text:
Uaxactun

Uaxactun (pronounced ) is an ancient sacred place of the Maya civilization, located in the Petén Basin region of the Maya lowlands, in the present-day department of Petén, Guatemala. The site lies some 12 mi north of the major center of Tikal. The name is sometimes spelled as Waxaktun.

History of discovery
With recent achievements in the decipherment of the ancient Maya hieroglyphic writing system, it has been determined that the ancient name for this site translates roughly as Siaan K'aan or "Born in Heaven". The name Uaxactun was given to the site by its rediscoverer, archaeologist Sylvanus Morley, in May 1916. He coined the name from Maya words Waxac and Tun, to mean "Eight Stones". The name has two meanings; Morley's stated reason for the name was to commemorate it as the first site where an inscription dating from the 8th Baktún of the Maya calendar was discovered (making it then the earliest known Maya date). The other meaning is a pun, since "Uaxactun" sounds like "Washington", the U.S. capital and home of the Carnegie Institute which funded Morley's explorations.

Morley's initial investigation of the site mostly focused on the hieroglyphic inscriptions; after that Uaxactun was not visited again until 1924, when Frans Blom made a more detailed investigation of the structures and mapped the site. The Carnegie Institution conducted archeological excavations there from 1926 through 1937, led by Oliver Ricketson. The excavations added greatly to knowledge of the early Classic and pre-Classic Maya. The remains of several badly ruined late Classic era temple-pyramids were removed, revealing well-preserved earlier temples underneath them.

For most of the Carnegie team's time at Uaxactun, communication with the outside world was via a four-day mule convoy to El Cayo, British Honduras. Towards the end of the time an airstrip was opened. Flights to Uaxactun continued and a small village grew there, as it became a center for gathering of chicle sap from the Petén jungle. In 1940 A. L. Smith and Ed Shook of the Carnegie project returned to make some additional excavations. In the late 1970s a rough road was opened, connecting Uaxactun to Tikal and thence to Flores, Guatemala. Air flights were discontinued. In 1984 the road was much improved. Shook returned again in 1974 to oversee consolidation and restoration of some architecture excavated earlier. In 1982 Guatemala's Tikal National Park was expanded to include the ruins of Uaxactun within its protected area. In 1990 the Maya Biosphere Reserve (MBR) was created, including Uaxactun as a part of the reserve's Multiple Use Zone.

In 2009 an excavation project of Slovak Archaeological and Historical Institute (SAHI) was started by professor Milan Kováč.

Conquest of Uaxactun by Siyaj K'ak'
Linda Schele, in A Forest of Kings, devotes an entire chapter to a war between Tikal and Uaxactun, in which Uaxactun was defeated by forces led by Fire is Born (Siyaj K'ak', formerly identified as Smoking Frog ) of Tikal. In this chapter, she also gives a brief overview of the known history of Uaxactun up to the final year of the war (378 AD) and of the Uaxactun kings who claimed descent from Fire is Born. The combined political entity of Tikal-Uaxactun dominated the Guatemalan Petén for the following 180 years.

Siyaj K'ak' might have come from Teotihuacán, been the general of the Teotihuacano ruler Spearthrower Owl, and conquered Tikal earlier the same year. This was a watershed moment of the Classic Maya. Some scholars suggested that new kings were installed at Tikal, Uaxactun, Rio Azul, El Peru, El Zapote and Bejucal during the Teotihuacan intrusions, new rituals and images were introduced, and a new order was established in the Maya Lowlands, while others suggested a less hegemonic role of Teotihuacan in its relationship with the Maya.

After 378 AD
After the conquest by Siyaj K'ak' in 378 AD, Uaxactun was still able to keep elite prerogatives of monument carving, temple erection, and rich burials during most of the Early Classic era. During the Hiatus period (about 600 AD) between Early Classic and Late Classic, Uaxactun experienced a lack of architectural activity and ceramic production, which coincided with the decline of the power of Teotihuacán and Tikal. There was no erection of dedicatory monuments between 554 AD and 711 AD. By the middle of Late Classic, Uaxactun showed evidences of population increase, new construction, remodeling of old structures, and appearance of new residential areas, plaza groups, and buildings. There was a time of distinctive population decrease towards the end of Late Classic. The last inscribed monument in Uaxactun is dated to 889. By the end of Terminal Classic, Uaxactun and Tikal were virtually abandoned.

References in popular culture
The Italian composer Giacinto Scelsi composed a piece in 1966 entitled Uaxuctum. It is subtitled: "The legend of the Maya city, destroyed by themselves for religious reasons".

Uaxactun was the location mentioned in the opening scene of the 1994 video game Pitfall: The Mayan Adventure. The game was released on various platforms from 1994 until 2009.

title: Uaxactun


text:
River Mole

The River Mole is a tributary of the River Thames in southern England. It rises in West Sussex near Gatwick Airport and flows northwest through Surrey for 80 km to the Thames at Hampton Court Palace. The river gives its name to the Surrey district of Mole Valley.

The Mole crosses the North Downs between Dorking and Leatherhead, where it cuts a steep-sided valley, known as the Mole Gap, through the chalk. Much of the catchment area lies on impermeable rock (including Weald Clay and London Clay), meaning that the river level responds rapidly to heavy rainfall.

During the second half of the 20th century, pollution levels in the river were high; however, since 1995 the water quality has improved dramatically and the Mole now boasts the greatest diversity of fish species of any river in England. Twelve Sites of Special Scientific Interest (SSSIs) that include wetland habitats are located within the Mole catchment area, and the stretch of river through Leatherhead has been designated a Local Nature Reserve. The Mole Gap forms part of a Special Area of Conservation and is an SSSI of European importance.

The river has captured the imagination of several authors and poets, particularly since in very hot summers the river channel can become dry between Dorking and Leatherhead, most recently in 2022. In John Speed's 1611 map of Surrey, this stretch of the river is denoted by a series of hills accompanied by the legend "The river runneth under". However the river's name is unlikely to have derived from this behaviour: The Oxford Dictionary of English Place Names suggests that Mole either comes from the Latin mola (a mill) or is a back-formation from Molesey (Mul's island). Domesday Book lists twenty mills on the river in 1086, of which Sidlow Mill was the oldest, dating from Saxon times.

Catchment area
The drainage area of the Mole is 477 km2 and forms 5% of the River Thames's catchment area above Teddington. The catchment area receives 761 mm of rain each year; the greatest average level of rainfall is 800 mm around Crawley. The Mole catchment reaches a maximum elevation of 265 m above Ordnance Datum at Leith Hill to the southwest of Dorking. There is only one aquifer in the drainage basin, at Fetcham, which means that the majority of the water in the river is from surface drainage, particularly from Gatwick Airport and the urban areas of Horley and Crawley, and that the flow rate responds rapidly to rainfall.

Upper Mole
The Mole rises in Baldhorns Copse 700 m to the south of the village of Rusper in West Sussex. It flows initially southwards for 1 km to a small lake at Baldhorns Park, before running eastwards through a largely rural area towards Crawley. The first tributaries to join the young river drain the northernmost part of St Leonard's Forest, between Horsham and Crawley, although much of the forest is in the catchment area of the River Arun. The Mole skirts the northern suburbs of Crawley where it is joined by its first major tributary, Ifield Brook, which drains Ifield Mill Pond.

The first gauging station on the Mole is south of Gatwick Airport (57 m above OD). The mean flow is 0.33 m3/s and the river ran dry at this point for the first time in the summer of 1995. The Mole runs under the airport runway in a culvert completed in 1985. The course of the Mole within the airport perimeter has been altered several times since commercial flights began in 1945; however the meanders visible on the 1839 tithe map in the 1.5 km stretch immediately north of the runway were reinstated in 1999, in a £1.2 million project to facilitate airport expansion.

The Mole enters Surrey to the south of Horley, where it meets the Gatwick Stream, a tributary draining Worth Forest to the southeast of Crawley. The second-largest Sewage Treatment Works (STW) in the Mole catchment is located on the Gatwick Stream 3 km upstream of the confluence with the Mole: Crawley STW discharges 15000 m3 of water per day, and in prolonged dry periods it accounts for up to 75% of the flow of the Mole downstream of the confluence. The mean flow measured at Horley gauging station (52 m above OD) is 1.40 m3/s.

The Mole passes Horley to the west, flowing north towards Sidlow and entering a largely rural area. 0.7 km south of Sidlow the mean flow is measured as 2.21 m3/s at Kinnersley Manor gauging station (48 m above OD). The Earlswood Brook, a tributary draining the urban area of Reigate and Redhill, joins the Mole at Sidlow. The largest STW in the Mole catchment (Reigate STW) discharges up to 118,500 m3 per day into the Earlswood Brook.

From Sidlow, the Mole turns northwest towards Brockham. A number of minor tributaries join the river from the west and are typically second order streams draining the woodland and arable land between Horsham and Dorking. The 18th-century weir at Betchworth was modified in 2004 to facilitate the installation of two 27.5 kW low-head hydro turbines. About 90% of the energy generated is fed into the regional electricity grid, while the remainder is used to supply the Betchworth Park Estate, where the weir is situated. The river leaves the Weald Clay at Brockham, passing Betchworth Castle and flowing briefly across greensand and Gault Clay to Pixham, 1 km northeast of Dorking. A mean flow of 3.74 m3/s is measured at a fourth gauging station, located at Castle Mill (39 m above OD). At Pixham the Mole meets the Pipp Brook, a tributary draining the northeastern slopes of Leith Hill.

Mole Gap
Between Dorking and Leatherhead the Mole cuts a steep-sided valley (the Mole Gap) though the North Downs, carving a 170 m river cliff on the western flank of Box Hill and a smaller 50 m cliff at Ham Bank in Norbury Park. The sudden change from impermeable Weald Clay to permeable chalk and the increased gradient of the river, allow the water table to drop below the bed of the river. Water is therefore able to flow out of the river through swallow holes in the bed and banks, decreasing the volume of water carried in the main channel.

The course of the river through Norbury Park was partially straightened when the Epsom to Horsham railway was built in 1837, with the removal of a small meander north of Westhumble. The meander was reinstated in 1997, in an attempt to enhance this area of the Park, although it has since become blocked by silt. The entirety of the Mole Gap lies within the Surrey Hills Area of Outstanding Natural Beauty.

Lower Mole
At Leatherhead, the Mole leaves the chalk and turns northwestwards to flow across impermeable London Clay, winding towards Cobham. The water table rises at this point and much of the water which drained out of the channel through the chalk returns through springs in the riverbed. The aquifer at Fetcham is the only one in the entire catchment area. The next major tributary, The Rye, (which drains Ashtead) joins to the north of Leatherhead, before the river is crossed by the M25 motorway. At Cobham the river swings round in a pronounced axehead meander skirting Painshill Landscape Garden, where a 35 ft diameter waterwheel raises water 5 m from the river to feed the ornamental lake in the park.

From Painshill Park the river flows northeastwards to the Thames, passing to the west of Esher Commons and then forming the boundary between Hersham and Esher, where the mean flow of the river is 5.43 m3/s. In response to heavy flooding of East Molesey and Thames Ditton in September 1968, the river was modified downstream of Albany Bridge to the Thames and new flood defences were constructed. Finally the Mole splits into two branches at the Island Barn Reservoir near Molesey: the northern (and smaller) branch continues as the River Mole and the southern branch is known as the River Ember. The two rivers flow either side of the reservoir, before running side by side in a northeasterly direction, merging 400 m before the confluence with the River Thames, on the reach above Teddington Lock. (For the purposes of the remainder of this article, the River Mole and the River Ember are treated as a single entity.)

Prior to the last ice age, the River Thames followed a more northerly route to the North Sea, from Reading via Marlow, Chorleywood, St Albans, Hertford and along the present Suffolk–Essex border. During this period, the Mole is thought to have merged with the River Wey near Byfleet and then flowed in a north-easterly direction via Richmond to meet the proto-Thames near Ware in Hertfordshire. Today the Mole and Wey are less than 2 km apart at their closest point near Painshill Park. During the Middle Pleistocene period, a large ice sheet built up across much of the East of England, reaching as far south as St Albans and Chelmsford, blocking the path of the proto-Thames. Glacial meltwater from the Anglian ice sheet caused the Thames to divert southwards and flood the valley of the Mole–Wey river, thus adopting its present route through London.

