Economic globalization vs mercantilism

Jacques Fontanel

Conference Economists Against Arms Race American Economic Association Congress New Orleans, USA, 7 January 2001.

For mercantilism, the main objective of economic action is to increase the power of the Prince (Machiavelli). With the rise of capitalism and the market economy, liberal economists strongly criticized state management of the national economy. The centralization of political power was often seen as a major brake on the market economy and thus on economic development. Since 1990, the history of capitalism seems to have stopped being written within national borders. We are moving from the wealth of nations to the wealth of the world. However, despite the existence of the World Trade Organization, there is in fact no supranational authority capable of imposing rules on multinational markets, which are often speculative and interested mainly in short-term profit, thus threatening the sovereignty of nations. Yet market system generalization is often perceived as a factor of democracy. This statement is debatable. As states have lost most of their economic power, the electorate is the victim of a democratic illusion comparable to Keynes' monetary illusion. The risk is the progressive establishment of international plutocratic systems which, within each state, will defend private interests, sometimes in competition from state to state. War and economic war are not over.

Pour le mercantilisme, l'objectif principal de l'action économique est d'accroître le pouvoir du Prince (Machiavel). Avec l'essor du capitalisme et de l'économie de marché, les économistes libéraux ont fortement critiqué la gestion étatique de l'économie nationale. La centralisation du pouvoir politique était souvent considérée comme un frein majeur à l'économie de marché et donc au développement économique. Depuis 1990, l'histoire du capitalisme semble avoir cessé de s'écrire à l'intérieur des frontières nationales. On passe de la richesse des nations à la richesse du monde. Cependant, malgré l'existence de l'Organisation mondiale du commerce, il n'existe en fait aucune autorité supranationale capable d'imposer des règles aux marchés multinationaux, souvent spéculatifs et intéressés principalement par le profit à court terme, menaçant ainsi la souveraineté des nations. Pourtant, la "marchandisation" est souvent perçue comme un facteur de démocratie. Cette affirmation est discutable. Les Etats ayant perdu l'essentiel de leur pouvoir économique, l'électorat est victime d'une illusion démocratique comparable à l'illusion monétaire de Keynes. Le risque est la mise en place progressive de systèmes ploutocratiques internationaux qui, au sein de chaque Etat, défendront des intérêts privés, parfois en concurrence d'Etat à Etat. La guerre et la guerre économique ne sont pas terminées.

Etat, mercantilisme, économie de marché, economic war, Organisation Mondiale du Commerce State, mercantilism, market economy, economic war, World Trade Organization For mercantilism, the main objective of economic action is to increase the power of the Prince (Machiavelli). The economy develops thanks to a strong state and in return the economic policy of the Prince favors the national economic development. This theory did not always work well. Colbert, a good mercantilist, banned grain exports from the kingdom, so as not to strengthen the economy of his potential enemies, which caused a depression in national production (obviously contrary to his objective). This policy would be followed, much later, by the United States against its systemic opponent, the USSR. With the criticism of state management and the return of hard-line liberalism, the state gradually gained a "bad reputation" as an economic agent. The centralization of political power was often judged to be fatal to the market economy and thus to economic development [START_REF] Baechler | Les origines du capitalisme[END_REF]). However, it is not so much the strength of this power that has condemned it as its rentier nature, its permanent abuse. The state does not encourage the production of wealth. It even creates increasing obstacles to it. A strong central power is not in contradiction with the development of the economy.

After the "revival" of state action during the Keynesian revolution, the end of the Trente Glorieuses led to the development of more liberal slogans, such as "less state" for "better state", or "modern state, modest state" [START_REF] Crozier | Etat modeste[END_REF]. Now, with the criticisms levelled at its leaders, the state is no more than a pressure group, certainly still dominant in the constituted nations, but often dominated by other "entryist" groups. The question is whether the nation-state has become a prisoner of forces beyond its control. Spectacular changes have taken place: international trade has grown faster than national production; markets are globalizing quantitatively and qualitatively; firms are proposing to regulate sectors through agreements, to avoid conflicts. The MNCs are seeking to establish private law rules for international competition, without calling on international economic organizations. The states have always had, whether they like it or not, a particular responsibility in the development of the national economy. This is the great problem of the end of the 20th century, which has lost its humanist values and which accepts a state whose main economic action is to listen to managers who defend the interests of liberalism. When economic relations systematically go beyond the borders of the nation-state, the potential for macro-economic regulation is reduced, as internationalization limits society's capacity to civilize its economy. There is no supranational sovereignty to control the international economy integrated into fragmented sovereignties, which reduces the weight of politics in relation to economics. But supranationality reduces democracy and places the participation of citizens in bureaucratic rules. Economic policy, especially industrial and technological policy, is greatly compromised by globalization. From now on, it has no other ambition than to encourage or prevent relocation, which is likely to call into question the theory of comparative advantage. The notion of competitive advantage developed by Michael Porter is interesting, but it does not indicate to which competitiveness one should refer, that of economic agents in national areas or that of multinational or global firms.

Keynesian states had accepted responsibility for managing employment and maintaining high levels of consumption. In crisis situations, they took control of the national economy by innovating economic policies through the threefold channel of public expenditure and revenue, market legislation and social support for citizens. They had learned to solve major macroeconomic problems, but the social policy that was accepted everywhere and by everyone is now sacrificed on the altar of the necessary competitiveness in a globalized economic world. It is true that political power must establish order and that this activity is not, a priori, guided by the objective of maximizing profit in a market. However, for neoliberals, the state is first and foremost the defender of private property and the order it implies. The development of free-trade zones places companies in good conditions, first by escaping the regulatory control of a single state and by facilitating the mobility of capital. The importance of the international market is favored by the fact that there are no supranational authorities that can enforce the discipline of multinational markets, which threaten the sovereignty of nations. The European Union is struggling to organize the economic, political and social integration of Europe and the vacuum created is filled by market forces. Multinational firms are then able to move into the interstices. With the system of zones, there is no state to impose any discipline other than that of respect for private property, and international organizations are not always equipped with sufficient instruments to ensure democratic regulation of the economy. There is even a "ratchet effect", which leads to going in the sole direction of privatization, without conceiving the opposite path. But privatization is not only a change of ownership, it is also a complete transformation of the activity itself, in favor of profit and solvent consumers.

The end of Keynesian policy has not completely dissuaded states from intervening in economic life. They still have a significant role to play. Generally speaking, they seek to :

-transform national markets into international markets; industrial policies then only make sense in the European context, and not at the national level, where such action appears to be an intolerable restriction on market competition; -to coordinate national economic policy within the framework of integrated markets, which also implies privatization and deregulation; it is noteworthy that in the convergence and criteria defined by the European Union, nothing has been decided concerning the maximum level of unemployment or the minimum level of social protection, which is very significant of the will not to hinder the markets; -Restoring and defending national competitiveness in an international economy, in the face of relocation and the search for international investments, through the implementation of aid plans, special administrative procedures, actions concerning risk perception, and the definition of rules of good competition at the international level.

According to the United Nations World Investment Report, there were more than 35,000 transnational companies and 200,000 subsidiaries in 1992. This type of management of the economy reduces the importance of democracy and hardly promotes growth with stability. While state intervention is often rejected, it is nonetheless necessary and international cooperation of governments is inescapable. The role of the state is not only that of power, but also that of permissiveness. With its growing indebtedness, especially in the social democratic countries, the state has been judged to be a bad manager of the economy. But the lessons of Keynes should be reviewed in the light of the new economic situation. A distinction must be made between capital and ordinary public spending, because the former creates opportunities for the future. Robert Eisner shows that the U.S. deficit would have been eliminated simply by accepting the simple idea that public capital spending is spending that produces long-term benefits. The intergenerational transfer creates a dilemma. Later generations will receive what others have built, even with their debt.

For Barro and Ricardian equivalence, today's debts are tomorrow's taxes, which implies that today's debts do not produce any wealth. This is obviously not correct, unless one imagines that public infrastructure or education have no positive effect on long-term economic growth. Cutting public spending is not always the answer. Tax cuts often benefit shareholders when they should be benefiting businesses. More precise measurement tools are needed.

Globalization creates a fundamental short circuit between national political institutions and the control of the economy. Globalization implies that international geo-economic forces dictate national economic policies. Governments lose their levers of control. The US effort to control the dollar led to the Eurodollar market. Every effort at national regulation provokes a powerful reaction from transnational groups. The Japanese government tried to prevent trading in complex financial derivatives that moved the Nikkei index in Tokyo. The trade was exported to Singapore. The era of national regulation is over. Business goes where there is no regulation. The weak control of financial movements can be dangerous. National laws are then put in competition (tax law, labor law, commercial law, banking law, etc.). Globalization creates pressures that are always downward, never upward. With satellites and electronics, the world has probably passed the point of no return for the maintenance of cultural exception. Multinational firms could gradually replace public funding.

