clear when s > 1. To say that it continues true when s = 0, I suspect you'd have to talk about "summation methods" or the like.  I'd be very surprised if anything as simple as this argument could be used to prove the Riemann hypothesis.  Whether, and in what sense, it makes sense to say that M(∞) = −2 is something I'll leave to those who know more that what little I know about this. Michael Hardy 21:48, 24 January 2006 (UTC)
 * I was assuming it would continue to work. I wonder...  Yes I thought I fixed the error about Merten's function, maybe I didn't click save.  The reason I said that it might help to prove the RH is that this shows that M doesn't go off to infinity as it increases.  I don't think it will but there's a chance, allbeit a very very small one.  It is still a strange result though, there are more -1 values than 1 values, weird...Ozone 02:07, 25 January 2006 (UTC)

The correct way to compute $$\sum_n^\infty f(n)$$ of any arithmetic function f is to compute one of the following limits (which are all identical):
 * $$\lim_{t\to 0} \sum_n f(n) e^{-tn}$$

or
 * $$\lim_{s\to 0} \sum_n f(n) n^{-s}$$

Here, you need to substitute M(n) for f(n), and not μ(n) for f(n). See zero-point energy for a real-life example, or also zeta function regularization. linas 14:42, 25 January 2006 (UTC)
 * But wouldn't this way still work, Your way may be the "official way" but how would you calculate M(infinty) your way? On the other hand I think I have found reason to believe that I am wrong.  Riemann Zeta Function says that this only works for s>1.  Also it gives very strange results as you use lower values of S.  Ozone(sorry wrong computer for sig.)

 * Perhaps I was too breif. M(infty) should really be understood to be given by either one or the other of the two sums above. The exponential sum will have divergences at t=0, these are to be "ignored", if you just want the finite part. The dirichlet series which is given in zeta function regularization will have simple poles at places other than s=0, and so taking the s=0 value is a valid thing to do. These two are related by a Mellin transform, again, as explained on the zeta function regularization page.

 * I am not sure what you thought the Riemann Zeta Function page may have said. The zeta can be analytically continued to s values less than 1. The zeta function has a pole at s=1, but nothing else. I suspect that what you really need to do is to study analytic continuation in greater detail: it is a technique for defining functions on the complex plane even if the series that defines them is divergent or infinite in some way. The very very simplest example is

 * $$\sum_{n=0}^\infty x^n$$

 * which fails to converge for any value of x greater than 1. Indeeded, for x=1, we have 1+1+1+1+... and for x=2, we have 2+4+8+16+.... yet, famously, we know that this sum is just plain old 1/(1-x). and so there we have it: 2+4+8+16+...= -1. linas 01:54, 26 January 2006 (UTC)

 * Oops, I had made several misleading statements last time, which are now corrected, I beleive. linas 02:09, 26 January 2006 (UTC)

 * Also, I did not mean to imply you were incorrect or wrong, or anything like that. Your maniplation was entirely valid, and you got the right answer. I guess I'm pointing out the technique has a whole bundle of terminology and theorems wrapped around it, and that one must be careful to use this termiology correctly: otherwise you get "silliness" like 2+4+8+16+...= -1 or perhaps more interestingly, 1+2+3+...=-1/12, etc. linas 02:09, 26 January 2006 (UTC)

 * So Linas, you agree that M(infinty)=-2? I wonder if there are any results from this.  (Yes I knew about the analytic continuation, I was just using the value of zeta(0)=-1/2 from a website.)

Exponential generating function?
Silly me, the exponential generating function
 * $$f(x)=\sum_{n=1}^\infty \mu(n) e^{-nx}$$

seems to be non-trivial? linas 03:28, 2 July 2006 (UTC)

This form is especially inteeresting,ssince its Mellin transform is just the Gamma function dvivided by the Riemann zeta function (note that I have changed the independent variable in the above formula from 'y' to 'x') 81.102.15.200 (talk) 13:27, 1 May 2009 (UTC)

Moebius of 0
I understand, of course, that the sentence claiming that μ(0) is generally left undefined is an undisputable fact, so instead I wish to argue the reasoning behind that belief. It would seems to me it would be obvious that the Maple system's definition of -1 is correct. 0 is clearly not square-free, since its divisible by all spmplex numbers. That leaves only 1 and -1. Since 0 is divisible by all complex numbers, and inherently all primes, it has an infinte number of primes divisors. Since infinity is not an integer, and it is also half of itself, it can be shown not to be even. Thus, 0 is non-square-free integer with and odd number of distinct prime divisors, which gives us μ(0) = -1. I may have overlooked something, but if not I simply wonder why it is generally left undefined. -- He Who Is[ Talk ] 22:58, 5 July 2006 (UTC)

As you say, 0 is not square-free, so the value of Moebius of 0 should be 0.