Water quality
Standard water quality of Mole and its tributaries has improved markedly since the 1990s. In 1990 the Environment Agency assessed 23% of the watercourses as Grade B (good) or better. In 2002 this figure was 60%. Investment in the Sewage Treatment Works in the catchment area has improved the quality of the discharges into the river, and modifications to the runway and apron drainage systems at Gatwick Airport mean that surface water is diverted to aerated pollution control lagoons and balancing ponds for treatments, including acid/alkali neutralisation, before release into the river.

In 1972 a sub-debate on the future cleanliness of the Mole was had by the House of Lords, involving a member of the South-East Strategic Committee of the Thames Conservancy (succeeded by the Environment Agency) and four others. In this it was said, "...I can see no future for [The Mole]. The Mole Valley has been polluted. The Mole where I used to tickle trout in my youth is a drain, and it will remain a drain." Rebuttals included that the Mole is a "charming Surrey steam" and "If Lord Lytton went to fish in the River Mole – and there are still angling clubs there – he would find that he would still be able to catch a trout".

Highly polluting discharges have become less common but have taken place since 2000. In 2003, Gatwick Airport Ltd pleaded guilty to charges of allowing chemical pollution to enter the River Mole after a detergent, used to clean rubber and oil from the runway, was washed into Crawters Brook by airport workers. The Environment Agency estimated that up to 5200 fish of 14 different species were killed as the pollution drained downstream. The airport was fined £30,000 by Lewes Crown Court. In May 2003, sewage leaking from a pump operated by Thames Water leaked into the Stanford Brook, killing coarse fish in the Gatwick stream.

Water quality of the River Mole in 2019:

Biodiversity
The marsh frog (Pelophylax ridibundus, a non-native species introduced from Europe in the 1930s) is now commonly found in the upper Mole and its tributaries around Newdigate and Gatwick. There is no evidence that the presence of the frogs has had a deleterious effect on indigenous amphibians. A second non-native species, the edible frog (Pelophylax kl. esculentus) was introduced to a site at Newdigate in the early 1900s. It has in 2008 been recorded in tributaries of the River Mole at Capel and Brockham.

The River Mole has the most diverse fish population of any river in England. The Gatwick Stream is dominated by coarse fish such as brown trout, brook lamprey, and eel. In 2003, the upper River Mole near Meath Green Lane, Horley, was enhanced to create a gravel spawning area to encourage chub and dace in addition to roach. In 1974 zander, a non-indigenous coarse fish native to Europe, were introduced legally to Old Bury Hill Lake which supplies the Pipp Brook. Zander have been caught in the Lower Mole below Dorking since the 1980s.

In the Mole Gap between Dorking and Leatherhead the river supports populations of chub, dace, barbel, and brown trout. Both barbel and brown trout are extremely sensitive to water quality and pollution. Below Leatherhead the river has historically supported larger predatory fish including chub, perch, pike, and eels, however in recent years chub and eel numbers have begun to decline. North of Esher the old river channel is dominated by floating pennywort, a highly invasive weed, which cuts off all light to the river bed, reducing oxygen levels and resulting in a poor habitat for fish. The Ember flood relief channel has a diverse fish population, including chub, dace, roach, bleak, large pike and barbel. At the confluence of the Mole and the River Thames at Hampton Court Palace it is possible to catch a wide variety of coarse fish species.

The geographical distribution of many species of invertebrate in the river reflects the geology of the catchment area. Viviparid snails and water scorpions (of the genus Nepidae) are commonly found where the river runs over the London Clay. Crayfish are common in areas associated with high alkalinity, particularly around Brockham, and the tributaries which run over the Weald Clay provide an excellent habitat for stoneflies, caddisflies, fast swimming mayflies and riffle beetles. The beautiful demoiselle (Calopteryx virgo) disappeared from the River Mole during the 1960s owing to deteriorating water quality, but has since recolonised. The white-legged damselfly (Platycnemis pennipes) is also found along the river, and the downy emerald (Cordulia aenea) is found between Box Hill and Leatherhead. The Mole is one of only three locations in England where the river shingle beetle (Meotica anglica) is found.

A nationally scarce species which is locally common on the River Mole is the greater dodder (Cuscuta europaea), a parasitic plant.

Conservation
The Mole catchment area includes twelve Sites of Special Scientific Interest (SSSIs) that include wetland habitats, The Mole Gap forms part of a Special Area of Conservation and is an SSSI of European importance.

The stretch of river between Thorncroft Manor (1 km south of Leatherhead) and River Lane in Fetcham has been designated a Local Nature Reserve. Although much of the surrounding land has been taken by residential and commercial development, this section of the Mole supports 20 different mammal species, 20 butterfly species and 15 species of dragonfly. The geology of the local area is complex, since the river leaves the chalk of the Mole Gap at this point and flows over gravel and clay, creating a patchwork of different habitats including scrub, woodland, hedges, banks, and meadows as well as the water itself. The river also provides a corridor for wildlife through the centre of Leatherhead.

The West End Common forms part of the Esher Commons, owned and managed by Elmbridge Borough Council. The River Mole forms the western boundary of the common, flowing past a steep sandy area known locally as The Ledges. Plant species typical of ancient woodland are found in this area, including bluebells (Hyacinthoides non-scriptus), marsh marigolds (Caltha palustris) and golden saxifrages (Chrysosplenium sp.). large bittercress (Cardamine amara) and the non-native Himalayan balsam (Impatiens glandulifera).

Molesey Heath Local Nature Reserve lies on the west bank of the Mole, west of the Island Barn Reservoir. The Heath is an area of rough scrubland and, despite its name, is a reclaimed landfill site hence its large but artificial mound and small plateau. The site is a rich habitat for birds including the redshank and little ringed plover.

Upper Mole (geology)
The Mole rises south of Rusper in West Sussex, where an outcrop of the Hastings Beds sandstone dips below the impermeable Weald Clay. From the source to Dorking, the river drains an area of 340 km2, of which approximately 60% is on Wealden or Atherfield Clay, 20% is on Tunbridge Wells Sand and 20% is on greensand. Brickearth deposits are common in the valley around Betchworth and east of Dorking.

The upper Mole catchment is dominated by a single broad terrace, which runs continuously from Tilgate Forest to the entry to the Mole Gap. Above Meath Green (near Horley), the terrace corresponds to the flood plain of the river, however from this point downstream to Brockham, the river cuts a narrow trench no more than 150 m wide. A similar trench is runs up several of the Mole's tributaries, notably along Deanoak Brook which follows a local strike. The presence of the trench protects the land from flooding, rendering it suitable for agriculture.

Mole Gap (geology)
Between Dorking and Leatherhead the Mole cuts a steep-sided valley through the North Downs, creating a 170 m river cliff on the western flank of Box Hill. The bedrock is permeable chalk and the water table lies permanently below the level of the riverbed, allowing water to drain out of the river through swallow holes in the bed and banks. The amount of water lost from the river is significant and in very hot summers the channel can become dry between Mickleham and Thorncroft Manor; this was recorded most recently in 1949, 1976 and 2022. At Leatherhead, the river leaves the chalk and flows across impermeable London Clay. At this point, the water table rises enough for the water to flow back into the main river channel.

In a survey in 1958, the geologist C. C. Fagg identified 25 active swallow holes between Dorking and Mickleham; the majority were only a few centimetres in diameter and were located in the vertical banks of the river below the water line. Most holes were difficult to observe in times of normal or heavy flow and were susceptible to silting up as new holes were continually being formed. A few much larger swallow holes were also observed separated from the main river by a channel of about a metre. About six of these larger swallow holes were found to the west of the Burford Bridge Hotel, along the course of the A24 Mickleham Bypass during its construction in 1936. Initially the surveyors tried to fill the holes with rubble to prevent the foundations of the new road subsiding. However this proved to be impractical and they were instead covered by concrete domes, up to 18 m in diameter, each fully supported by the surrounding chalk and provided with a manhole and access shaft to allow periodic inspection. In the late 1960s the domes were reopened and inspected and the alluvium in the largest swallow hole was observed to have subsided by 1.5 m under the centre of one of the domes. When the Dorking to Leatherhead railway was constructed in 1859, a fossilised swallow hole was discovered in the cutting at the south end of Box Hill & Westhumble railway station, suggesting that even in its early history, the river had swallow holes.

The author Daniel Defoe, who attended school in Dorking and probably grew up in the village of Westhumble, described the swallow holes in the River Mole in his book A tour thro' the Whole Island of Great Britain (first published in 1724): "...the current of the river being much obstructed by the interposition of those hills, called Box Hill ... it forces the waters as it were to find their way through as well as they can; and in order to do this, beginning, I say, where the river comes close to the foot of the precipice of Box-Hill, called the Stomacher, the waters sink insensibly away, and in some places are to be seen (and I have seen them) little channels which go out on the sides of the river, where the water in a stream not so big as would fill a pipe of a quarter of an inch diameter, trills away out of the river, and sinks insensibly into the ground. In this manner it goes away, lessening the stream for above a mile, near two, and these they call the Swallows."

Not all of the water removed from the river by the swallow holes is returned to the channel at Leatherhead. The chalk aquifer also feeds the springs at the southern end of Fetcham Mill Pond, which have never been known to run dry. A survey in March 1883 estimated that the Fetcham springs were producing about 3.6 e6impgal every day. A second survey in 1948 estimated that the same springs were yielding about 5 e6impgal a day. The water table in the chalk of the Wey Gap is significantly higher than might be expected from natural rainwater percolation alone. It has been suggested that a proportion of the excess water originates from the Mole Gap.

Lower Mole (geology)
At Leatherhead the river leaves the chalk bedrock, moving onto London Clay. The river meanders across an alluvial plain between 400 and(-) wide towards Cobham, where it begins to descend to a lower flood plain, which broadens as the river turns in an axehead meander. At the east end of Painshill Park, the flood plain narrows into a trench about 60 m wide, in which the river runs northwards for 6.5 km towards Hersham, where the river enters the flood plain of the River Thames.

Between Cobham and Esher, the Mole's historic courses have deposited gravel on top of the London Clay. The depth of the deposits generally varies from 2.5 to(-); the lower layers are generally highly compacted and cemented together with brick-red iron oxide, whereas the upper layers are loosely packed with angular flints and sand. Remains of a further gravel terrace, containing cherts and flints to a depth of 4 m, line the east side of St George's Hill.

Etymology
The river is first recorded in the Red Book of Thorney in AD 983 as Emen and in the AD 1005 Cartulary of the Abbey of Eynsham as both Emen and Æmen. Variations in the name are recorded throughout the Middle Ages and the river appears as Amele in the Domesday Book of 1086, and subsequently as Emele in 12th- and 13th-century Court Rolls. This name is probably derived from the Old English word æmen meaning misty or causing mists, and the name of the River Ember probably has its origins in this name.

The name Mole does not appear until the 16th century, first occurring as Moule in Harrison's Description of Britain of 1577. The antiquarian William Camden uses the Latinized form Molis in the 1586 edition of Britannia and Michael Drayton is the first to use Mole in his poem Poly-Olbion published in 1613. In John Speed's 1611 map of Surrey this stretch of the river is denoted by a series of hills accompanied by the legend "The river runneth under". However the river's name is unlikely to have derived from this behaviour: The Oxford Dictionary of English Place Names suggests that Mole either comes from the Latin mola (a mill) or is a back-formation from Molesey (Mul's island). In John Rocque's 1768 map of Surrey, the name Moulsey River is used.

Archaeology
In common with much of the rest of the Weald, the earliest evidence of human settlement along the Upper Mole is from the Mesolithic Period (20,000–7000 BC). Mesolithic sites at Wonham, Flanchford and Sidlow. Finds at Wonham include arrowheads and a plano-convex knife. The Lower Mole appears to have been settled during the same period and a flint axe dating from Mesolithic period found on spit of land close to River Mole in Cobham in 1965. Remains of a flat-bottomed dug-out canoe were found at the confluence of the Mole and Thames in 1877 by a local boatman. The canoe is preserved at the Pitt Rivers Museum in Oxford.