Financial shocks are increasingly powerful, even though economic theory assumes that they are impossible, given the rationality of agents and the application of flexible exchange rates. Governments have fewer and fewer instruments to control international monetary and financial flows. On a normal day, more than 1.3 trillion dollars are exchanged on the international markets, while world exports do not exceed 10 billion dollars per day. The crisis in Mexico could just as easily have occurred in the Philippines, Indonesia or Brazil. But a chain of circumstances created the conditions for the crisis in Mexico, which had to follow the rules and policies dictated by the IMF and the World Bank, leading immediately to economic recession. After the crisis, inflation was 60%, 13 years of growth in living standards were sacrificed to introduce market reforms, more than 4 million people had to reduce their hours by 15 hours a week and 500,000 additional unemployed were created. The loss of purchasing power was 33%, and even 50% for the middle classes. The crisis has been terrible. The United States now controls the Mexican oil revenues in a FED account and guarantees the assets of Mexican "non-citizens". Mexico's economic defaults were not large, and were mainly due to the private sector, as the public sector was not a net borrower. The price of restoring the confidence of international investors was very high.

Structural adjustment has often been tried in indebted countries in Asia and Latin America. It proposes a contraction of the budget deficit, devaluation of the currency (to modify the terms of internal and external trade between tradable and non-tradable goods), monetary rigor (selectivity of private sector loans, increase in interest rates), modification of the production incentive system (increase in agricultural prices, rationalization of state interventions in the markets, privatization), reduction of industrial protection by forcing the application of comparative advantages and a reduction in costs. Applied early enough, this policy allowed the re-establishment of the great balances and the restoration of the confidence of foreign operators, at the cost of a high social cost (lost decade of development in Latin America for [1980][1981][1982][1983][1984][1985][1986][1987][1988][1989][1990].

Internationalization calls for a renewed democratic national policy. The question is whether we are witnessing the end of economic ideologies (socialist, with the USSR, and socio-democratic with Sweden). Economic activity is then presented as obeying eternal and universal laws, and any action by the state is then to the detriment of the economy. An international financial community can exercise more and more political control over the states, which have the exclusive responsibility for democratic institutions. But globalization also implies NAFTA or the European Union, which depend on political decisions that establish the rules of free competition, capital mobility and minimal state intervention. Deregulation is also an economic policy. There is a fragmentation of public power in a unified economy. The Treaties set severe constraints on the action of national public authorities, and Europe does not replace the interventionist state at the higher level. The main state tools have become the establishment of rights and incentives rather than obligations and constraints, thus reproducing the anarchic voluntarism of the international system at the national level. Nevertheless, the state still monopolizes sovereignty and public power. Markets, like societies, need direction and guidance. Most firms sell locally. Markets should be controlled by democratic procedures, not by the managers of highly volatile markets.

The state must respond to two policy challenges:

-The first challenge comes from market enlargement. States must carefully define their trade policy, because the free market can lead in the short term to dramatic solutions (such as the "beggar-thy-neighbor" policy) when things go wrong, especially in terms of employment. At present, while a return to protectionism is not on the agenda, it remains an effective instrument for protecting the losers or for setting up a new industry. States can react to globalization through regionalization, by strengthening their own economic policy instruments, but also by fully exercising the negotiating powers they have in international bodies. The state should be able to promote the dissemination of human rights, democracy and the improvement of working conditions, national economic development through sustainable overall economic growth and the improvement of the social welfare of all workers. This is why in a context of globalization, international economic organizations are in principle so useful. However, they are often rendered ineffective by too much bureaucracy. Introducing a social clause in these bodies is always a lost cause. Developed countries even use the threat of competition from workers in developing countries to undermine the gains made by workers. International coordination is a long apprenticeship, which cannot work effectively without political will.

-The second challenge raises the question of the active role of the global liberalization order on the economic future of the nation. Distinct national systems are capable of redefining the rules of the game by responding to different incentive environments. States can then ask themselves what actions to take to turn a potential comparative advantage into an economic production advantage. National answers are increasingly difficult to find. Human capital becomes one of the only resources of competitive advantage in a global economy, given the mobility of other factors of production and the constraints of trade treaties. The aim of policies is therefore first to create an infrastructure and a climate that strengthen the country's competitiveness, then to provide the instruments to support workers in the adjustments required as a result of globalization, and finally to protect the quality of working life, health, education and training of citizens.

New functions of the state have been proposed. Community economic development is one example. There are two alternative scenarios.

-In the first, the state integrates post-Fordist changes into state administrative processes without making a political transition to more democracy, developing decentralization, improving public services and listening to consumers and citizens. The political goal is then to preserve public services through a combination of innovation and centralized supervision. National politics implies fragmented sovereignty. The return to the market presupposes, at least during the transition period, a high level of state activism, especially in terms of regulation.

-In the second, the state not only wants to transform the idea and implementation of public service, it also proposes to make democratic representation stronger. Power is shifting. The State subcontracts, while keeping the possibility of resuming its delegation, certain collective responsibilities to associations or organized groups. The initiative of collective actions emerge, unequally, from a great diversity of groups, with different interests and capacities.

Economic globalization favors the private sphere over public service. All activities are in competition with each other and the World Trade Organization will have great difficulty in enforcing in the long term, through the DSB (Dispute Settlement Body), the maintenance of rules that could reduce the role of competition. Public enterprises formed as monopolies to provide a national public service have an uncertain future, which will depend largely not on citizens directly, but on the application of competitive rules supported by the private sector in other countries. The Washington Consensus approved by the International Monetary Fund gives an example of this by always privileging the private sector over the citizens' idea of a public service less open to the notion of profit and the satisfaction of the less well-off users.

Nations have played a determining role in the development of capitalism, and national capitalism has not disappeared, even if it is no longer the only coherent form of capital organization. MNFs have not yet taken over the whole of national economies and national defense still has a meaning. The nation-state, which alone defines a sufficient democratic space, is not necessarily always too small to respond to today's challenges, even if the Chernobyl accident underlines the growing distortion between the model of the sovereign state and technological and environmental realities. The national level remains significant, but it is no longer always the main strategic interlocutor for the key actors of scientific development, technological innovation and socioeconomic growth. The current economic globalization does not imply its equivalent in the political domain. The history of capitalism has ceased to be written within borders. It moves from the wealth of nations to the wealth of the world. Marketization" is often perceived as a factor of democracy. This statement is debatable. The values of individual interest and profit are valued, and the great private money powers can free themselves from all the tutelage of the States by putting them in competition, notably for personnel costs, the location of head offices, and corporate taxation. Under these conditions, the state itself can only become the custodian of the interests of the large multinational firms. Since the states have lost most of their economic power, the electorate is the victim of a democratic illusion comparable to Keynes' monetary illusion. The risk is the progressive establishment of international plutocratic systems, which, within each state, will defend private interests, sometimes in competition from state to state. War and economic war are not over. Bibliography


Fran De Aquino

A New Design for the Gravelectric Generator

Keywords: Gravitational Electromotive Force, Gravitational Energy, Electrical Energy, Generation of Electrical Energy

. Here we show a new design for the Gravelectric Generator. This system can have individual outputs powers of several tens of kW or more.

INTRODUCTION

The electrical current arises in a conductor when an outside force acts upon the free electrons of the conductor. This force is called, in a generic way, of electromotive force (EMF). Usually, it has electrical nature. In a previous paper we have shown that this force can have gravitational nature (Gravitational Electromotive Force), and we have proposed a system to produce Gravitational Electromotive Force, called Gravelectric Generator, which converts Gravitational Energy directly into Electrical Energy [START_REF] De Aquino | The Gravelectric Generator: Conversion of Gravitational Energy Directly Into Electrical Energy[END_REF].

A new design for the Gravelectric Generator is shown in this paper. This system can have individual outputs powers of several tens of kW. It is easy to be built, and can easily be transported.