Physics connection needs citation
the connection w/physics and pauli exclusion principle sounds neat but it's possibly a clever physicist messing with our heads.It was put in by 70.16.52.150 on 10/26/05. At about same time that editor stated similar stuff on Mobius inversion formula, later removed by Linas, I think.Rich 10:34, 25 October 2006 (UTC)
 * I think the current version is correct, and that the primon gas article provides the citations, as well as enough details to verify it yourself. linas (talk) 13:13, 6 May 2009 (UTC)

Strictly positive natural numbers
It seems like this phrase near the beginning of the article overspecifies the numbers we're talking about. If we want to avoid the 0-v-no-0 business mentioned at the natural number page, wouldn't it make more sense to say "positive integers," and just avoid the poorly-defined label "natural" in the first place? I didn't want to change it myself, because there may be a reason I'm not aware of for keeping it. 129.2.56.220 (talk) 14:09, 15 February 2008 (UTC)

 * Thanks. I agree, and I changed it.  You may want to read WP:BOLD.  -- Dominus (talk) 14:36, 15 February 2008 (UTC)

Distribution at $$\infty$$
What is the overall distribution of the values of the mobius function? Is it -1 1/3 of the time 0 1/3 of the time and 1 1/3 of the time? Woscafrench (talk) 22:28, 29 April 2010 (UTC)

I'm (quite possibly mistakenly) under the impression that the Mertens function is chaotic. Therefore, I think that that implies that as n goes to infinity, the chances of the mobius function being any one of the three values is equal, namely 1/3.My 2 Cents' Worth (talk) 12:13, 25 May 2010 (UTC)

 * The article Square-free integer has a short proof that the density of square-free integers is asymptotically $$\frac{6}{\pi^2}.$$ These numbers have non-zero values of μ. I do not know if they are equally divided between plus and minus one. Thus $$1-\frac{6}{\pi^2}$$ of the integers have μ equal to zero. Virginia-American (talk) 00:43, 4 January 2011 (UTC)

0 is even?
What makes 0 even? Is it because 0/2 = 0? 0 is the absence of a number, so can you give to 0 properties of a real number? My 2 Cents' Worth (talk) 12:16, 25 May 2010 (UTC)

 * 0 is commonly considered as an integer, and hence a real number. 0 is not the absence of a number, it is the number that represents the absence of something. If you have 1 apple, and you eat it, you can still ask what number of apples you have, the answer being that you have zero apples. As for the evenness of zero, I'll refer you to the wiki article Parity of zero for a long discussion (though I'll also add that zero is divisible by two (since 2×0=0), so it satisfies one of the defining properties of even numbers). RobHar (talk) 18:00, 25 May 2010 (UTC)

Hello
"There is a formula for calculating the Möbius function without knowing the factorization of its argument"

That formula requires knowing the k's such that gcd(k,n) = 1, which requires knowing the factorization of the argument.

" μ(n) is the sum of the primitive nth roots of unity."

This is misleading because the sum of the primitive nth roots of unity is zero which is not always μ(n) [edit: for n>1]

77.1.191.252 (talk) 18:14, 3 January 2011 (UTC)

 * To address your first point: finding a gcd is easier than factoring, see Euclidean algorithm. To address your second point: the sum of all nth roots of unity is zero, but the sum of the primitive nth roots of unity need not be. Take for example n = 3, then the three third roots of unity are 1, $$\frac{-1+i\sqrt{3}}{2}, \frac{-1-i\sqrt{3}}{2}$$, and only the last two are primitive. Their sum is clearly −1, which is μ(3). Anyway, there is a reference for this statement in the article, namely (16.6.4) of Hardy and Wright's book, so you should take a look there before claiming the formula is wrong. Cheers. RobHar (talk) 19:56, 3 January 2011 (UTC)

 * There is a proof of this formuula in the article Root of unity
 * Virginia-American (talk) 00:52, 4 January 2011 (UTC)

Incoherent sentence
Toward the end of the article this passage occurs:

"The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. In this theory, the fundamental particles or "primons" have energies log p. Under second-quantization, multiparticle excitations are considered; these are given by log n for any natural number n. This follows from the fact that the factorization of the natural numbers into primes is unique. In the free Riemann gas, any natural number can occur, if the primons are taken as bosons."

Aha! So any natural number can occur. As what????? Readers trying to understand the last sentence of this passage will not be able to get even a clue as to its meaning if an essential piece of information is omitted. It makes as much sense as if someone had written just: "In mathematics, any natural number can occur."Daqu (talk) 04:46, 10 February 2013 (UTC)

Product formula relation
"The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function

\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}. This is easy to see from its Euler product"

Why is it clear? Am i being stupid here? — Preceding unsigned comment added by 109.153.163.53 (talk) 23:23, 29 September 2013 (UTC)

Unsourced material
I have removed material sourced to a blog post on math.stackexchange.com. Material added must be verified by independent reliable sources. This does not qualify. Deltahedron (talk) 15:46, 12 April 2014 (UTC)

Möbius function of 1
I'm a little confused as to why the function returns positive 1 for the input 1. Wouldn't it be considered to have repeating prime factors (1x1x1x...) and output to 0? Or simply an odd number of factors, a single factor, which is "1". I'm not understanding how 1 has an even number of prime factors. 67.186.128.135 (talk) 06:07, 28 January 2020 (UTC)
 * 1 is not prime...--Jasper Deng (talk) 06:14, 28 January 2020 (UTC)

Typographical consistency
Regarding this revert...

To answer the question in your edit summary, "more consistent with what?", the consistency is between the inline and standalone formulas. For example, the μ in these formulas:

 * $$\sum_{d\mid n} \mu(d) = \begin{cases}1 & \text{if } n=1, \\ 0&\text{if } n>1.\end{cases}$$
 * $$\sum_{d\mid n} \mu(d) = \begin{cases}1 & \text{if } n=1, \\ 0&\text{if } n>1.\end{cases}$$

They look slightly different. It's possible some question is left in the mind of the reader as to whether or not they represent the same thing. Sometimes typography does make a difference in math formulas. For example, consider the potential differences and non-differences among R, R, ℝ, r, and r. It would be better if math symbols that mean the same thing retain the same appearance, so the reader has fewer questions to puzzle through. -- Beland (talk) 14:19, 21 September 2020 (UTC)

title: Talk:Möbius function