The Mole Gap and North Downs do not appear to have been settled until the late Stone Age: A large axe, typical of a "rough-out" axe produced during the Neolithic period, which was discovered in 1952 during building work in Westhumble, A flint mine of the same period has been discovered at East Horsley along with Neolithic flakes of flint at Fetcham and Headley Heath.

Significant Bronze Age finds include a bronze sword found close to the river north of Amberley Farm near Charlwood and a small hoard of weaponry consisting of two palstave axes and a scabbard chape was discovered in 2003 in Norbury Park close to Ham Bank.

Navigation
It is not clear to what extent the Mole was used for navigation in the past: In the late 13th century, Thorncroft Manor (south of Leatherhead) purchased a shout, a type of boat up to 16 m in length used to carry produce to market and it has been suggested that stone cut from quarries in Reigate was transported to London via the river.

During the 17th century, two bills came before parliament to make sections of the river navigable. The first, in 1663, was passed by the House of Commons but was defeated in the House of Lords. During the following year (1664), an act was passed by both Houses of Parliament to make the River Mole navigable from Reigate to the River Thames, but was never executed. The only Surrey river to have been made fully navigable is the combined lower section of the River Wey.

In 1798 William Marshall advocated the canalisation of a short stretch of the River Mole between Betchworth and Dorking to facilitate the movement of chalk from quarry to market. In 1810 the engineer John Rennie proposed a canal linking the River Medway to Portsmouth which was to have a branch to London following the Mole for much of its length. Between 1825 and 1828 the architect and civil engineer Nicholas Wilcox Cundy proposed a Grand Imperial Ship Canal from Deptford to Chichester passing through the Mole Gap, however he was unable to attract sufficient financial interest in his scheme.

Today the Mole is navigable for the 400 m from the confluence with the River Thames to Molember Weir at East Molesey where there is a private mooring facility. The river may be paddled all year by canoe between Brockham and Fetcham with an appropriate licence from the British Canoe Union.

Second World War defences
During the Second World War several sites along the course of the Mole were fortified for the defence of London against invasion from the south. GHQ Line B ran along the North Downs from Farnham and Guildford, crossing the Mole Gap to the north of Dorking and following the river to Horley. Between Betchworth and Box Hill, the north bank of the River Mole was stabilised and made steeper to prevent wheeled vehicles from crossing. At Boxhill Farm, where access to the river from the north bank was required for the herd of dairy cows, a row of twelve concrete cylinders were cast as an anti-tank measure. Gun mounts were also installed to protect both Boxhill and Deepdene bridges and several pillboxes were installed. An anti-tank ditch was dug from the Stepping Stones eastwards across the fields belonging to Bradley Farm (now Denbies vineyard).

The river crossing at Sidlow Bridge was heavily defended and a line of pill boxes was constructed on the north bank of the river. Concrete anti-tank dragon's teeth were built on both sides of the river, a short distance upstream of the bridge, as an obstacle to armoured vehicles.

Mole Gap
The North Downs Way crosses the river at Box Hill via seventeen hexagonal stepping stones, which are frequently submerged after heavy rainfall. The current stones were dedicated in September 1946 by the then Prime Minister Clement Attlee, replacing those destroyed during the Second World War as an anti-invasion measure. The location is popular with anglers and families, but swimming is strongly discouraged as the water is polluted in places. The stones give their name to the pub in the nearby village of Westhumble.

When the Burford Bridge was rebuilt in 1937, excavations revealed a "flint-surfaced approach to [a] ford at low level having all the signs of Roman workmanship" suggesting that Stane Street (which ran from London to Chichester via Dorking) crossed the river at this point. In Defoe's time, there was a footbridge at this point, but carts and waggons had to cross the river by a ford.

Leatherhead
There are three listed bridges in Leatherhead of which the 14 arch Leatherhead Town Bridge is the oldest. The first bridge on this site was probably constructed during the late Middle Ages and was rebuilt and enlarged by the county surveyor, George Gwilt in 1782–83. The bridge is a long structure of 14 segmental arches resting on low piers, finished in red brick in Flemish bond, with some Portland stone dressing.

A few metres downstream of Leatherhead Bridge stands the Grade II listed railway viaduct which carries the Leatherhead to Dorking line over the river. The bridge was constructed in 1867 and, at the insistence of Thomas Grissell the landowner, was given lavish architectural treatment.

The single-span Shell Bridge stands in the grounds of Thorncroft Manor, to the south of the Town Centre. The bridge was constructed as an ornamental feature in parkland laid out by Capability Brown in the late 18th century. The bridge is largely composed of flint and has a large inverted scallop shell in the place of the keystone and similar shells in the spandrels at each side. The bridge was renovated and strengthened in 1999 by a group of Canadian engineers working on the Terra Nove project. The restoration project was dedicated to the memory of the Canadian servicemen who were billeted in the manor during the Second World War.

Cobham
A wooden bridge is thought to have existed on the site of Cobham Bridge since the 12th century, the upkeep of which was the responsibility of the adjacent landowners. The present bridge was constructed by George Gwilt in 1792, after responsibility for maintenance had been transferred to the county council by Act of Parliament. The bridge has nine low arches and is primarily built of red brick with stone coping. The parapets were rebuilt in 1914 and the structure was given a Grade II listing in 1953.

Watermills
Domesday Book listed twenty mills on the River Mole in 1086.

Upper Mole
Horley Mill was first mentioned in a deed of the early 13th century. The most recent mill was demolished in 1959, although the mill house still stands.

The first mill at Sidlow was built during Saxon times. The final mill on the site was demolished in 1790, however remains of the mill leat are still visible.

Mention is made of a mill at Brockham in 1634 and remains of the mill race are still visible.

Castle Mill at Pixham is a Grade II listed building. It was a corn mill, built in the early 19th century, and has been converted into a bed and breakfast hotel.

Lower Mole
Slyfield Mill near Stoke d'Abernon is first mentioned in Domesday Book. It was used for fulling woollen cloth and milling corn.

Five of the mills mentioned in Domesday Book were in the borough of Elmbridge.

Downside Mill, Cobham was the mill of the manor of Downe. Until the Dissolution of the Monasteries it was owned by Chertsey Abbey. It has been used for many purposes including the processing of corn, paper, iron, tinplate and flock and the generation of electricity. The present building dates from the 18th century but it is inaccessible to the public.

Cobham Mill, downstream of Leatherhead, consisted of two mills used for grinding corn. In 1953 the larger mill was demolished by Surrey County Council to allieviate traffic congestion on Mill Road. The remaining red brick mill dates from the 1822 and was in use until 1928. It was restored to full working order by the Cobham Mill Preservation Trust, and is now open to the public from 2 pm to 5 pm on the second Sunday of each month (between April and October).

Esher Mill also known as Royal Mill was at the end of Mill Road in Lower Green, where there is now an industrial estate. It was used to process corn, brass wire, iron, paper, linoleum, and books. For many years there may have been two mills on the site for corn grinding and industrial use. There were a series of fires over a century and after the last in 1978 the buildings were demolished.

East Molesey Upper Mill was associated with the manor of Molesey Matham. It was used to produce gunpowder from the time of the Commonwealth until about 1780. The island where it stood now forms part of the ornamental gardens of a housing development called "The Wilderness".

East Molesey Lower Mill, also known as Sterte Mill, was associated with the manor of Molesey Prior. During the Commonwealth it was used for gunpowder manufacture, but after the restoration of the monarchy in 1660 it reverted to corn milling. An old timber structure was replaced by a brick building in the 1820s which can be seen from the bridge over the Ember in Hampton Court Way.

In addition there was Ember Mill, which stood on the banks of the old course of the River Ember near Hampton Court Way.

River Ember Flood Relief Channel and confluence with the Thames
The River Mole originally flowed into the River Thames at the point where the present Hampton Court bridge now crosses the Thames (approximately 500 m upstream of the present confluence, on the reach above Teddington Lock).

However, during the early 1930s, when Hampton Court Way and the bridge were built, the River Mole was redirected to flow into the River Ember and both rivers now enter the Thames in a single widened and straightened channel once occupied only by the River Ember. There have been further alterations to the courses of these two rivers in a major flood prevention scheme since serious flooding in the area in 1947 and 1968.

Literature
The river has captured the imagination of several authors and poets, particularly since in very hot summers the river channel can become dry between Dorking and Leatherhead (most recently during the 1976 drought).

In The Faerie Queene (first published in 1590) Edmund Spenser wrote of the river:
 * And Mole, that like a nousling mole doth make
 * His way still under ground till Thamis he overtake.

In Poly-Olbion (first published in 1612) the poet Michael Drayton described the journey taken by the River Thames to the sea:
 * As still his goodly traine yet every houre increast,
 * And from the Surrian shores cleer Wey came down to meet
 * His Greatnes, whom the Tames so gratiously doth greet
 * That with the Fearne-crown'd Flood he Minion-like doth play:
 * Yet is not this the Brook, entiseth him to stay.
 * But as they thus, in pompe, came sporting on the shole,
 * Gainst Hampton-Court he meets the soft and gentle Mole.
 * Whose eyes so pierc't his breast, that seeming to foreslowe
 * The way which he so long intended was to go,
 * With trifling up and down, he wandreth here and there;
 * And that he in her sight, transparent might appeare,
 * Applyes himselfe to Fords, and setteth his delight,
 * On that which might make him gratious in her sight.

 * But Tames would hardly on: oft turning back to show,
 * For his much loved Mole how loth he was to go.
 * The mother of the Mole, old Holmsdale, likewise beares
 * Th'affection of her childe, as ill as they do theirs:
 * But Mole respects her words, as vaine and idle dreames,
 * Compar'd with that high joy, to be belov'd of Tames:
 * And head-long holds her course, his company to win.
 * Mole digs her selfe a path, by working day and night
 * (According to her name, to shew her nature right)
 * And underneath the Earth, for three miles space doth creep:
 * Till gotten out of sight, quite from her mothers keep,
 * Her foreintended course the wanton Nymph doth run;
 * As longing to imbrace old Tame and Isis son...

He writes in the appendix to Song XVII This Mole runnes into the earth, about a mile from Darking in Surrey, and after some two miles sees the light againe, which to be certaine hath been affirmed by Inhabitants thereabout reporting triall made of it.

John Milton (c. 1562–1647) described the river as
 * sullen Mole that runneth underneath

In a similar vein, Alexander Pope (1688–1744) wrote in his poem Windsor Forest (first published 1713)
 * And sullen Mole that hides his diving flood

Robert Bloomfield (1766–1823) writes the following lines about the Mole Valley in his 1806 poem Wild Flowers.
 * Sweet Health, I seek thee! Hither bring
 * Thy balm that softens human ills;
 * Come on the long drawn clouds that fling
 * Their shadows o'er the Surry-Hills.
 * Yon green-topt hills, and far away
 * Where late as now I freedom stole,
 * And spent one dear delicious day
 * On thy wild banks romantic Mole.

 * Ay there's the scene! Beyond the sweep
 * Of London's congregated cloud,
 * The dark-brow'd wood, the headlong steep,
 * And valley paths without a crowd!
 * Here Thames I watch thy flowing tides,
 * Thy thousand sails am proud to see;
 * But where the Mole all silent glides
 * Dwells Peace - and Peace is wealth to me.

Extract from The River Mole or Emlyn Stream by Mary Drinkwater Bethune, which was published in 1839.
 * Who may count back that forgotten time
 * When first the waters forced an outlet here:
 * When the foundations of these stedfast hills
 * Were shaken, and the long imprisoned stream
 * Flowed through the yawning chasm? That awful day
 * Yet leaves its trace. The waters find their way,
 * Now laughing in the sun - now swallowed up
 * In caverns pervious to their course alone,
 * They leave their channel dry, and hide awhile
 * Their silent flow; like bitter tears, unshed
 * From the dim eye, before a careless world
 * Unheeding of our grief; but swelling still
 * In the full heart, which leaves unsoothed, unseen,
 * And broods o'er ruined hopes, and days gone by.