2.THEORY

Consider a coil with iron core. Through the coil passes a electrical current i , with frequency H f . Thus, there is a magnetic field with frequency H f through the iron core. If the system is subject to a gravity acceleration g , then the gravitational forces acting on electrons    

1 0 g m a m F e Be e ge e      2 0 g m a m F p Bp p gp p      3 0 g m a m F n Bn n gn n    ge m ,

 , Bp

 and Bn  are deduced in the paper [START_REF] De Aquino | Superconducting State generated by Cooper Pairs bound by Intensified Gravitational Interaction[END_REF] and Appendix of [START_REF] De Aquino | The Gravelectric Generator: Conversion of Gravitational Energy Directly Into Electrical Energy[END_REF], and are given by It is known that, in some materials, called conductors, the free electrons are so loosely held by the atom and so close to the neighboring atoms that they tend to drift randomly from one atom to its neighboring atoms. This means that the electrons move in all directions by the same amount. However, if some outside force acts upon the free electrons their movement becomes not random, and they move from atom to atom at the same direction of the applied force. This flow of electrons (their electric charge) through the conductor produces the electrical current, which is defined as a flow of electric charge through a medium [START_REF] Valkengurg | Basic Electricity[END_REF]. This charge is typically carried by moving electrons in a conductor, but it can also be carried by ions in an electrolyte, or by both ions and electrons in a plasma [START_REF] Fischer-Cripps | The electronics companion[END_REF].

  4 1 10 5 . 1 1 2 1 1 56 . 45 1 2 1 2 4 23 2 2 2 2 0 4 4 22 2                                               f B f c m B r k rms e rms e xe Be      5 1 56 . 45 1 2 1 2 2 2 2 0 4 4 2                       f c m B r p rms p Bp      6 1 56 . 45 1 2 1 2 2 2 2 0 4 4 2                       f c m B r
Thus, the electrical current arises in a conductor when an outside force acts upon its free electrons. This force is called, in a generic way, of electromotive force (EMF). Usually, it is of

electrical nature   eE F e 
. However, if the nature of the electromotive force is gravitational V is applied on its ends, the electrical current through the wire is i . Electrodynamics tell us that the electric field, E , through the wire is uniform, and correlated with V and l by means of the following expression [START_REF] Quevedo | Eletromagnetismo[END_REF]  

9    El l d E V  
Since the current i and the area S are constants, then the current density

J  is also constant. Therefore, it follows that     10 .     S l V ES S d J i    
By substitution of E , given by Eq.( 9), into Eq.( 8 17) shows that the Gravelectric Generator is able to produce much more than this value. † This value is based on the well-know fact that, a modern 60 Hz power transformer will probably have a magnetic flux density between 1 and 2T inside the core. Thus, 1.2T can be easily obtained in the device here proposed.

                              If   T B

Ferromagnetic Pin Aluminum pin

Note that: A modern well-designed 60 Hz power transformer will probably have a magnetic flux density between 1 and 2 T inside the core. 4, the Micro Gravelectric Generator produces the electrical current i at any position of the mobile phone, in respect to Earth's surface. Also note that, the direction of the Gravitational Electromotive Force (vector in blue) is always in opposition to gravity (g), because

Be  is negative.

Finally, the most important fact to be observed here is that the electrical energy, which comes from the Gravelectric Generator, is directly converted from the Gravitational Energy, which is a type of renewable energy, always available for use, and that never ends.

FF

  and neutrons   p of the Iron core, are respectively expressed by the following relations[2]

  the inertial masses at rest of the electrons, protons and neutrons.The expressions of the correlation factors Be



  According to Eq. (1) we can rewrite Eq. (7) as follows Now consider a wire with length l ; cross-section area S and electrical conductivity  . When a voltage

VFig. 1 -

 1 Fig. 1 -The voltage V between the ends of a metallic cylinder when it is subjected to a uniform magnetic field H B with frequency

(Fig. 2 -

 2 Fig. 2 -Schematic diagram of a more compact and powerful type of Gravelectric Generator.

Fig. 4 -

 4 Fig. 4 -Schematic diagram of aMicroGravelectricGenerator inside a mobile phone. Note that the Gravelectric Generator produces the electrical current i at any position of the phone, in respect to Earth's surface. Also note that, the direction of the Gravitational Electromotive Force (vector in blue) is always in opposition to gravity (g), because Be  is


Bei Li
email: bei.li@szu.edu.cn

Robin Roche
email: robin.roche@utbm.fr

Sizing a 100% renewable energy based power supply system based on multi-objective optimization

Keywords: zero carbon emission, cost-effective, multiobjective, microgrid, hydrogen storage

Nowadays, large amounts of renewable energy have been installed around the world, and building a zero carbon emission energy supply system has been put on the agenda. Then, how to size the zero carbon emission energy system to achieve cost-effective is an essential problem. In this paper, we build a 100% renewable energy based generating station and microgrid clusters to supply the load demands. Three objectives are considered, namely, minimizing the total cost, minimizing the total exchanged energy, and maximizing the installed PV panels. Genetic algorithm is adopted to solve the problem. The simulation results show that: 1) the hydrogen storage operates as the core device to build the 100% renewable energy based power supply system; 2) when the investment cost is decreasing, the total exchanged energy is increasing, and the installed PV panels are decreasing; when the PV panels are decreasing, the total exchanged energy is decreasing; 3) the volume of the hydrogen tanks in generating station is larger than that in microgrids, because it needs to cover the demands from microgrids.

I. INTRODUCTION

Nowadays, renewable energy has been installed widely around the world. Based on the renewable energy generation, current energy supply system is increasingly changing to low carbon emissions system, and in the future, achieving zero carbon emission energy system is the ultimate goal [START_REF] Colbertaldo | Impact of hydrogen energy storage on california electric power system: Towards 100renewable electricity[END_REF]. In addition, photovoltaics have been verified as an effective way to reduce greenhouse gas emission in Europe union [START_REF] Jäger-Waldau | How photovoltaics can contribute to ghg emission reductions of 55% in the eu by 2030[END_REF]. In the generation side, large amounts of renewable energy resources are integrated with storage systems to form zero carbon emission generating station. In the customer side, large numbers of microgrids are built to absorb local renewable energy and reduce buying costs from utility grids. The zero carbon emission generating station and microgrid clusters can be seen in Fig. 1.

The generating station and the microgrids are different. Generating station is often located in remote areas which has This work has been supported by the "Guangdong Basic and Applied Basic Research Foundation" (2019A1515110641), the "Fundamental Research Funds for the Shenzhen university" (000002110235), the EIPHI Graduate School (contract ANR-17-EURE-0002), and the Region Bourgogne-Franche-Comté. abundant renewable energy resources, and a large capacity of renewable energy is installed. Microgrid is often located in customer side, and due to the geographical limitation, limited renewable energy might be installed.

In fact, due to the intermittent and uncertainty of the renewable energy resources, energy storage system is a necessary component [START_REF] Chang | Review on distributed energy storage systems for utility applications[END_REF]. In general, hydrogen storage system has high 978-1-6654-3597-0/21/$31.00 ©2021 IEEE energy density, medium power density; battery has medium energy and power density [START_REF] Chang | Review on distributed energy storage systems for utility applications[END_REF]. Then, a combined hydrogen and battery storage system is a better choice to respond to renewable energy uncertainty. This is because the energy and power density will be always in a high level. In this paper, the combined hydrogen and battery storage system is deployed to build the generating station and microgrids, and the structure can be seen in Fig. 2. In the hydrogen storage system, fuel cell uses H 2 to produce electricity and heat; electrolyzer uses electricity to produce H 2 ; H 2 is stored in hydrogen tanks. Then how to size the above zero carbon emission energy system to achieve cost-effective is an essential problem [START_REF] Hannan | Review of optimal methods and algorithms for sizing energy storage systems to achieve decarbonization in microgrid applications[END_REF]. On the one hand, we can install a large capacity of renewable energy resources and storage system, which can certainly satisfy the load demands, but the total investment cost is not economical. On the other hand, if we satisfy the economical budget (namely, install limited capacity of renewable energy), the total load demands may not be supplied, where a load shedding concern may happen. So, the sizing of such system should consider both the investment cost and operation efficiency.

Authors in [START_REF] Hannan | Review of optimal methods and algorithms for sizing energy storage systems to achieve decarbonization in microgrid applications[END_REF] review the optimization sizing methods for energy storage systems, in which the different methods can be divided into four categories: probabilistic, analytical, artificial intelligence and hybrid methods. Among them, the artificial intelligence method can give a reasonable sizing values with limited searching time, but be with probabilities to converge to the local optimums [START_REF] Hannan | Review of optimal methods and algorithms for sizing energy storage systems to achieve decarbonization in microgrid applications[END_REF]. And artificial intelligence technique is popular used in various researches.

In this paper, we focus on sizing a zero carbon emission energy system. Several objective functions are considered, including minimizing the total investment cost and operation cost, minimizing the exchanged energy, and maximizing the installed renewable energy. It is actually a multi-objective optimal sizing problem.

In fact, papers using multi-objective algorithm to optimal design renewable energy based microgrids have also been published. Authors in [START_REF] Durmaz | Multi-objective optimization of sustainable biomass supply chain network design[END_REF] address the optimal design of the biomass supply chain system. And analytic hierarchy process is adopted to firstly decide the candidate location of biogas facilities. Then a multi-objective mixed integer linear programming model (namely, maximize the profit, and minimize total distance between poultry farms and biogas facilities) is presented to determine the biogas facility capacities. In [START_REF] Ramli | Optimal sizing of pv/wind/diesel hybrid microgrid system using multi-objective selfadaptive differential evolution algorithm[END_REF], authors adopt multi-objective self-adaptive differential evolution algorithm to size a hybrid microgrid system. Loss of power supply probability and cost of electricity are adopted as the two objective functions. In [START_REF] Giallanza | A sizing approach for stand-alone hybrid photovoltaic-wind-battery systems: A sicilian case study[END_REF], authors present an optimal sizing for stand-alone hybrid microgrid. Two objective functions, i.e., loss of load probability and seasonal loss of load probability ratio are considered. In [START_REF] Abdelkader | Multiobjective genetic algorithm based sizing optimization of a stand-alone wind/pv power supply system with enhanced battery/supercapacitor hybrid energy storage[END_REF], authors adopt a multi-objective genetic algorithm to size the microgrid with hybrid storage system. Discrete Fourier transform is used to split energy to battery and supercapacitor. Total costs of electricity and loss of power supply probability two objective functions are deployed. In [START_REF] Duchaud | Multi-objective particle swarm optimal sizing of a renewable hybrid power plant with storage[END_REF], authors optimally size a renewable hybrid power plant with storage. Multi-objective particle swarm optimization is used to minimize two objective functions, i.e., the annualized cost of system and the amount of energy imported from non renewable sources. In [START_REF] Fioriti | Economic multiobjective approach to design off-grid microgrids: A support for business decision making[END_REF], authors present a multi-objective optimal design for typical rural microgrids in developing countries. Non-dominated sorting genetic algorithm (NSGA-II) is adopted, where there are two objective functions: net present value and modified internal rate of return are considered. In [START_REF] Yousefi | Ga/ahp-based optimal design of a hybrid cchp system considering economy, energy and emission[END_REF], authors present an optimal design of the integration of a hybrid CCHP system into a commercial building. Firstly, genetic algorithm is adopted to optimize three objective functions separately, namely, annual operating cost ratio, primary energy saving ratio and carbon emission reduction ratio. Then, an analytic hierarchy process is used to choose the best answer. In [START_REF] Zheng | A minlp multi-objective optimization model for operational planning of a case study cchp system in urban china[END_REF], authors present a method to determine the best combination of technologies in district buildings. Three objective functions (i.e., net present value, low carbon emissions, and low energy bill) are transferred to a single weighted function based on entropy weight method. In [START_REF] Ding | Multi-objective optimization for an integrated renewable, power-to-gas and solid oxide fuel cell/gas turbine hybrid system in microgrid[END_REF], authors present a planning model for a wind/hydrogen based hybrid microgrid system. Genetic algorithm is adopted to minimize system cost and wind curtailment rate.

However, the above papers all study the sizing of microgrids, and they did not consider the 100% renewable energy based generating station, which is used to supply energy to microgrids. With the extended microgrids and 100% renewable energy based generating station, a new zero carbon emission energy system can be built.

A. Contributions

In our previous paper, we studied the sizing of the full electric microgrid [START_REF] Li | Microgrid sizing with combined evolutionary algorithm and milp unit commitment[END_REF] and multi-energy supply microgrid [START_REF] Li | Sizing of a stand-alone microgrid considering electric power, cooling/heating, hydrogen loads and hydrogen storage degradation[END_REF], but they are single-objective optimization. In this paper, we build a zero carbon emission generating station and microgrid clusters using multi-objective optimization to achieve three goals: minimizing total cost, minimizing exchanged energy, and maximizing installed renewable energy. Compare to previous works, the contribution of this paper can be concluded as follows:

• First, a zero carbon emission energy supply system (100% renewable energy based generating station and microgrid clusters) is built;

• Second, three objective functions (minimizing total cost, minimizing exchanged energy, and maximizing installed renewable energy) are considered, and NSGA-II is adopted to obtain the optimal results; • Last, it can be seen that the volume of the hydrogen tanks in generating station is larger than that in microgrids.

Because it needs to cover the exchanged energy from microgrids. In addition, due to the geographical location of the generating station, the sizing value of PV number is larger than that in microgrids. The remainder of this paper is organized as follows. Section II describes the operation of hydrogen-based storage system, and Section III the sizing method based on genetic algorithm. Section IV the simulation results. Finally, Section V concludes the paper.

II. OPERATION OF HYDROGEN-BASED STORAGE SYSTEM

In the generating station and microgrid, the core device is the hydrogen-based storage system, which is used to respond to renewable energy intermittent. In fact, there are two common methods to operate the hybrid storage system, namely, rule-based strategy and optimization strategy. With different strategies, the operation of the storage is significantly different. Here, we adopt the basic rule-based strategy to control the operation of the hydrogen-based storage. More advanced operation strategies are out the scope of this paper, and will be further studied in the future.

The operation rules are set as follows: Algorithm 1 Rule-based strategy 1: Input PV P P V and load demand P load dataset; 2: for t = 1 : T do 3:

if P P V > P load then 4:
firstly stored in hydrogen storage system, considering sizing limitations: P ch hy < P ch,max hy (= P ele ), V H2 < V tanks ; firstly utilize the hydrogen storage system, considering sizing limitations: P dis hy < P dis,max hy

(= P f c ), V H2 < V tanks ; 10:
secondly use battery storage system, considering sizing limitations: P dis ba < P t=t+1; 14: Output: P curt , P ls ; When the generated PV power P P V is larger than the load demands P load , the surplus energy P P V -P load is firstly stored in hydrogen storage system; if the hydrogen storage is full, then secondly stored in battery storage; if the two storages are all fully charged, the remaining surplus energy is curtailed.

When the generated PV power P P V is smaller than the load demands P load , the shortage energy P load -P P V is firstly supplied by hydrogen storage system; if the hydrogen storage is insufficient, battery storage system is secondly used; if the two storages are all used up, the remaining load demands are shed. Here, P ch,dis,max ba = 0.9 • C ba . P ele , P f c , V tanks are the sizing values of fuel cell, electrolyzer, and tanks, respectively. The shed loads P ls in microgrids are then supplied by generating station, and P ls is the exchanged energy P M Gi ex in each microgrid. In fact, there are many rule-based strategies, it depends on the storage conditions and user's goal. Here, we just show a reasonable operation strategy.

Based on the above storage system operation strategy, we can then develop the operation of the microgrid clusters and generating station, which is shown in the following:

Algorithm 2 Operation strategy of the whole system are the installed PV panels in generating station and microgrid n. Firstly, each microgrid runs the operation strategy based on Algorithm 1, and the exchanged energy P M Gi ex with generating station can be obtained; then, the total exchanged energy n i=1 P M Gi ex are then submitted to generating station; thirdly, generating station runs its operation strategy based on Algorithm 1; at last, the curtailed power P base curt , load shedding P base ls , battery charging/discharging power Z ch,dis , and fuel cell/electrolyzer ON/OFF state onof f f c,ele of the generating station are obtained.

III. SIZING METHOD BASED ON GENETIC ALGORITHM

Based on the above operation strategy of the whole system, we can then develop the sizing method. Here, we adopt the genetic algorithm (NSGA-II) to search for the best sizing values. The developed sizing algorithm can be seen as follows:

Genetic algorithm is an iterative search method based on biological evolution to find optimal solutions [START_REF] Ding | Multi-objective optimization for an integrated renewable, power-to-gas and solid oxide fuel cell/gas turbine hybrid system in microgrid[END_REF]. Firstly, genetic algorithm generates sizing values for each component. Then, based on the generated sizing values, operation of the whole system is executed, namely, run Algorithm 2. Based on the running output of Algorithm 2, the operation cost C op , Algorithm 3 Developed sizing method 1: for j = 1 : it max do 2:

generate initial sizing values, generating station: N base P V ,P base f c , P base ele , V base tanks , C base ba ; microgrids:

N M Gi P V , P M Gi f c , P M Gi ele , V M Gi tanks , C M Gi ba , i = 1, • • • , n; 3:
run Algorithm 2; . After that, based on the fitness function, process NSGA-II operators [START_REF] Abdelkader | Multiobjective genetic algorithm based sizing optimization of a stand-alone wind/pv power supply system with enhanced battery/supercapacitor hybrid energy storage[END_REF], the newly sizing values are updated. The sizing algorithm is repeatedly running until the stopping criteria is satisfied. Here, the stopping criteria is the maximum iteration number it max .