Tributaries
The major tributaries of the River Mole are the Ifield Brook, Gatwick Stream, Earlswood Brook, Pipp Brook and The Rye, which drains Ashtead. A full list of the tributaries is given in the table below.

Distributary

 * River Ember

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

title: River Mole


text:
Dirichlet character

In analytic number theory and related branches of mathematics, a complex-valued arithmetic function $$\chi:\mathbb{Z}\rightarrow\mathbb{C}$$ is a Dirichlet character of modulus $$m$$ (where $$m$$ is a positive integer) if for all integers $$a$$ and $$b$$: \chi(a) \begin{cases} =0 &\text{if } \gcd(a,m)>1\\ \ne 0&\text{if }\gcd(a,m)=1. \end{cases}$$ (gcd is the greatest common divisor) The simplest possible character, called the principal character, usually denoted $$\chi_0$$, (see Notation below) exists for all moduli:
 * 1) $$\chi(ab) = \chi(a)\chi(b);$$ that is, $$\chi$$ is completely multiplicative.
 * 1) $$\chi(a + m) = \chi(a)$$; that is, $$\chi$$ is periodic with period $$m$$.

\chi_0(a)= \begin{cases} 0 &\text{if } \gcd(a,m)>1\\ 1 &\text{if } \gcd(a,m)=1. \end{cases}$$ The German mathematician Peter Gustav Lejeune Dirichlet—for whom the character is named—introduced these functions in his 1837 paper on primes in arithmetic progressions.

Notation
$$\phi(n)$$ is Euler's totient function.

$$\zeta_n$$ is a complex primitive n-th root of unity:

\zeta_n^n=1,$$ but $$\zeta_n\ne 1, \zeta_n^2\ne 1, ... \zeta_n^{n-1}\ne 1.$$

$$(\mathbb{Z}/m\mathbb{Z})^\times$$ is the group of units mod $m$. It has order $$\phi(m).$$

$$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}$$ is the group of Dirichlet characters mod $$m$$.

$$p, p_k,$$ etc. are prime numbers.

$$(m,n)$$ is a standard abbreviation for $$\gcd(m,n)$$

$$\chi(a), \chi'(a), \chi_r(a),$$ etc. are Dirichlet characters. (the lowercase Greek letter chi for "character")

There is no standard notation for Dirichlet characters that includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters of different moduli appear. Where appropriate this article employs a variation of Conrey labeling (introduced by Brian Conrey and used by the LMFDB).

In this labeling characters for modulus $$m$$ are denoted $$\chi_{m, t}(a)$$ where the index $$t$$ is described in the section the group of characters below. In this labeling, $$\chi_{m,\_}(a)$$ denotes an unspecified character and $$\chi_{m,1}(a)$$ denotes the principal character mod $$m$$.

Relation to group characters
The word "character" is used several ways in mathematics. In this section it refers to a homomorphism from a group $$G$$ (written multiplicatively) to the multiplicative group of the field of complex numbers:
 * $$\eta: G\rightarrow \mathbb{C}^\times,\;\;\eta(gh)=\eta(g)\eta(h),\;\;\eta(g^{-1})=\eta(g)^{-1}.$$

The set of characters is denoted $$\widehat{G}.$$ If the product of two characters is defined by pointwise multiplication $$\eta\theta(a)=\eta(a)\theta(a),$$ the identity by the trivial character $$\eta_0(a)=1$$ and the inverse by complex inversion $$\eta^{-1}(a)=\eta(a)^{-1}$$ then $$\widehat{G}$$ becomes an abelian group.

If $$A$$ is a finite abelian group then there are 1) an isomorphism $$A\cong\widehat{A}$$ and 2) the orthogonality relations:

 * $$\sum_{a\in A} \eta(a)=

\begin{cases} 0&\text{ if } \eta\ne\eta_0 \end{cases} $$    and      $$\sum_{\eta\in\widehat{A}}\eta(a)= \begin{cases} 0&\text{ if } a\ne 1. \end{cases} $$
 * A|&\text{ if } \eta=\eta_0\\
 * A|&\text{ if } a=1\\

The elements of the finite abelian group $$(\mathbb{Z}/m\mathbb{Z})^\times$$ are the residue classes $$[a]=\{x:x\equiv a\pmod m\}$$ where $$(a,m)=1.$$

A group character $$\rho:(\mathbb{Z}/m\mathbb{Z})^\times\rightarrow \mathbb{C}^\times$$ can be extended to a Dirichlet character $$\chi:\mathbb{Z}\rightarrow \mathbb{C}$$ by defining

\chi(a)= \begin{cases} 0 &\text{if } [a]\not\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)> 1\\ \rho([a])&\text{if } [a]\in(\mathbb{Z}/m\mathbb{Z})^\times&\text{i.e. }(a,m)= 1, \end{cases}$$

and conversely, a Dirichlet character mod $$m$$ defines a group character on $$(\mathbb{Z}/m\mathbb{Z})^\times.$$

Paraphrasing Davenport Dirichlet characters can be regarded as a particular case of Abelian group characters. But this article follows Dirichlet in giving a direct and constructive account of them. This is partly for historical reasons, in that Dirichlet's work preceded by several decades the development of group theory, and partly for a mathematical reason, namely that the group in question has a simple and interesting structure which is obscured if one treats it as one treats the general Abelian group.

Elementary facts
4) Since $$\gcd(1,m)=1,$$ property 2) says $$\chi(1)\ne 0$$ so it can be canceled from both sides of  $$\chi(1)\chi(1)=\chi(1\times 1) =\chi(1)$$:

 * $$\chi(1)=1.$$

5) Property 3) is equivalent to

 * if $$a \equiv b \pmod{m}$$  then $$\chi(a) =\chi(b).$$

6) Property 1) implies that, for any positive integer $$n$$
 * $$\chi(a^n)=\chi(a)^n.$$

7) Euler's theorem states that if $$(a,m)=1$$ then $$a^{\phi(m)}\equiv 1 \pmod{m}.$$ Therefore,
 * $$\chi(a)^{\phi(m)}=\chi(a^{\phi(m)})=\chi(1)=1.$$

That is, the nonzero values of $$\chi(a)$$ are $$\phi(m)$$-th roots of unity:

\chi(a)= \begin{cases} 0 &\text{if } \gcd(a,m)>1\\ \zeta_{\phi(m)}^r&\text{if } \gcd(a,m)=1 \end{cases}$$

for some integer $$r$$ which depends on $$\chi, \zeta,$$ and $$a$$. This implies there are only a finite number of characters for a given modulus.

8) If $$\chi$$ and $$\chi'$$ are two characters for the same modulus so is their product $$\chi\chi',$$ defined by pointwise multiplication:
 * $$\chi\chi'(a) = \chi(a)\chi'(a)$$  ($$\chi\chi'$$ obviously satisfies 1-3).

The principal character is an identity:

\chi\chi_0(a)=\chi(a)\chi_0(a)= \begin{cases} 0 \times 0 &=\chi(a)&\text{if } \gcd(a,m)>1\\ \chi(a)\times 1&=\chi(a) &\text{if } \gcd(a,m)=1. \end{cases}$$

9) Let $$a^{-1}$$ denote the inverse of $$a$$ in $$(\mathbb{Z}/m\mathbb{Z})^\times$$. Then
 * $$\chi(a)\chi(a^{-1})=\chi(aa^{-1})=\chi(1)=1,

$$ so $$\chi(a^{-1})=\chi(a)^{-1},$$ which extends 6) to all integers.

The complex conjugate of a root of unity is also its inverse (see here for details), so for $$(a,m)=1$$
 * $$\overline{\chi}(a)=\chi(a)^{-1}=\chi(a^{-1}). $$  ($$\overline\chi$$ also obviously satisfies 1-3).

Thus for all integers $$a$$

\chi(a)\overline{\chi}(a)= \begin{cases} 0 &\text{if } \gcd(a,m)>1\\ 1 &\text{if } \gcd(a,m)=1 \end{cases}; $$  in other words  $$\chi\overline{\chi}=\chi_0$$.

10) The multiplication and identity defined in 8) and the inversion defined in 9) turn the set of Dirichlet characters for a given modulus into a finite abelian group.

The group of characters
There are three different cases because the groups $$(\mathbb{Z}/m\mathbb{Z})^\times$$ have different structures depending on whether $$m$$ is a power of 2, a power of an odd prime, or the product of prime powers.

Powers of odd primes
If $$q=p^k$$ is an odd number $$(\mathbb{Z}/q\mathbb{Z})^\times$$ is cyclic of order $$\phi(q)$$; a generator is called a primitive root mod $$q$$. Let $$g_q$$ be a primitive root and for $$(a,q)=1$$ define the function $$\nu_q(a)$$ (the index of $$a$$) by
 * $$a\equiv g_q^{\nu_q(a)}\pmod {q},$$
 * $$0\le\nu_q<\phi(q).$$

For $$(ab,q)=1,\;\;a \equiv b\pmod{q}$$ if and only if $$\nu_q(a)=\nu_q(b).$$ Since
 * $$\chi(a)=\chi(g_q^{\nu_q(a)})=\chi(g_q)^{\nu_q(a)},$$  $$\chi$$ is determined by its value at $$g_q.$$

Let $$\omega_q= \zeta_{\phi(q)}$$ be a primitive $$\phi(q)$$-th root of unity. From property 7) above the possible values of $$ \chi(g_q)$$ are $$ \omega_q, \omega_q^2, ... \omega_q^{\phi(q)}=1.$$ These distinct values give rise to $$\phi(q)$$ Dirichlet characters mod $$q.$$ For $$(r,q)=1$$ define $$\chi_{q,r}(a)$$ as

\chi_{q,r}(a)= \begin{cases} 0 &\text{if } \gcd(a,q)>1\\ \omega_q^{\nu_q(r)\nu_q(a)}&\text{if } \gcd(a,q)=1. \end{cases}$$

Then for $$(rs,q)=1$$ and all $$a$$ and $$b$$
 * $$\chi_{q,r}(a)\chi_{q,r}(b)=\chi_{q,r}(ab),$$ showing that $$\chi_{q,r}$$ is a character and
 * $$\chi_{q,r}(a)\chi_{q,s}(a)=\chi_{q,rs}(a),$$ which gives an explicit isomorphism $$\widehat{(\mathbb{Z}/p^k\mathbb{Z})^\times}\cong(\mathbb{Z}/p^k\mathbb{Z})^\times.$$

Examples m = 3, 5, 7, 9
2 is a primitive root mod 3. ($$\phi(3)=2$$)
 * $$2^1\equiv 2,\;2^2\equiv2^0\equiv 1\pmod{3},$$

so the values of $$\nu_3$$ are

\begin{array}{|c|c|c|c|c|c|c|} a & 1 & 2 \\ \hline \nu_3(a) & 0 & 1\\ \end{array} $$. The nonzero values of the characters mod 3 are

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 2 \\ \hline \chi_{3,1} & 1 & 1 \\ \chi_{3,2} & 1 & -1 \\ \end{array} $$

2 is a primitive root mod 5. ($$\phi(5)=4$$)
 * $$2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 3,\;2^4\equiv2^0\equiv 1\pmod{5},$$

so the values of $$\nu_5$$ are

\begin{array}{|c|c|c|c|c|c|c|} a & 1 & 2 & 3 & 4 \\ \hline \nu_5(a) & 0 & 1 & 3 & 2 \\ \end{array} $$. The nonzero values of the characters mod 5 are

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 2 & 3    &  4  \\ \hline \chi_{5,1} & 1 & 1  & 1   &  1 \\ \chi_{5,2} & 1 & i  & -i  & -1\\ \chi_{5,3} & 1 & -i & i   & -1\\ \chi_{5,4} & 1 & -1 & -1  &  1\\ \end{array} $$

3 is a primitive root mod 7. ($$\phi(7)=6$$)
 * $$3^1\equiv 3,\;3^2\equiv 2,\;3^3\equiv 6,\;3^4\equiv 4,\;3^5\equiv 5,\;3^6\equiv3^0\equiv 1\pmod{7},$$

so the values of $$\nu_7$$ are

\begin{array}{|c|c|c|c|c|c|c|} a       & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \nu_7(a) & 0 & 2 & 1 & 4 & 5 & 3 \\ \end{array} $$. The nonzero values of the characters mod 7 are ($$\omega=\zeta_6, \;\;\omega^3=-1$$)

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 2        & 3         & 4        & 5          & 6   \\ \hline \chi_{7,1} & 1 & 1        & 1         & 1        & 1          & 1   \\ \chi_{7,2} & 1 & -\omega  & \omega^2  & \omega^2 & -\omega    & 1   \\ \chi_{7,3} & 1 & \omega^2 & \omega    & -\omega  & -\omega^2  & -1  \\ \chi_{7,4} & 1 & \omega^2 & -\omega   & -\omega  & \omega^2   & 1   \\ \chi_{7,5} & 1 & -\omega  & -\omega^2 & \omega^2 & \omega     & -1  \\ \chi_{7,6} & 1 & 1        & -1        & 1        & -1         & -1  \\ \end{array} $$.