The operation cost C op is shown as follows:

T t=1 Baop(Z ch (t) + Z dis (t)) + Hyop(onof f f c (t) + onof f ele (t)) + Hyst(ST f c (t) + ST ele (t)) + αP ls (t) + βPcurt(t) ( 1
)
where T is the time horizon, here T = 8760h represents one year; Ba op = Binv B lif e is the battery utilization cost, B inv is the battery investment cost, B lif e is the battery lifetime; Z ch , Z dis are the charging and discharging power of battery; Hy op = f c,ele inv f c,ele lif e + O&m is the hydrogen storage utilization cost, f c, ele inv is the fuel cell/electrolyzer investment cost, O&m is the operation and maintenance cost; onof f f c,ele are the ON/OFF state of fuel cell/electrolyzer; Hy st is the start up cost of hydrogen storage; ST f c,ele are start up state of fuel cell/electrolyzer; α is the penalty cost of load shedding; P ls is the load shedding; β is the penalty cost of curtailed renewable energy; P curt is the curtailed power. Here, α and β are arbitrarily chosen as 10 20 , in order to reduce load shedding and curtailed power. When the value of the objective function is larger than 10 20 , it means that load shedding and curtailed power occur.

The investment cost C inv is shown as follows:

C inv =P V inv N P V + f c inv P f c + ele inv P ele + T ank inv V tanks + Ba inv C ba (2)
where P V inv , T ank inv are the investment cost of PV panel and hydrogen tanks, respectively.

At last, the three objective functions can be described as:

f 1 = C base op + C base inv + n i=1 (C M Gi op + C M Gi inv ) (3)
f 2 = n i=1 P M Gi ex (4)
f 3 = N base P V + n i=1 N M Gi P V (5)
The multi-objective sizing problem can be represented as:

minf 1 , minf 2 , maxf 3 (6)
Namely, minimizing the operation and investment cost, minimizing the exchanged energy with generating station, maximizing the installed PV panels. It can be seen that when the investment cost is decreasing (f 1 ↓), the exchanged energy is increasing (f 2 ↑), and the installed PV panels are decreasing (f 3 ↓).

IV. SIMULATION RESULTS

Based on the above proposed multi-objective sizing algorithm, we then deploy a typical simulation case. In the case, four microgrids are considered, including industrial park microgrid, smart building microgrid, parking lot, and residential area microgrid. These four microgrids are supplied by a zero carbon emission generating station. Our goal is to find the best sizing value for each component to achieve the above three objective functions. The cost coefficients are presented in Tab. I. The data is obtained from [START_REF] Li | Microgrid sizing with combined evolutionary algorithm and milp unit commitment[END_REF] [16]. One unit PV panel rated power is 10kW . The solar radiation data profile in microgrids area is shown in Fig. 3. The load demands in generating station is presented in Fig. 4. Load demands in residential microgrid is presented in Fig. 5. Due to the page limitation, load demands in the other microgrids are not presented. One simulation results based on genetic algorithm is shown in Fig. 6. The average time of one simulation running is about 1289 seconds. It can be seen that one objective function results (Total costs) in some candidates are larger than 10 20 . Here, we should notice that if the operation costs are larger than 10 20 , it means that load shedding or curtailed PV power happen, then we should abandon this candidate. In order to obtain possible candidates that no load shedding and curtailed power occur, we run large numbers of simulations. At last, the simulation results are presented in Fig. 7. Here, the installed PV numbers are negative values, because the third objective function is maximization, and we need to transfer it into the minimization using the negative operator. The simulation results did not show obvious Pareto fronts, because the candidate solutions are not enough, and more simulations should be done. We then adopt the scatter plot to present the relationship between each sizing component, which can be seen in Fig. 8. It can be seen that the values of the components are falling in a respectful range, which means the sizing results are effective.

A. Results analysis

We choose one candidate solution from Fig. 7 as an example, and analyse the sizing results. The chosen sizing solution is presented in Tab. II. Pie plot is adopted to show the sizing results of residential, parking lot, industrial, and smart building microgrid in Fig. 9. It can be seen that the volume of tanks and the capacity of battery are the important constituent parts of the microgrid. The sizing for generating stations can be seen in Fig. 10, where "fc" represents fuel cell, "ele" represents electrolyzer, "tanks" represents hydrogen tanks. Compared with microgrids, it can be seen that the volume of the hydrogen tanks in generating station is six times larger than that in microgrids. In addition, large numbers of PV panels can be installed, because the generating station is located far away from city centre, thus a plenty of lands can be utilized. At last, based on the above candidate sizing results, we run the operation of the whole system for one year. And the changes of the levels of the hydrogen in different parts can be seen in Fig. 11. It shows that at the end of the current year, the hydrogen in residential microgrid is used up; in addition, the volume of the hydrogen at the end of the year is less than the volume of the hydrogen at the beginning of the year. This means that along the whole year, the produced hydrogen is less than the consumed hydrogen, and the extra hydrogen should be charged in each part for preparing the next year operation.

V. CONCLUSION

In this paper, we have developed a zero carbon emission generating station and microgrid clusters, to achieve a zero carbon emission energy supply system. In order to consider different aspects, a multi-objective optimization is adopted to achieve three goals simultaneously, namely, minimizing total cost, minimizing total exchanged energy, and maximizing installed PV panels. The simulation results show several insights. 1) It is a reasonable idea to build the zero carbon emission generating station to supply energy to microgrids.

The core device is the hydrogen-based storage system. The scatter distribution shows that the sizing values are all in a respectful range. 2) When the investment cost is decreasing, the total exchanged energy is increasing, and the installed PV panels are decreasing; when the PV panels are decreasing, the total exchanged energy is decreasing. 3) The volume of the hydrogen tanks in generating station is larger than that in microgrids, because it needs to cover the exchanged energy from microgrids.

Fig. 1 .

 1 Fig. 1. Zero carbon emission generating station and microgrid clusters.

Fig. 2 .

 2 Fig. 2. Combined hydrogen and battery storage system.

Fig. 3 .

 3 Fig. 3. The solar radiation data profile in microgrids area.

Fig. 4 .

 4 Fig. 4. The load demands in generating station.

Fig. 5 .

 5 Fig. 5. Load demands in residential microgrid.

Fig. 6 .

 6 Fig. 6. One simulation results based on genetic algorithm.

Fig. 7 .

 7 Fig. 7. Simulation results.

Fig. 8 .

 8 Fig. 8. Scatter plot to present the relationship between each sizing component.

Fig. 9 .

 9 Fig. 9. The sizing results for residential, parking lot, industrial, and smart building microgrid.

Fig. 10 .

 10 Fig. 10. The sizing for generating station.

Fig. 11 .

 11 Fig. 11. Changes of the levels of the hydrogen in different parts.

TABLE II ONE

 II CANDIDATE SOLUTION.

	Variables	PV numbers	fuel cell [kW ]	electrolyzer [kW ]	hydrogen tanks [N.m 3 ]	battery [kW h]
	Generating station	500	5191	14362	28138	1284
	Residential	111	363	598	4834	484
	Parking lot	78	323	451	2670	4494
	Industrial	57	28	754	4950	4011
	Smart building	55	377	306	4980	4679


Jouni Parkkonen

Frédéric Paulin

Sectorial Mertens and Mirsky formulae for imaginary quadratic number fields

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the standard Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences.

Introduction

Let K be a number field of degree n K , with ring of integers O K , number of real places r 1 , number of complex conjugated places r 2 , regulator R K , class number h K , number of units ω K , discriminant D K and Dedekind zeta function ζ K (see for instance [Nar]). Let I K be the semigroup of nonzero ideals of O K , let ϕ K : I K Ñ N be the Euler function of K, and let N : I K Ñ N be the norm, with ϕ K paq " ϕ K paO K q and Npaq " NpaO K q for every a P O K ´t0u. As usual, p below ranges over prime ideals in I K . The functions Op¨q below depend only on K.

Our first result (see Section 2) is a Mertens formula with congruences for number fields. Though probably well-known at least when m " O K , we provide a proof for lack of reference (compare with [START_REF] Grotz | Mittelwert der Eulerschen ϕ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern[END_REF]Satz 2], [START_REF] Cosentino | Equidistribution of parabolic fixed points in the limit set of Kleinian groups[END_REF]§4.3], [START_REF] Parkkonen | On the arithmetic of cross-ratios and generalised Mertens' formulas. Numéro Spécial "Aux croisements de la géométrie hyperbolique et de l'arithmétique[END_REF]Theo. 3.1]) since arguments of its proof will be useful for our next result. For every m P I K , let c m " Npmq ź p|m p1 `1 Nppq q .