2 is a primitive root mod 9. ($$\phi(9)=6$$)
 * $$2^1\equiv 2,\;2^2\equiv 4,\;2^3\equiv 8,\;2^4\equiv 7,\;2^5\equiv 5,\;2^6\equiv2^0\equiv 1\pmod{9},$$

so the values of $$\nu_9$$ are

\begin{array}{|c|c|c|c|c|c|c|} a & 1 & 2 &4 & 5&7&8 \\ \hline \nu_9(a) & 0 & 1 & 2 & 5&4&3 \\ \end{array} $$. The nonzero values of the characters mod 9 are ($$\omega=\zeta_6, \;\;\omega^3=-1$$)

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 2        & 4         & 5         &7          & 8 \\ \hline \chi_{9,1} & 1 & 1        & 1         & 1         & 1         & 1 \\ \chi_{9,2} & 1 & \omega   & \omega^2  & -\omega^2 & -\omega   & -1 \\ \chi_{9,4} & 1 & \omega^2 & -\omega   & -\omega   & \omega^2  & 1  \\ \chi_{9,5} & 1 & -\omega^2 & -\omega  & \omega    & \omega^2  & -1  \\ \chi_{9,7} & 1 & -\omega  &  \omega^2 & \omega^2  & -\omega   & 1  \\ \chi_{9,8} & 1 & -1       & 1         & -1        & 1         & -1  \\ \end{array} $$.

Powers of 2
$$(\mathbb{Z}/2\mathbb{Z})^\times$$ is the trivial group with one element. $$(\mathbb{Z}/4\mathbb{Z})^\times$$ is cyclic of order 2. For 8, 16, and higher powers of 2, there is no primitive root; the powers of 5 are the units $$\equiv 1\pmod{4}$$ and their negatives are the units $$\equiv 3\pmod{4}.$$ For example
 * $$5^1\equiv 5,\;5^2\equiv5^0\equiv 1\pmod{8}$$
 * $$5^1\equiv 5,\;5^2\equiv 9,\;5^3\equiv 13,\;5^4\equiv5^0\equiv 1\pmod{16}$$
 * $$5^1\equiv 5,\;5^2\equiv 25,\;5^3\equiv 29,\;5^4\equiv 17,\;5^5\equiv 21,\;5^6\equiv 9,\;5^7\equiv 13,\;5^8\equiv5^0\equiv 1\pmod{32}.$$

Let $$q=2^k, \;\;k\ge3$$; then $$(\mathbb{Z}/q\mathbb{Z})^\times$$ is the direct product of a cyclic group of order 2 (generated by −1) and a cyclic group of order $$\frac{\phi(q)}{2}$$ (generated by 5). For odd numbers $$a$$ define the functions $$\nu_0$$ and $$\nu_q$$ by
 * $$a\equiv(-1)^{\nu_0(a)}5^{\nu_q(a)}\pmod{q},$$
 * $$0\le\nu_0<2,\;\;0\le\nu_q<\frac{\phi(q)}{2}.$$

For odd $$a$$ and $$b, \;\;a\equiv b\pmod{q}$$ if and only if $$\nu_0(a)=\nu_0(b)$$ and $$\nu_q(a)=\nu_q(b).$$ For odd $$a$$ the value of $$ \chi(a)$$ is determined by the values of $$ \chi(-1)$$ and $$\chi(5).$$

Let $$\omega_q = \zeta_{\frac{\phi(q)}{2}}$$ be a primitive $$\frac{\phi(q)}{2}$$-th root of unity. The possible values of $$ \chi((-1)^{\nu_0(a)}5^{\nu_q(a)})$$ are $$ \pm\omega_q, \pm\omega_q^2, ... \pm\omega_q^{\frac{\phi(q)}{2}}=\pm1.$$ These distinct values give rise to $$\phi(q)$$ Dirichlet characters mod $$q.$$ For odd $$r $$ define $$\chi_{q,r}(a)$$ by

\chi_{q,r}(a)= \begin{cases} 0 &\text{if } a\text{ is even}\\ (-1)^{\nu_0(r)\nu_0(a)}\omega_q^{\nu_q(r)\nu_q(a)}&\text{if } a \text{ is odd}. \end{cases}$$ Then for odd $$r$$ and $$s$$ and all $$a$$ and $$b$$
 * $$\chi_{q,r}(a)\chi_{q,r}(b)=\chi_{q,r}(ab)$$ showing that $$\chi_{q,r}$$ is a character and
 * $$\chi_{q,r}(a)\chi_{q,s}(a)=\chi_{q,rs}(a)$$ showing that $$\widehat{(\mathbb{Z}/2^{k}\mathbb{Z})^\times}\cong (\mathbb{Z}/2^{k}\mathbb{Z})^\times.$$

Examples m = 2, 4, 8, 16
The only character mod 2 is the principal character $$\chi_{2,1}$$.

−1 is a primitive root mod 4 ($$\phi(4)=2$$)

\begin{array}{|||} a   & 1 & 3  \\ \hline \nu_0(a) & 0 & 1 \\ \end{array} $$

The nonzero values of the characters mod 4 are

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 3   \\ \hline \chi_{4,1} & 1 & 1       \\ \chi_{4,3} & 1 & -1 \\

\end{array} $$

−1 is and 5 generate the units mod 8 ($$\phi(8)=4$$)

\begin{array}{|||} a   & 1 & 3 & 5 & 7  \\ \hline \nu_0(a) & 0 & 1 & 0 & 1 \\ \nu_8(a) & 0 & 1 & 1 & 0 \\ \end{array} $$.

The nonzero values of the characters mod 8 are

\begin{array}{|c|c|c|c|c|c|c|} & 1 & 3 &  5  & 7     \\ \hline \chi_{8,1} & 1 & 1 &  1  & 1      \\ \chi_{8,3} & 1 & 1 & -1  & -1 \\ \chi_{8,5} & 1 & -1 & -1 & 1    \\ \chi_{8,7} & 1 & -1 & 1  & -1     \\ \end{array} $$

−1 and 5 generate the units mod 16 ($$\phi(16)=8$$)

\begin{array}{|||} a   & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 \\ \hline \nu_0(a) & 0 & 1 & 0 & 1 & 0 & 1 & 0  & 1  \\ \nu_{16}(a) & 0 & 3 & 1 & 2 & 2 & 1 & 3  & 0  \\ \end{array} $$.

The nonzero values of the characters mod 16 are

\begin{array}{|||} & 1 & 3 & 5  & 7  & 9  & 11  & 13 & 15 \\ \hline \chi_{16,1}   & 1 & 1  & 1  & 1  & 1  & 1   & 1  & 1  \\ \chi_{16,3}   & 1 & -i & -i & 1  & -1 & i   & i  & -1   \\ \chi_{16,5}   & 1 & -i & i  & -1 & -1 & i   & -i & 1  \\ \chi_{16,7}   & 1 & 1  & -1 & -1 & 1  & 1   & -1 & -1  \\ \chi_{16,9}   & 1 & -1 & -1 & 1  & 1  & -1  & -1 & 1  \\ \chi_{16,11}  & 1 & i  & i  & 1  & -1 & -i  & -i & -1  \\ \chi_{16,13}  & 1 & i  & -i & -1 & -1 & -i  & i  & 1  \\ \chi_{16,15}  & 1 & -1 & 1  & -1 & 1  & -1  & 1  & -1  \\

\end{array} $$.

Products of prime powers
Let $$m=p_1^{m_1}p_2^{m_2} \cdots p_k^{m_k} = q_1q_2 \cdots q_k$$ where $$ p_1<p_2< \dots < p_k$$ be the factorization of $$m$$ into prime powers. The group of units mod $$m$$ is isomorphic to the direct product of the groups mod the $$q_i$$:
 * $$(\mathbb{Z}/m\mathbb{Z})^\times \cong(\mathbb{Z}/q_1\mathbb{Z})^\times \times(\mathbb{Z}/q_2\mathbb{Z})^\times \times \dots \times(\mathbb{Z}/q_k\mathbb{Z})^\times .$$

This means that 1) there is a one-to-one correspondence between $$a\in (\mathbb{Z}/m\mathbb{Z})^\times$$ and $$k$$-tuples $$(a_1, a_2,\dots, a_k)$$ where $$a_i\in(\mathbb{Z}/q_i\mathbb{Z})^\times $$ and 2) multiplication mod $$m$$ corresponds to coordinate-wise multiplication of $$k$$-tuples:
 * $$ab\equiv c\pmod{m}$$ corresponds to
 * $$(a_1,a_2,\dots,a_k)\times(b_1,b_2,\dots,b_k)=(c_1,c_2,\dots,c_k)$$ where $$c_i\equiv a_ib_i\pmod{q_i}.$$

The Chinese remainder theorem (CRT) implies that the $$a_i$$ are simply $$a_i\equiv a\pmod{q_i}.$$

There are subgroups $$ G_i<(\mathbb{Z}/m\mathbb{Z})^\times$$ such that
 * $$G_i\cong(\mathbb{Z}/q_i\mathbb{Z})^\times $$ and
 * $$G_i\equiv

\begin{cases} (\mathbb{Z}/q_i\mathbb{Z})^\times &\mod q_i\\ \{1\}&\mod q_j, j\ne i. \end{cases} $$

Then $$(\mathbb{Z}/m\mathbb{Z})^\times \cong G_1\times G_2\times...\times G_k$$ and every $$a\in (\mathbb{Z}/m\mathbb{Z})^\times$$ corresponds to a $$k$$-tuple $$(a_1, a_2,...a_k)$$ where $$a_i\in G_i $$ and $$a_i\equiv a\pmod{q_i}. $$ Every $$a\in (\mathbb{Z}/m\mathbb{Z})^\times$$ can be uniquely factored as $$a =a_1a_2...a_k.$$

If $$\chi_{m,\_}$$ is a character mod $$m,$$ on the subgroup $$G_i$$ it must be identical to some $$\chi_{q_i,\_}$$ mod $$q_i$$ Then
 * $$\chi_{m,\_}(a)=\chi_{m,\_}(a_1a_2...)=\chi_{m,\_}(a_1)\chi_{m,\_}(a_2)...=\chi_{q_1,\_}(a_1)\chi_{a_2,\_}(a_2)...,$$

showing that every character mod $$ m$$ is the product of characters mod the $$q_i$$.