Theorem 1.1 For every m P I K , if n K ě 2, then as x Ñ `8, we have

ÿ aPI K : Npaqďx, m|a ϕ K paq " 2 r 1 `r2 ´1 π r 2 R K h K ω K a |D K | ζ K p2q c m x 2 `O `x2´1 n K ˘.
Assume in the remaining part of this introduction that K is imaginary quadratic and that O K is principal. By Dirichlet's unit theorem, these assumptions are more or less necessary (besides K " Q) for the following sums to be well defined and finite.

We give in Section 3 a version in angular sectors of the Mertens formula given by Theorem 1.1, that will be needed in [START_REF] Parkkonen | Pair correlations of logarithms of complex lattice points[END_REF]. For all z P C ˆ, θ P s0, 2πs and R ě 0, we consider the truncated angular sector Cpz, θ, Rq "

! ρ e it z : t P s ´θ 2 , θ 2 s, 0 ă ρ ď R |z| ) . (1)
It is important that the function Op¨q in the following result is uniform in m, z and θ.

Theorem 1.2 Assume that K is imaginary quadratic with O K principal. For all m P I K , z P C ˆand θ P s0, 2πs, as x Ñ `8, we have

ÿ aPmXCpz,θ,xq ϕ K paq " θ 2 a |D K | ζ K p2q c m x 4 `Opx 3 q .
Lastly, we give a uniform asymptotic formula for the sum in angular sectors in C of angle θ of the products of two shifted Euler functions with congruences, that will be needed in [START_REF] Parkkonen | Pair correlations of logarithms of complex lattice points[END_REF]. When K " Q (the sectorial restriction is then meaningless), this formula is due to Mirsky [START_REF] Mirsky | Summation formulae involving arithmetic functions[END_REF]Thm. 9,Eq. (30)] without congruences, and to Fouvry [START_REF] Parkkonen | On the statistics of pairs of logarithms of integers[END_REF]Appendix] with congruences. For simplicity, we give a version without congruences and without an error term in this introduction, see Section 4 Theorem 4.1 for the general statement.

Theorem 1.3 For all z P C ˆ, θ P s0, 2πs and k P O K , as x Ñ `8, we have

ÿ aPO K XCpz,θ,xq ϕ K paq ϕ K pa `kq " θ 3 a |D K | ź p `1 ´2 Nppq 2 ˘ź p | kO K `1 `1 NppqpNppq 2 ´2q ˘x6 .
Theorems 1.2 and 1.3 are used in [START_REF] Parkkonen | Pair correlations of logarithms of complex lattice points[END_REF] in order to study the correlations of pairs of complex logarithms of Z-lattice points in the complex line at various scalings, when the weights are defined by the Euler function, proving the existence of pair correlation functions. We prove in op. cit. that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. A geometric application is given in op. cit. to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi manifolds PSL 2 pO K qzH 3 R .

Recall that the Dedekind zeta function ζ K : ts P C : Repsq ą 1u Ñ C of K is defined (see for instance [START_REF] Narkiewicz | Elementary and analytic theory of algebraic numbers[END_REF]§7.1]) equivalently by

ζ K psq " ÿ aPI K 1 Npaq s " ź p `1 ´1 Nppq s ˘´1 .
We denote by ϕ K : I K Ñ N the Euler function of K, defined (see for instance [START_REF] Narkiewicz | Elementary and analytic theory of algebraic numbers[END_REF]page 13]) equivalently by

@ a P I K , ϕ K paq " Card `pO K {aq ˆ˘" Npaq ź p|a `1 ´1 Nppq ˘.
For every a P O K ´t0u, we define ϕ K paq " ϕ K paO K q. Note that the Euler function ϕ K is multiplicative2 by the Chinese remainder theorem. We have

Npaq "

ÿ b|a ϕ K pbq , (2)
as checked by telescopic sum when a is a power of a prime ideal, and by multiplicativity. We denote by µ K : I K Ñ Z the Möbius function of K, defined by

@ a P I K , µ K paq " $ ' ' ' & ' ' ' % 1 if a " O K 0 if p 2 |
a for some prime ideal p p´1q m if a " p 1 . . . p m for pairwise distinct prime ideals p 1 , . . . , p m and m P N ´t0u .

For every a P O K ´t0u, we define µ K paq " µ K paO K q. We have (see for instance [Sha]) the Möbius inversion formula: for all f, g :

I K Ñ C, f paq " ÿ b|a gpbq if and only if gpaq " ÿ b|a µ K pbqf pab ´1q . (3)
In particular, since the norm is completely multiplicative and by Equation (2), we have

@ a P I K , ϕ K paq Npaq " ÿ b|a µ K pbq Npbq . (4)
Proof of Theorem 1.1. In this proof, all functions Op¨q depend only on K. Let

ρ K " 2 r1 p2πq r2 R K h K ω K a |D K | . ( 5
)
Recall (see for instance [START_REF] Murty | Counting integral ideals in a number field[END_REF]Theo. 5]) that, as x Ñ `8, we have

Cardta P I K : Npaq ď xu " ρ K x `Opx 1´1 n K q . ( 6
)
By Abel's summation formula, as y Ñ `8, we have

ÿ a PI K : Npaqďy Npaq " ÿ 1ďnďy n Cardta P I K : Npaq " nu " ρ K 2 y 2 `Opy 2´1 n K q . (7)
Furthermore, we have

Cardta P I K : Npaq " yu " Opy 1´1 n K q .
This formula implies since Nppb, mqq ď Npmq that

ˇˇÿ bPI K : Npbqěx µ K pbq Nppb, mqq Npbq 2 ˇˇ" O ´Npmq ÿ něx n 1´1 n K n 2 ¯" OpNpmq x ´1 n K q . ( 8
)
Let us denote by S m pxq the sum on the left hand side in the statement of Theorem 1.1. Note that by the Gauss lemma, for all m, b, c P I K , we have m | bc if and only if mpm, bq ´1 | c. Then by Equation ( 4), by the change of variable c " mpm, bq ´1a, by the complete multiplicativity of the norm, by Equation ( 7) with y " Nppb,mqqx Npbq Npmq , since Nppb, mqq ď Npmq, and by Equation ( 8), we have

S m pxq " ÿ aPI K : Npaqďx, m|a ÿ b,c PI K : bc"a µ K pbq Npcq " ÿ bPI K : Npbqďx µ K pbq ÿ c PI K : Npcqď x Npbq , m|bc Npcq " ÿ bPI K : Npbqďx µ K pbq ÿ a PI K : Npaqď Nppb,mqqx Npbq Npmq Npmq Nppb, mqq Npaq " ´ÿ bPI K : Npbqďx µ K pbq Nppb, mqq Npbq 2 ¯ρK 2 Npmq x 2 `Op x 2´1 n K q " ´ÿ bPI K µ K pbq Nppb, mqq Npbq 2 ¯ρK 2 Npmq x 2 `Op x 2´1 n K q . ( 9
)
By decomposing a nonzero integral ideal b into powers of prime ideals, by the definition of the Möbius function, and by the Euler product formula for the Dedekind zeta function, we have

ÿ bPI K µ K pbq Nppb, mqq Npbq 2 " ź p m p1 ´1 Nppq 2 q ź p | m p1 ´1 Nppq q " 1 ζ K p2q ź p | m Nppq 1 `Nppq .
Equations ( 9) and ( 5) hence imply Theorem 1.1. l

A sectorial Mertens formula

Assume in the remaining part of this paper that K is imaginary quadratic and that O K is principal (or equivalently factorial (UFD)). By Dirichlet's unit theorem, the group of units O K , whose order we denote by |O K |, is finite if and only if pr 1 , r 2 q is equal to p1, 0q or p0, 1q. This justifies our restriction, the case K " Q being well-known. With the notation of the beginning of the introduction, we then have (see for instance [Nar]) D K P t´4, ´8, ´3, ´7, ´11, ´19, ´43, ´67, ´163u, and

r 1 " 0, r 2 " 1, n K " 2, R K " 1, ω K " |O K | and h K " 1 . ( 10
)
Given a Z-lattice Λ in the Euclidean space C (that is, a discrete (free abelian) subgroup of pC, `q), we denote by covol Λ " VolpC{ Λq the area of a fundamental parallelogram F Λ for Λ and by diam Λ the diameter of F Λ . Note that every element m