For $$(t,m)=1$$ define
 * $$ \chi_{m,t}=\chi_{q_1,t}\chi_{q_2,t}...$$

Then for $$(rs,m)=1$$ and all $$a$$ and $$b$$
 * $$\chi_{m,r}(a)\chi_{m,r}(b)=\chi_{m,r}(ab),$$ showing that $$\chi_{m,r}$$ is a character and
 * $$\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a),$$ showing an isomorphism $$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.$$

Examples m = 15, 24, 40
$$(\mathbb{Z}/15\mathbb{Z})^\times\cong(\mathbb{Z}/3\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.$$

The factorization of the characters mod 15 is

\begin{array}{|c|c|c|c|c|c|c|} & \chi_{5,1} & \chi_{5,2} &  \chi_{5,3} &  \chi_{5,4}   \\ \hline \chi_{3,1} & \chi_{15,1} & \chi_{15,7} & \chi_{15,13} & \chi_{15,4} \\ \chi_{3,2} & \chi_{15,11} & \chi_{15,2} & \chi_{15,8} & \chi_{15,14} \\ \end{array} $$ The nonzero values of the characters mod 15 are

\begin{array}{|||} & 1 & 2  & 4   & 7   & 8    & 11  & 13 & 14 \\ \hline \chi_{15,1}   & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\ \chi_{15,2}   & 1 & -i  & -1  & i   & i   & -1  & -i  & 1   \\ \chi_{15,4}   & 1 & -1  & 1   & -1  & -1   & 1   & -1  & 1  \\ \chi_{15,7}   & 1 &  i  & -1  & i   & -i   & 1   & -i  & -1  \\ \chi_{15,8}   & 1 &  i  & -1  & -i  & -i   & -1  & i   & 1  \\ \chi_{15,11}  & 1 & -1  & 1   & 1   &  -1  & -1  & 1  & -1  \\ \chi_{15,13}  & 1 & -i   & -1  & -i  & i   & 1  & i   & -1  \\ \chi_{15,14}  & 1 & 1   & 1   & -1   & 1    & -1  & -1   & -1  \\

\end{array} $$.

$$(\mathbb{Z}/24\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/3\mathbb{Z})^\times.$$ The factorization of the characters mod 24 is

\begin{array}{|c|c|c|c|c|c|c|} & \chi_{8,1} & \chi_{8,3} &  \chi_{8,5} &  \chi_{8,7}   \\ \hline \chi_{3,1} & \chi_{24,1} & \chi_{24,19} & \chi_{24,13} & \chi_{24,7} \\ \chi_{3,2} & \chi_{24,17} & \chi_{24,11} & \chi_{24,5} & \chi_{24,23} \\ \end{array} $$

The nonzero values of the characters mod 24 are

\begin{array}{|||} & 1 & 5  & 7   & 11   & 13    & 17  & 19 & 23 \\ \hline \chi_{24,1}   & 1 & 1   & 1   & 1   & 1    & 1   & 1   & 1  \\ \chi_{24,5}   & 1 & 1  & 1  & 1   & -1   & -1  & -1  & -1   \\ \chi_{24,7}   & 1 & 1  & -1   & -1  & 1   & 1   & -1  & -1  \\ \chi_{24,11}   & 1 &  1  & -1  & -1   & -1   & -1   & 1  & 1  \\ \chi_{24,13}   & 1 &  -1  & 1  & -1  & -1   & 1  & -1   & 1  \\ \chi_{24,17}  & 1 & -1  & 1   & -1   &  1  & -1  & 1  & -1  \\ \chi_{24,19}  & 1 & -1   & -1  & 1  & -1   & 1  & 1   & -1  \\ \chi_{24,23}  & 1 & -1   & -1   & 1   & 1    & -1  & -1   & 1  \\

\end{array} $$.

$$(\mathbb{Z}/40\mathbb{Z})^\times\cong(\mathbb{Z}/8\mathbb{Z})^\times\times(\mathbb{Z}/5\mathbb{Z})^\times.$$ The factorization of the characters mod 40 is

\begin{array}{|c|c|c|c|c|c|c|} & \chi_{8,1}  &  \chi_{8,3}  &  \chi_{8,5}  &  \chi_{8,7}   \\ \hline \chi_{5,1} & \chi_{40,1} & \chi_{40,11} & \chi_{40,21} & \chi_{40,31}  \\ \chi_{5,2} & \chi_{40,17} & \chi_{40,27} & \chi_{40,37} & \chi_{40,7} \\ \chi_{5,3} & \chi_{40,33} & \chi_{40,3} & \chi_{40,13} & \chi_{40,23}  \\ \chi_{5,4} & \chi_{40,9} & \chi_{40,19} & \chi_{40,29} & \chi_{40,39} \\ \end{array} $$

The nonzero values of the characters mod 40 are

\begin{array}{|||} & 1 & 3 & 7   & 9   & 11   & 13  & 17 & 19 & 21 & 23  & 27  & 29 & 31 & 33  & 37 & 39 \\ \hline \chi_{40,1}   & 1 & 1  & 1   & 1   & 1    & 1   & 1  & 1  & 1  & 1   & 1   & 1  & 1  & 1   & 1  & 1  \\ \chi_{40,3}   & 1 & i  & i   & -1  & 1    & -i  & -i & -1 & -1 & -i  & -i  & 1  & -1 & i   & i  & 1  \\ \chi_{40,7}   & 1 & i  & -i  & -1  & -1   & -i  & i  & 1  & 1  & i   & -i  & -1 & -1 & -i  & i  & 1  \\ \chi_{40,9}   & 1 & -1 & -1  & 1   & 1    & -1  & -1 & 1  & 1  & -1  & -1  &  1 & 1  & -1  & -1 & 1  \\ \chi_{40,11}  & 1 & 1  & -1  & 1   & 1    & -1  & 1  & 1  & -1 & -1  & 1   & -1 & -1 & 1   & -1 & -1 \\ \chi_{40,13}  & 1 & -i & -i  & -1  & -1   & -i  & -i & 1  & -1 & i   & i   & 1  &  1 & i   & i  & -1 \\ \chi_{40,17}  & 1 & -i & i   & -1  & 1    & -i  & i  & -1 & 1  & -i  & i   & -1 &  1 & -i  & i  & -1 \\ \chi_{40,19}  & 1 & -1 & 1   & 1   & 1    & 1   & -1 & 1  & -1 & 1   & -1  & -1 & -1 & -1  & 1 & -1  \\ \chi_{40,21}  & 1 & -1 & 1   & 1   & -1   & -1  & 1  & -1 & -1 & 1   & -1  & -1 &  1 & 1   & -1 & 1  \\ \chi_{40,23}  & 1 & -i & i   & -1  & -1   & i   & -i & 1  & 1  & -i  & i   & -1 & -1 & i   & -i & 1  \\ \chi_{40,27}  & 1 & -i & -i  & -1  & 1    & i   & i  & -1 & -1 & i   & i   & 1  & -1 & -i  & -i & 1  \\ \chi_{40,29}  & 1 & 1  & -1  & 1   & -1   & 1   & -1 & -1 & -1 & -1  & 1   & -1 &  1 & -1  & 1  & 1  \\ \chi_{40,31}  & 1 & -1 & -1  & 1   & -1   & 1   & 1  & -1 & 1  & -1  & -1  & 1  & -1 & 1   & 1  & -1 \\ \chi_{40,33}  & 1 & i  & -i  & -1  & 1    & i   & -i & -1 &  1 & i   & -i  & -1 & 1  & i   & -i & -1 \\ \chi_{40,37}  & 1 & i  & i   & -1  & -1   & i   & i  & 1  & -1 & -i  & -i  &  1 &  1 & -i  & -i & -1 \\ \chi_{40,39}  & 1 & 1  & 1   & 1   & -1   & -1  & -1 & -1 & 1  & 1   & 1   & 1  & -1 & -1  & -1 & -1 \\ \end{array} $$.

Summary
Let $$m=p_1^{k_1}p_2^{k_2}\cdots = q_1q_2 \cdots$$, $$p_1<p_2< \dots $$ be the factorization of $$m$$ and assume $$(rs,m)=1.$$

There are $$\phi(m)$$ Dirichlet characters mod $$m.$$ They are denoted by $$\chi_{m,r},$$ where $$\chi_{m,r}=\chi_{m,s}$$ is equivalent to $$r\equiv s\pmod{m}.$$ The identity $$\chi_{m,r}(a)\chi_{m,s}(a)=\chi_{m,rs}(a)\;$$ is an isomorphism $$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times.$$

Each character mod $$m$$ has a unique factorization as the product of characters mod the prime powers dividing $$m$$:
 * $$\chi_{m,r}=\chi_{q_1,r}\chi_{q_2,r}...$$

If $$m=m_1m_2, (m_1,m_2)=1$$ the product $$\chi_{m_1,r}\chi_{m_2,s}$$ is a character $$\chi_{m,t}$$ where $$t$$ is given by $$t\equiv r\pmod{m_1}$$ and $$t\equiv s\pmod{m_2}.$$

Also, $$ \chi_{m,r}(s)=\chi_{m,s}(r)$$

Orthogonality
The two orthogonality relations are
 * $$\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi(a)=

\begin{cases} \phi(m)&\text{ if }\;\chi=\chi_0\\ 0&\text{ if }\;\chi\ne\chi_0 \end{cases} $$    and      $$\sum_{\chi\in\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}}\chi(a)= \begin{cases} \phi(m)&\text{ if }\;a\equiv 1\pmod{m}\\ 0&\text{ if }\;a\not\equiv 1\pmod{m}. \end{cases} $$

The relations can be written in the symmetric form
 * $$\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi_{m,r}(a)=

\begin{cases} \phi(m)&\text{ if }\;r\equiv 1\\ 0&\text{ if }\;r\not\equiv 1 \end{cases} $$    and      $$\sum_{r\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi_{m,r}(a)= \begin{cases} \phi(m)&\text{ if }\;a\equiv 1\\ 0&\text{ if }\;a\not\equiv 1. \end{cases} $$

The first relation is easy to prove: If $$\chi=\chi_0$$ there are $$\phi(m)$$ non-zero summands each equal to 1. If $$\chi\ne\chi_0$$there is some $$a^*,\; (a^*,m)=1,\;\chi(a^*)\ne1.$$  Then
 * $$\chi(a^*)\sum_{a\in(\mathbb{Z}/m\mathbb{Z})^\times} \chi(a)=\sum_{a}\chi(a^*) \chi(a)=\sum_{a} \chi(a^*a)=\sum_{a} \chi(a),

$$  implying
 * $$(\chi(a^*)-1)\sum_{a} \chi(a)=0.$$  Dividing by the first factor gives $$\sum_{a} \chi(a)=0,$$ QED. The identity $$\chi_{m,r}(s)=\chi_{m,s}(r)$$ for $$(rs,m)=1$$ shows that the relations are equivalent to each other.

The second relation can be proven directly in the same way, but requires a lemma
 * Given $$a \not\equiv 1\pmod{m},\;(a,m)=1,$$ there is a $$ \chi^*,\; \chi^*(a)\ne1.$$

The second relation has an important corollary: if $$(a,m)=1,$$ define the function
 * $$f_a(n)=\frac{1}{\phi(m)} \sum_{\chi} \bar{\chi}(a) \chi(n). $$  Then
 * $$f_a(n)

= \frac{1}{\phi(m)} \sum_{\chi} \chi(a^{-1}) \chi(n) = \frac{1}{\phi(m)} \sum_{\chi} \chi(a^{-1}n) = \begin{cases} 1, & n \equiv a \pmod{m} \\ 0, & n\not\equiv a\pmod{m},\end{cases}$$ That is $$f_a=\mathbb{1}_{[a]}$$ the indicator function of the residue class $$[a]=\{ x:\;x\equiv a \pmod{m}\}$$. It is basic in the proof of Dirichlet's theorem.

Conductor; Primitive and induced characters
Any character mod a prime power is also a character mod every larger power. For example, mod 16

\begin{array}{|||} & 1 & 3 & 5  & 7  & 9  & 11  & 13 & 15 \\ \hline \chi_{16,3}   & 1 & -i & -i & 1  & -1 & i   & i  & -1   \\

\chi_{16,9}   & 1 & -1 & -1 & 1  & 1  & -1  & -1 & 1  \\

\chi_{16,15}  & 1 & -1 & 1  & -1 & 1  & -1  & 1  & -1  \\

\end{array} $$

$$\chi_{16,3}$$ has period 16, but $$\chi_{16,9}$$ has period 8 and $$\chi_{16,15}$$ has period 4:  $$\chi_{16,9}=\chi_{8,5}$$ and  $$\chi_{16,15}=\chi_{8,7}=\chi_{4,3}.$$

We say that a character $$\chi$$ of modulus $$q$$ has a quasiperiod of $$d$$ if $$\chi(m)=\chi(n)$$ for all $$m$$, $$n$$ coprime to $$q$$ satisfying $$m\equiv n$$ mod $$d$$. For example, $$\chi_{2,1}$$, the only Dirichlet character of modulus $$2$$, has a quasiperiod of $$1$$, but not a period of $$1$$ (it has a period of $$2$$, though). The smallest positive integer for which $$\chi$$ is quasiperiodic is the conductor of $$\chi$$. So, for instance, $$\chi_{2,1}$$ has a conductor of $$1$$.

The conductor of $$\chi_{16,3}$$ is 16, the conductor of $$\chi_{16,9}$$ is 8 and that of $$\chi_{16,15}$$ and $$\chi_{8,7}$$ is 4. If the modulus and conductor are equal the character is primitive, otherwise imprimitive. An imprimitive character is induced by the character for the smallest modulus: $$\chi_{16,9}$$ is induced from $$\chi_{8,5}$$ and $$\chi_{16,15}$$ and $$\chi_{8,7}$$ are induced from $$\chi_{4,3}$$.

A related phenomenon can happen with a character mod the product of primes; its nonzero values may be periodic with a smaller period.

For example, mod 15,

\begin{array}{|||} & 1 & 2 &3 & 4  &5&6 & 7   & 8 &9&10   & 11&12  & 13 & 14 &15 \\ \hline

\chi_{15,8}   & 1 &  i &0 & -1 &0&0 & -i  & -i &0&0  & -1  &0& i   & 1 &0 \\ \chi_{15,11}  & 1 & -1 &0 & 1  &0&0 & 1   & -1 &0&0 & -1  &0& 1  & -1  &0\\ \chi_{15,13}  & 1 & -i &0 & -1 &0&0 & -i  & i  &0&0 & 1 &0 & i   & -1  &0\\

\end{array} $$.

The nonzero values of $$\chi_{15,8}$$ have period 15, but those of $$\chi_{15,11}$$ have period 3 and those of $$\chi_{15,13}$$ have period 5. This is easier to see by juxtaposing them with characters mod 3 and 5:

\begin{array}{|||} & 1 & 2  & 3 & 4  & 5  & 6 & 7   & 8  & 9  & 10 & 11 & 12  & 13 & 14 &15\\ \hline \chi_{15,11}  & 1  & -1 & 0 & 1  & 0  & 0 & 1   & -1 & 0  & 0  & -1 & 0  & 1  & -1  &0\\ \chi_{3,2}    & 1  & -1 & 0 & 1  & -1 & 0 & 1   & -1 & 0  & 1  & -1 & 0  & 1  & -1  &0\\ \hline \chi_{15,13}  & 1  & -i & 0 & -1 &  0 & 0 & -i  & i  & 0  & 0  & 1  & 0  & i  & -1  &0\\ \chi_{5,3}    & 1  & -i & i & -1 &  0 & 1 & -i  & i  & -1 & 0  & 1  & -i & i  & -1  &0\\ \end{array} $$.

If a character mod $$m=qr,\;\; (q,r)=1, \;\;q>1,\;\; r>1$$ is defined as
 * $$ \chi_{m,\_}(a)=

\begin{cases} 0&\text{ if }\gcd(a,m)>1\\ \chi_{q,\_}(a)&\text{ if }\gcd(a,m)=1 \end{cases} $$,  or equivalently as $$ \chi_{m,\_}= \chi_{q,\_} \chi_{r,1},$$ its nonzero values are determined by the character mod $$q$$ and have period $$q$$.

The smallest period of the nonzero values is the conductor of the character. For example, the conductor of $$\chi_{15,8}$$ is 15, the conductor of $$\chi_{15,11}$$ is 3, and that of $$\chi_{15,13}$$ is 5.

As in the prime-power case, if the conductor equals the modulus the character is primitive, otherwise imprimitive. If imprimitive it is induced from the character with the smaller modulus. For example, $$\chi_{15,11}$$ is induced from $$\chi_{3,2}$$ and $$\chi_{15,13}$$ is induced from $$\chi_{5,3}$$

The principal character is not primitive.

The character $$\chi_{m,r}=\chi_{q_1,r}\chi_{q_2,r}...$$ is primitive if and only if each of the factors is primitive.

Primitive characters often simplify (or make possible) formulas in the theories of L-functions and modular forms.

Parity
$$\chi(a)$$ is even if $$\chi(-1)=1$$ and is odd if $$\chi(-1)=-1.$$

This distinction appears in the functional equation of the Dirichlet L-function.

Order
The order of a character is its order as an element of the group $$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}$$, i.e. the smallest positive integer $$n$$ such that $$\chi^n= \chi_0.$$ Because of the isomorphism $$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}\cong(\mathbb{Z}/m\mathbb{Z})^\times$$ the order of $$\chi_{m,r}$$ is the same as the order of $$r$$ in  $$(\mathbb{Z}/m\mathbb{Z})^\times. $$ The principal character has order 1; other real characters have order 2, and imaginary characters have order 3 or greater. By Lagrange's theorem the order of a character divides the order of $$\widehat{(\mathbb{Z}/m\mathbb{Z})^\times}$$ which is $$\phi(m)$$

Real characters
$$\chi(a)$$ is real or quadratic if all of its values are real (they must be $$0,\;\pm1$$); otherwise it is complex or imaginary.

$$\chi$$ is real if and only if $$\chi^2=\chi_0$$; $$\chi_{m,k} $$ is real if and only if $$k^2\equiv1\pmod{m}$$; in particular, $$\chi_{m,-1} $$ is real and non-principal.

Dirichlet's original proof that $$L(1,\chi)\ne0$$ (which was only valid for prime moduli) took two different forms depending on whether $$\chi$$ was real or not. His later proof, valid for all moduli, was based on his class number formula.

Real characters are Kronecker symbols; for example, the principal character can be written $$\chi_{m,1}=\left(\frac{m^2}{\bullet}\right)$$.

The real characters in the examples are:

Principal
If $$m=p_1^{k_1}p_2^{k_2}...,\;p_1<p_2<\;...$$ the principal character is $$\chi_{m,1}=\left(\frac{p_1^2p_2^2...}{\bullet}\right).$$

$$\chi_{16,1}=\chi_{8,1}=\chi_{4,1}=\chi_{2,1}=\left(\frac{4}{\bullet}\right)$$ $$\chi_{9,1}=\chi_{3,1}=\left(\frac{9}{\bullet}\right)$$ $$\chi_{5,1}=\left(\frac{25}{\bullet}\right)$$ $$\chi_{7,1}=\left(\frac{49}{\bullet}\right)$$ $$\chi_{15,1}=\left(\frac{225}{\bullet}\right)$$ $$\chi_{24,1}=\left(\frac{36}{\bullet}\right)$$ $$\chi_{40,1}=\left(\frac{100}{\bullet}\right)$$

Primitive
If the modulus is the absolute value of a fundamental discriminant there is a real primitive character (there are two if the modulus is a multiple of 8); otherwise if there are any primitive characters they are imaginary.

$$\chi_{3,2}=\left(\frac{-3}{\bullet}\right)$$ $$\chi_{4,3}=\left(\frac{-4}{\bullet}\right)$$ $$\chi_{5,4}=\left(\frac{5}{\bullet}\right)$$ $$\chi_{7,6}=\left(\frac{-7}{\bullet}\right)$$ $$\chi_{8,3}=\left(\frac{-8}{\bullet}\right)$$ $$\chi_{8,5}=\left(\frac{8}{\bullet}\right)$$ $$\chi_{15,14}=\left(\frac{-15}{\bullet}\right)$$ $$\chi_{24,5}=\left(\frac{-24}{\bullet}\right)$$ $$\chi_{24,11}=\left(\frac{24}{\bullet}\right)$$ $$\chi_{40,19}=\left(\frac{-40}{\bullet}\right)$$ $$\chi_{40,29}=\left(\frac{40}{\bullet}\right)$$

Imprimitive
$$\chi_{8,7}=\chi_{4,3}=\left(\frac{-4}{\bullet}\right)$$ $$\chi_{9,8}=\chi_{3,2}=\left(\frac{-3}{\bullet}\right)$$ $$\chi_{15,4}=\chi_{5,4}\chi_{3,1}=\left(\frac{45}{\bullet}\right)$$ $$\chi_{15,11}=\chi_{3,2}\chi_{5,1}=\left(\frac{-75}{\bullet}\right)$$ $$\chi_{16,7}=\chi_{8,3}=\left(\frac{-8}{\bullet}\right)$$ $$\chi_{16,9}=\chi_{8,5}=\left(\frac{8}{\bullet}\right)$$ $$\chi_{16,15}=\chi_{4,3}=\left(\frac{-4}{\bullet}\right)$$

$$\chi_{24,7}=\chi_{8,7}\chi_{3,1}=\chi_{4,3}\chi_{3,1}=\left(\frac{-36}{\bullet}\right)$$ $$\chi_{24,13}=\chi_{8,5}\chi_{3,1}=\left(\frac{72}{\bullet}\right)$$ $$\chi_{24,17}=\chi_{3,2}\chi_{8,1}=\left(\frac{-12}{\bullet}\right)$$ $$\chi_{24,19}=\chi_{8,3}\chi_{3,1}=\left(\frac{-72}{\bullet}\right)$$ $$\chi_{24,23}=\chi_{8,7}\chi_{3,2}=\chi_{4,3}\chi_{3,2}=\left(\frac{12}{\bullet}\right)$$

$$\chi_{40,9}=\chi_{5,4}\chi_{8,1}=\left(\frac{20}{\bullet}\right)$$ $$\chi_{40,11}=\chi_{8,3}\chi_{5,1}=\left(\frac{-200}{\bullet}\right)$$ $$\chi_{40,21}=\chi_{8,5}\chi_{5,1}=\left(\frac{200}{\bullet}\right)$$ $$\chi_{40,31}=\chi_{8,7}\chi_{5,1}=\chi_{4,3}\chi_{5,1}=\left(\frac{-100}{\bullet}\right)$$ $$\chi_{40,39}=\chi_{8,7}\chi_{5,4}=\chi_{4,3}\chi_{5,4}=\left(\frac{-20}{\bullet}\right)$$

L-functions
The Dirichlet L-series for a character $$\chi$$ is

 * $$L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.$$

This series only converges for $$\mathfrak{R}s >1$$; it can be analytically continued to a meromorphic function

Dirichlet introduced the $$L$$-function along with the characters in his 1837 paper.

Modular forms and functions
Dirichlet characters appear several places in the theory of modular forms and functions. A typical example is

Let $$\chi\in\widehat{(\mathbb{Z}/M\mathbb{Z})^\times}$$ and let $$\chi_1\in\widehat{(\mathbb{Z}/N\mathbb{Z})^\times}$$ be primitive.

If define
 * $$f(z)=\sum a_nq^n\in M_k(M,\chi)$$
 * $$f_{\chi_1}(z)=\sum\chi_1(n)a_nz^n$$,

Then
 * $$f_{\chi_1}(z)\in M_k(MN^2,\chi\chi_1^2)$$. If $$f$$ is a cusp form so is $$f_{\chi_1}.$$

See theta series of a Dirichlet character for another example.

Gauss sum
The Gauss sum of a Dirichlet character modulo $N$ is
 * $$G(\chi)=\sum_{a=1}^N\chi(a)e^\frac{2\pi ia}{N}.$$

It appears in the functional equation of the Dirichlet L-function.

Jacobi sum
If $$ \chi$$ and $$\psi$$ are Dirichlet characters mod a prime $$p$$ their Jacobi sum is

 * $$ J(\chi,\psi) = \sum_{a=2}^{p-1} \chi(a) \psi(1 - a). $$

Jacobi sums can be factored into products of Gauss sums.