P I K is a Z-lattice in C with covol m " Npmq covol O K " Npmq a |D K | 2 and diam m " Op a |D K | Npmq q (11) since diam O K " |1 `?D K 2 | if D K " 0 mod 4 and diam O K " | 3`?D K
With the notation of Equation ( 1), note that for every z 1 P C ˆ, we have

z 1 Cpz, θ, Rq " Cpzz 1 , θ, R |z 1 | q . ( 12
)
Proof of Theorem 1.2. Let z P C ˆ, θ P s0, 2πs and y ą 0. Since AreapCpz, θ, yqq " θ 2 y 2 , the standard Gauss counting argument, the finiteness of the number of imaginary quadratic number fields with class number 1, and the equality on the left of Formula (11) give

Card `OK X Cpz, θ, yq ˘" AreapCpz, θ, yqq covol O K `O `diam O K y covol O K " θ a |D K | y 2 `Opyq .
Since the map z 1 Þ Ñ |z 1 | 2 " Npz 1 q takes only integral values on O K , by Abel's summation formula, as y Ñ `8, we have

ÿ dPO K XCpz,θ,yq |d| 2 " ÿ 1ďnďy 2 n Cardtd P O K X Cpz, θ, yq : |d| 2 " nu " θ 2 a |D K | y 4 `Opy 3 q . ( 13
|m| |b| q, Npmq Nppb, mqq |d| 2 " θ Nppb, mqq 2 a |D K | Npmq Npbq 2 x 4 `O ´x3 Npbq 3{2 ¯.
Let us denote by S m,z,θ pxq the sum on the left hand side in the statement of Theorem 1.2. Then by Equation ( 4), we have

S m,z,θ pxq " ÿ aPmXCpz,θ,xq ϕ K paO K q " ÿ aPmXCpz,θ,xq ÿ b,c PI K : bc"aO K µ K pbq Npcq " ÿ bPI K : Npbqďx 2 µ K pbq S b " ´ÿ bPI K : Npbqďx 2 µ K pbq Nppb, mqq Npbq 2 ¯θ 2 a |D K | Npmq x 4 `Opx 3 q .
The proof then proceeds exactly as in the proof of Theorem 1.1. l

A sectorial Mirsky formula

We now give a uniform asymptotic formula for the sum in angular sectors of the products of shifted Euler functions with congruences. For all z P C ˆ, θ P s0, 2πs, k P O K , m P I K and x ě 1, let

S z,θ,k,m pxq " ÿ aPmXCpz,θ,xq ϕ K paq ϕ K pa `kq . ( 14
)
Theorem 4.1 Assume that K is imaginary quadratic with O K principal. There exists a universal constant C ą 0 such that for all k P O K and m P I K , there exists c m,k P s0, 1s such that for all z P C ˆ, θ P s0, 2πs and x ě 1, we have

ˇˇSz,θ,k,mpxq ´θ c m,k 3 a |D K |
x 6 ˇˇď C `p1 `aNpkq q x 5 `Npkq x 4 ˘.

We will prove Theorem 4.1 at the end of this Section after giving a number of Lemmas required for the proof. We fix k P O K and m " mO K P I K , and we define h " kO K , which is a possibly zero integral ideal. We start by giving the first definition and a simpler formula for the constant c m,k that appears in the statement of Theorem 4.1. We define

c m,k " ÿ b,cPI K pb,cq | h, pcpb,mq,mpb,cqq | hb µ K pbq µ K pcq N `pcpb, mq, mpb, cqq Npbq 2 Npcq 2 Npmq , (15)
and

c 1 m " inf kPO K c m,k .
Lemma 4.2 The series in Equation (15) defining c m,k converges absolutely. We have c m,k ď 1 and c 1 m ą 0. Furthermore, we have

c m,k " 1 Npmq ź p pp,mq | h `1 ´Nppp, mqq Nppq 2 ˘ź p `1 ´κm,h ppq κ 1 h ppq Nppp, mqq Nppq 2 ˘, ( 16
)
where

κ m,h ppq " ! p1 ´Nppp,mqq Nppq 2 q ´1 if pp, mq | h 1 otherwise and κ 1 h ppq " " 1 ´1 Nppq if p | h 1 otherwise. ( 17
)
In the special case m " O K , Equation ( 16) becomes

c O K ,k " ź p `1 ´1 Nppq 2 ˘ź p | h ´1 ´p1 ´1 Nppq 2 q ´1p1 ´1 Nppq q Nppq 2 ¯ź p h ´1 ´p1 ´1 Nppq 2 q ´1 Nppq 2 " ź p `1 ´1 Nppq 2 ˘ź p | h ´1 ´Nppq ´1 NppqpNppq 2 ´1q ¯ź p `1 ´1 Nppq 2 ´1 ˘ź p | h ´1 ´1 Nppq 2 ´1 ¯´1 " ź p `1 ´2 Nppq 2 ˘ź p | h `1 `1 NppqpNppq 2 ´2q ˘. ( 18
)
Theorem 1.3 in the introduction follows from Theorem 4.1 and the above computation.

Proof. Let us prove that uniformly in x ě 1, we have

ÿ b,cPI K : Npbqěx, pb,cq | h, pcpb,mq,mpb,cqq | hb N `pcpb, mq, mpb, cqq Npbq 2 Npcq 2 Npmq " O `1 ? x ˘. (19)
This implies, by taking x " 1, that the first claim of Lemma 4.2 is satisfied, since the Möbius function has values in t0, ˘1u. Let us denote by Z m,h pxq the above sum. Since N `pcpb, mq, mpb, cqq ˘ď Npmpb, cqq, we have

Z m,h pxq ď ÿ b,cPI K : Npbqěx Nppb, cqq Npbq 2 Npcq 2 ď ÿ a,b 1 ,c 1 PI K Npb 1 qěx{ Npaq Npaq Npab 1 q 2 Npac 1 q 2 " ÿ c 1 PI K 1 Npc 1 q 2 ÿ aPI K 1 Npaq 5{2 ÿ b 1 PI K Npaq Npb 1 qěx 1 Npb 1 q 3{2 pNpaq Npb 1 qq 1{2 ď ζ K p2q ζ K p 5 2 q ζ K p 3 2 q 1 ? x .
Equation ( 19) follows, since there are only finitely many fields K satisfying the assumptions of Theorem 4.1.

The proof of Equation ( 16) that we now give is similar to Fouvry's proof of Equation ( 21) in [START_REF] Parkkonen | On the statistics of pairs of logarithms of integers[END_REF]Appendix].

For every b P I K , let χ b : I K Ñ t0, 1u be the characteristic function of the set of elements c P I K such that pc, bq | h. Let us define a map ψ b :

I K Ñ I K by ψ b : c Þ Ñ `c, m pb, mq pb, cq ˘. (20)
Note that the assertion pcpb, mq, mpb, cqq | b h is equivalent to the assertion

ψ b pcq | b pb, mq h .
For every b P I K , let χ b : I K Ñ t0, 1u be the characteristic function of the set of elements c P I K such that the above divisibility assertion is satisfied. Let us finally define a map C ˚: I K Ñ R (which depends on m and h) by

C ˚: b Þ Ñ ÿ cPI K µ K pcq Npcq 2 χ b pcq χ b pcq Npψ b pcqq . (21)
By the absolute convergence property, Equation ( 15) then becomes

c m,k " 1 Npmq ÿ bPI K µ K pbq Npbq 2 Nppb, mqq C ˚pbq . (22)
In order to transform the series C ˚pbq defined by Formula ( 21) into an Eulerian product and in order to analyse it, we will use the following two lemmas. Proof. We have ψ b pO K q " O K and χ b pO K q " χ b pO K q " 1. Let I, J P I K be coprime. The equality pIJ, bq " pI, bqpJ, bq and the fact that pI, bq and pJ, bq are coprime imply that χ b pIJq " χ b pIqχ b pJq.

In order to prove the multiplicativity of the map ψ b , we write ψ b pIJq " `IJ, m pb, mq pb, IJq ˘" `I, m pb, mq pI, bqpJ, bq ˘`J, m pb, mq pI, bqpJ, bq ˘.

Since I is coprime to pJ, bq and since J is coprime to pI, bq, we obtain as wanted the equality ψ b pIJq " ψ b pIq ψ b pJq. 21) may be written as an Eulerian product

C ˚pbq " ź p `1 ´χb ppq χ b ppq Npψ b ppqq Nppq 2 ˘" ź p χ b ppq χ b ppq"1 `1 ´Npψ b ppqq Nppq 2 ˘. (23)
By Equations ( 22) and ( 23), and by Lemma 4.4, we have

c m,k " 1 Npmq ÿ bPI K µ K pbq Npbq 2 Nppb, mqq ź p b, pp,mq | h `1 ´Nppp, mqq Nppq 2 ˘ź p | pb,hq `1 ´1 Nppq ˘. Let us define Γ m,h " ź p pp,mq | h `1 ´Nppp,mqq Nppq 2 ˘, so that c m,k " Γ m,h Npmq ÿ bPI K µ K pbq Npbq 2 Nppb, mqq ź p p | b, pp,mq | h `1 ´Nppp, mqq Nppq 2 ˘´1 ź p p | pb,hq `1 ´1 Nppq ˘.
This equation writes c m,k as a series

Γ m,h Npmq ř bPI K f pbq
N pbq 2 where f : I K Ñ R is a multiplicative function, which vanishes on the nontrivial powers of prime ideals. By Eulerian product, we have therefore proved Equation ( 16).

Let us now prove that 0 ď c m,k ď 1. Note that for every prime ideal p, we have

1 ď κ m,h ppq ď 2 and 1 2 ď κ 1 h ppq ď 1 . (24)
In particular all the factors of the two products over p in Equation ( 16) belong to r0, 1s, hence 0 ď c m,k ď x 6 `O `p1 `aNpkq q x 5 `Npkq x 4 ˘.

Replacing f `by ´gives the same minoration to S z,θ,k,m pxq, hence Theorem 4.1 follows. l

)

  For all x ě 1 and b P I K , let us fix b, m, pb, mq P O K ´t0u such that b " bO K , m " mO K and pb, mq " pb, mqO K . Since for every c P O K ´t0u we have m | bc if and only if m pb,mq | c, by the change of variable c " m pb,mq d, by Equation (12) and by Equation (13) applied with y " x|pb,mq| |m| |b| , if S b " ÿ cPI K , aPmXCpz,θ,xq : bc"aO K Npcq , we have S b " ÿ c P O K ´t0u, aPmXCpz,θ,xq : bc"a |c| 2 " ÿ c P O K ´t0u : bc P Cpz,θ,xq, m | bc |c| 2 " ÿ d P O K ´t0u : d P Cp zpb,mq m b , θ, x|pb,mq|

Lemma 4. 3

 3 For every b P I K , the maps χ b , χ b and ψ b on I K are multiplicative.

  Finally, the multiplicativity property χ b pIJq " χ b pIqχ b pJq of the function χ b is a consequence of the multiplicativity of the map ψ b and of the fact that ψ b pIq and ψ b pJq are coprime. l Lemma 4.4 For every prime ideal p and every b P I K , we have ψ b ppq " " p if p | b, pp, mq otherwise, and χ b ppq χ b ppq " 1 ô $ & % p | pb, hq or p b and pp, mq | h . Proof. The first formula follows from the definition of ψ b ppq (see Formula (20)) by considering the three cases ' p | b, ' p b and p | m, and ' p b and p m. The second formula follows from the first one, from the definitions of χ b ppq and χ b ppq, and from the fact that χ b ppq χ b ppq " 1 if and only if χ b ppq " χ b ppq " 1, by considering the two cases ' p | b and ' p b. l The arithmetic function c Þ Ñ µ K pcq χ b pcq χ b pcq Npψ b pcqq being multiplicative by Lemma 4.3 and the complete multiplicativity of the norm, and vanishing on the nontrivial powers of primes, the series defining C ˚pbq in Formula (

1

  Npmq ď 1. Let us finally prove that c 1 m ą 0. For every prime ideal p, let w p " 1{2 if p | h and p | m 1{6 if p | h and p m 1{2 if p h and p | m 1{3 if p h and p m and similarly Npa `kq ě Npaq `1 ´2b Npkq Npaq `Npkq Npaq ˘. Let us define the maps f ˘: r1, `8r Ñ R by t Þ Ñ t 2 ˘2a Npkq t 3{2 `Npkq t, so that their derivatives are f 1 ˘ptq " 2t ˘3a Npkq t 1{2 `Npkq and For all z P C ˆ, θ P s0, 2πs, x ě 1 and n P N ´t0u, let By the definition (14) of the sum S z,θ,k,m pxq and the inequalities (31), by Abel's summation formula, by applying twice Lemma 4.5, and since c m,k ď 1 by Lemma 4.2, we have S z,θ,k,m pxq ď

								f ´pNpaqq Npaq Npa `kq	ď 1 ď	f `pNpaqq Npaq Npa `kq	(31)
						a n "	ÿ aPmXCpz,θ,x q : Npaq"n	ϕ K paq Npaq	ϕ K pa `kq Npa `kq	,
								ÿ
	so that by Equation (25), we have r Spxq "	a n .
								1ďnďx 2
					ÿ 1ďnďx 2	a n f `pnq "	´ÿ 1ďnďx 2	a n ¯f`p x 2 q	´ż x 2 1 ´ÿ 1ďnďt	a n	¯f 1 `ptq dt
	"	´θ c m,k a |D K |	x 2 `Opxq ¯´x 4 `2a Npkq x 3 `Npkq x 2	ż
			x 2 1	´θ c m,k a |D K |	t `Opt 1{2 q ¯´2t	`3a Npkq t 1{2 `Npkq ¯dt
	"	3	θ c m,k a |D K |
								κ m,h ppq κ 1 h ppq Nppp,mqq Nppq 2	. By
	Formula (17), if Nppq " 2, we have
								$
								'
								'
								&
							w p "
								'
								'
								%

Keywords: Euler function, imaginary quadratic number field, Mertens formula, Mirsky formula. AMS codes: 11R04, 11N37, 11R11.

Recall that a function f : I K Ñ C ˆis multiplicative if f pOK q " 1 and if for all coprime integral ideals a, b in I K , we have f pabq " f paqf pbq.

| if D K " 1 mod 4.

This research was supported by the French-Finnish CNRS IEA PaCap. We thank E. Fouvry for his inspirational appendix in [PP2].

In particular 1 ´wp ‰ 0 if Nppq " 2. From the inequalities (24) and by Equation ( 16), we have

´Nppp, mqq Nppq 2 ˘ź p : Nppqě3 `1 ´2 Nppp, mqq Nppq 2 ˘ź p : Nppq"2 p1 ´wp q .

Since there are only finitely many primes ideals p dividing m, the term on the right hand side is bounded from below by a positive constant c 1 m " min kPO K c m,k ą 0. This concludes the proof of Lemma 4.2. l

Now that we understand the constant c m,k , we continue towards the proof of Theorem 4.1 by giving an asymptotic formula for the sum

Lemma 4.5 Uniformly in m P I K , k P O K , z P C ˆ, θ P s0, 2πs and x ě 1, we have

Proof. For all nonzero elements a and b in the factorial ring O K , we denote by pa, bq any fixed choice of gcd of a and b, and by ra, bs any fixed choice of lcm of a and b.

By Equation (4), for every a P O K ´t0u, we have

Let x ě 1. Applying twice this equality, since Npbq ď Npaq when b | a, we have by Fubini's theorem r Spxq " 1

Recall that a system of two congruences " n " α 0 mod α n " β 0 mod β with unknown n P O K , where α, β, α 0 , β 0 P O K and α, β ‰ 0, has a solution if and only if α 0 ´β0 " 0 mod pα, βq. Furthermore, if this congruence condition is satisfied, that is, if there exists n 0 , m 0 P O K such that α 0 ´β0 " βm 0 ´αn 0 , then n is a solution if and only if

This is equivalent to asking n to belong to the translate Λ α,β,α0,β0 " α 0 `αn 0 ` Λ α,β of the Z-lattice Λ α,β " rα, βsO K . Applying this with α " m pb,mq , β " c pb,cq , α 0 " 0 and β 0 " ´k Thus Equation ( 27) becomes, using Equation ( 12), r Spxq " 1

Let b, c be as in the index of the first sum above. Using again the standard Gauss counting argument, using Formula (11) for the second equality and the equation Nprα, βsq " Npαq Npβq Nppα,βqq for the last equality, we have, uniformly in b, c, m Using this with y " x |b| , which is at least 1 since |b| ď x, we have

By Equation (19) (replacing therein x by x 2 ), completing the first sum of the above equation with the indices b P O K ´t0u such that |b| ą x introduces an error of the form Op 1

x q (uniformly in m P O K ´t0u, k P O K and x ě 1). A computation similar to the one done for Equation (19) gives that the second sum in Equation ( 30) is actually bounded by 1 |O K | 2 ζ K p 3 2 q 2 ζ K p2q, which is uniform since there are only finitely many such fields K.

By the definition of the constant c m,k in Equation ( 15), this proves Equation ( 26