Kloosterman sum
If $$\chi$$ is a Dirichlet character mod $$q$$ and $$\zeta = e^\frac{2\pi i}{q}$$ the Kloosterman sum $$ K(a,b,\chi)$$ is defined as

 * $$K(a,b,\chi)=\sum_{r\in (\mathbb{Z}/q\mathbb{Z})^\times}\chi(r)\zeta^{ar+\frac{b}{r}}.$$

If $$b=0$$ it is a Gauss sum.

Sufficient conditions
It is not necessary to establish the defining properties 1) – 3) to show that a function is a Dirichlet character.

From Davenport's book
If $$\Chi:\mathbb{Z}\rightarrow\mathbb{C}$$ such that
 * 1)  $$\Chi(ab) = \Chi(a)\Chi(b),$$
 * 2)  $$\Chi(a + m) = \Chi(a)$$,
 * 3)  If $$\gcd(a,m)>1$$ then $$\Chi(a)=0$$, but
 * 4)  $$\Chi(a)$$ is not always 0,

then $$\Chi(a)$$ is one of the $$\phi(m)$$ characters mod $$m$$

Sárközy's Condition
A Dirichlet character is a completely multiplicative function $$f: \mathbb{N} \rightarrow \mathbb{C}$$ that satisfies a linear recurrence relation: that is, if $$a_1 f(n+b_1) + \cdots + a_kf(n+b_k) = 0

$$

for all positive integer $$n$$, where $$a_1,\ldots,a_k$$ are not all zero and $$b_1,\ldots,b_k$$ are distinct then $$f$$ is a Dirichlet character.

Chudakov's Condition
A Dirichlet character is a completely multiplicative function $$f: \mathbb{N} \rightarrow \mathbb{C}$$ satisfying the following three properties: a) $$f$$ takes only finitely many values; b) $$f$$ vanishes at only finitely many primes; c) there is an $$\alpha \in \mathbb{C}$$ for which the remainder

$$\left|\sum_{n \leq x} f(n)- \alpha x\right| $$

is uniformly bounded, as $$x \rightarrow \infty$$. This equivalent definition of Dirichlet characters was conjectured by Chudakov in 1956, and proved in 2017 by Klurman and Mangerel.

title: Dirichlet character


text:
Steinway (disambiguation)

Steinway may refer to:
 * Steinway & Sons, an American-German piano company

People

 * C.F. Theodore Steinway, son of Henry E. Steinway
 * Henry E. Steinway, piano technician and founder of Steinway & Sons
 * Henry Z. Steinway, piano technician and great-grandson of Henry E. Steinway
 * Theodore E. Steinway, piano technician, philatelist and son of William Steinway
 * William Steinway, piano technician, civil leader and son of Henry E. Steinway

Places

 * Steinway Hall, a building housing concert halls, showrooms and sales departments for Steinway & Sons pianos
 * Steinway Mansion, a historic home in Astoria, Queens, New York City, the United States
 * Steinway Street, a major street in Astoria, Queens, New York City, the United States
 * Steinway Street (IND Queens Boulevard Line), a subway station in Astoria, Queens, New York City, the United States
 * Steinway Tunnel, a subway tunnel in Manhattan and Queens, New York City, the United States
 * The Steinway Tower, a residential skyscraper in Manhattan, New York City, the United States

Other

 * Steinway Car, a subway car
 * Steinway Lyngdorf, audio systems
 * Steinway Musical Instruments, a musical instrument manufacturing conglomerate and parent company of Steinway & Sons
 * Steinway Omnibus, a bus company
 * Steinway Transit Corporation, a bus company

title: Steinway (disambiguation)


text:
List of tourist attractions in Ireland

The following list includes the tourist attractions on the island of Ireland which attract more than 100,000 visitors annually. It includes attractions in both Northern Ireland and the Republic of Ireland.

Multi-county destinations/routes

 * The Wild Atlantic Way
 * Ireland's Ancient East
 * Ireland's Hidden Heartlands

A to C

 * Antrim
 * Antrim Castle and Gardens
 * Carrick-a-Rede Rope Bridge
 * Belfast, capital of Northern Ireland, second largest city on the island
 * Titanic Quarter, including the Titanic Belfast visitor attraction, the SS Nomadic museum ship, and W5 science museum
 * Ulster Museum within the Botanic Gardens
 * Dark Hedges
 * Giant's Causeway, a geological phenomenon and a UNESCO World Heritage Site
 * Lagan Valley Regional Park
 * Old Bushmills Distillery, the oldest Irish whiskey distillery in existence
 * Armagh
 * Armagh city, ecclesiastical capital of Ireland and home to St Patrick's Cathedral, Armagh (Church of Ireland) and St Patrick's Cathedral, Armagh (Roman Catholic)
 * Lough Neagh, including Kinnego Marina and Oxford Island National Nature Reserve
 * Lurgan Park
 * Slieve Gullion
 * Carlow
 * Altamont House and Gardens
 * Clare
 * Bunratty Castle
 * The Burren, a karst landscape which is home to prehistoric monuments such as Poulnabrone dolmen
 * Cliffs of Moher
 * Cork
 * Blarney, including Blarney Castle the home of the Blarney Stone
 * Cork City, third largest city in all of Ireland and second city of the Republic of Ireland
 * Church of St Anne (Shandon)
 * Crawford Art Gallery
 * English Market
 * University College Cork campus
 * Doneraile Park
 * Fota Wildlife Park
 * Kinsale
 * Midleton, home of the Jameson Distillery and Experience centre

D

 * Donegal
 * Glenveagh National Park, including Glenveagh Castle
 * Malin Head, most northerly point on the mainland of Ireland
 * Slieve League sea cliffs
 * Down
 * Crawfordsburn Country Park
 * Dundonald International Ice Bowl, ice rink
 * Irish linen - Thomas Ferguson & Co Ltd, the last remaining Irish linen damask factory
 * Kilbroney Park near Rostrevor at the base of the Mourne Mountains
 * Portstewart Strand
 * Scrabo Tower and Scrabo Country Park
 * Tollymore Forest Park
 * Cathedral of SS Patrick & Colman, Newry
 * Dublin City, largest city on the island, capital and cultural and economic centre of the Republic of Ireland
 * Christ Church Cathedral, seat of Anglican Church of Ireland Archbishop of Dublin
 * Chester Beatty Library
 * Croke Park, one of Europe's largest stadiums, with the Museum of the Gaelic Athletic Association
 * Dublin Castle, former seat of British rule, now a major Irish government complex
 * Dublinia, museum and "historical recreation" attraction
 * EPIC The Irish Emigration Museum, diaspora museum
 * General Post Office building, headquarters of the 1916 Easter Rising rebels, on O'Connell Street, the main thoroughfare of Dublin's Northside
 * Glasnevin Cemetery, burial location of Éamon de Valera, Michael Collins, Roger Casement, and many others
 * Grafton Street, one of the main shopping streets in Dublin
 * Ha'penny Bridge, Victorian pedestrian bridge across the River Liffey
 * Hugh Lane Gallery
 * Irish Museum of Modern Art
 * Old Jameson Distillery, Smithfield
 * Kilmainham Gaol, a former prison where, among others, most of the rebels of 1916 were held and executed; now a museum
 * National Aquatic Centre, Blanchardstown
 * National Botanic Gardens, Glasnevin, Dublin (Northside)
 * National Gallery of Ireland, houses the Irish national collection of Irish and European art
 * National Library of Ireland, has a large quantity of Irish historical, literary and Irish-related material
 * National Museum of Ireland for Archaeology (in Kildare St) and Decorative Art and History (in the former Collins Barracks)
 * Phoenix Park, "largest inner city park in the world"; within the park are Farmleigh Estate and Dublin Zoo
 * Guinness Storehouse
 * St Patrick's Cathedral, Dublin, Ireland's "national cathedral"
 * St Stephen's Green, a landscaped inner-city centre public park in Dublin
 * Temple Bar, a mainly cobblestonequarter, directly on the Southern banks of the Liffey, popular for its cultural and nightlife spots
 * Trinity College, Dublin (also called the University of Dublin), Ireland's oldest university, home of the Book of Kells and the Book of Durrow
 * Dún Laoghaire–Rathdown
 * DLR Lexicon housing a library and cultural centre

F to K

 * Fermanagh
 * Belleek Pottery Visitor Centre
 * Fingal
 * Howth and Howth Head
 * Malahide, with the Norman Malahide Castle
 * Galway
 * Aran Islands, Gaeltacht islands in Galway Bay which are the location of Dún Aonghasa
 * Connemara, Irish Gaeltacht, a heathland area, including Connemara National Park
 * Galway City, a university city (seat of the NUI Galway)
 * Kylemore Abbey & Gardens
 * Kerry, scenic rural county in the south west
 * Dingle, main town of the Dingle Peninsula and home to the Dingle Oceanworld Aquarium
 * Killarney
 * Killarney National Park including Killarney Lakes and Muckross House and Gardens
 * Ring of Kerry, ring road around the Iveragh Peninsula passing through, among others, the village of Cahersiveen, the birthplace of Daniel O'Connell
 * Skellig Islands with the monastic site on Skellig Michael, a UNESCO World Heritage Site
 * Tralee, home of the Rose of Tralee festival and Aqua Dome
 * Kildare
 * Castletown House
 * Kildare town with Kildare Cathedral and nearby Irish National Stud & Japanese Gardens
 * Newbridge Silverware visitors centre and "Museum of Style Icons"
 * Kilkenny
 * Kilkenny City, a medieval city which is home to St Canice's Cathedral, Rothe House, and Kilkenny Castle

L to M

 * Laois
 * Emo Court and Parklands
 * Rock of Dunamase
 * Limerick
 * Adare, "Ireland's most beautiful village" with Adare Manor, Desmond Castle, a Franciscan and a Trinitarian abbey
 * Limerick city, historic Irish city and home to the Munster rugby team, and King John's Castle
 * Londonderry
 * City of Derry, only city in Ireland with intact and unbreached city walls (hence sometimes called 'the Maiden City'). The Guildhall, Derry attracted 350,000 visitors in 2017
 * Roe Valley Country Park
 * Louth
 * Carlingford, one of Ireland's best preserved mediaeval towns, on the edge of Carlingford Lough
 * Drogheda, formerly Ireland's largest walled town (formed when two separate towns united in 1412); site of Laurence's Gat, Millmount Museum in the castle taken by Cromwell in 1649
 * Mayo
 * Croagh Patrick, mountain place of pilgrimage from pagan times to the present day, near Westport
 * Museum of Country Life near Castlebar
 * Meath
 * Brú na Bóinne, location of the Knowth, Dowth and Newgrange neolithic tombs and monuments
 * Hill of Tara, seat of Ireland's ancient High Kings
 * Emerald Park, theme park
 * Trim Castle
 * Monaghan
 * Harvest Time Blues Festival

O to W

 * Offaly
 * Birr Castle, including its gardens and science centre
 * Clonmacnoise, monastic site on the banks of the River Shannon
 * Roscommon
 * Lough Key Lake and forest park
 * Sligo
 * Drumcliffe with its church and gravesite of William Butler Yeats, overlooked by Ben Bulben mountain
 * Sligo town with Sligo Abbey
 * Tipperary
 * Holy Cross Abbey, a restored Cistercian monastery
 * Rock of Cashel, traditional seat of the Kings of Munster
 * Tyrone
 * Dungannon Park
 * Ulster American Folk Park near Omagh
 * Waterford
 * Waterford, Ireland's oldest city, home to the Waterford Museum of Treasures (which includes Reginald's Tower and other museum sites, which collectively attracted 100,000 visitors in 2017)
 * Waterford Greenway, a cycling and hiking trail
 * Westmeath, centre of Ireland; lakelands
 * Belvedere House and Gardens, historic gardens near Mullingar
 * Wexford
 * Ferns Castle and Abbey
 * New Ross and the John F Kennedy Arboretum
 * Wicklow, "the garden of Ireland"
 * Glendalough, 6th-century monastic site with Irish round tower
 * Powerscourt Estate, house, grounds and gardens (together with the nearby Powerscourt Waterfall)
 * Russborough House, an example of Irish Palladian architecture, designed by Richard Cassels, built between 1741 and 1755

title: List of tourist attractions in Ireland