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Bettin–Conrey cotangent sums

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Numbers
$k$
$h$ 
$c_0(h/k)$
2
1:
0
comment: $c_0(1/2)=0$ because $\cot(\pi/2)=0$.
3
1:
0.1924500897298752548363829268339858185492005837567089586728674421613258907676444485645717985285831751
3
2:
-0.1924500897298752548363829268339858185492005837567089586728674421613258907676444485645717985285831751
4
1:
1/2
comment: $c_0(1/4)=1/2$ because $\cot(\pi/4)=1$ and $\cot(3\pi/4)=-1$.
4
3:
-1/2
comment: $c_0(3/4)=-1/2$ by reflection.
5
1:
0.8908130915292853881555000315895595007065202959091902180809664031987916964524527807041093429554351255
5
2:
-0.08032456635449091194791906905309685693223778428874667248845232164181529817698762801957795735383500819
5
3:
0.08032456635449091194791906905309685693223778428874667248845232164181529817698762801957795735383500819
5
4:
-0.8908130915292853881555000315895595007065202959091902180809664031987916964524527807041093429554351255
6
1:
1.347150628109126783854680487837900729844404086296962710710072095129281235373511139952002589700082226
6
5:
-1.347150628109126783854680487837900729844404086296962710710072095129281235373511139952002589700082226
7
1:
1.857610089128434951163195776019074438499620669488671163027227687106366123810796261981255606303923651
7
2:
0.1751593035241762036688669033433271013992197196595921476143636962906889864559422274064066947916274054
7
3:
-0.6129819184119794356695530742768588858114187939341868697267430758040137129635451383727305853789925545
7
4:
0.6129819184119794356695530742768588858114187939341868697267430758040137129635451383727305853789925545
7
5:
-0.1751593035241762036688669033433271013992197196595921476143636962906889864559422274064066947916274054
7
6:
-1.857610089128434951163195776019074438499620669488671163027227687106366123810796261981255606303923651
8
1:
2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
8
3:
0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
8
5:
-0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
8
7:
-2.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
9
1:
3.011054187492305334650747252573366340220116505432997700785803557271525381383385535194952690540152748
9
2:
0.5271535360753661117781664521513872449364878125243437585610611095032542550035847017086968891789017766
9
4:
-1.329200113037687693854283239418064183988425190368400190187537794800315781773934142098825010189751921
9
5:
1.329200113037687693854283239418064183988425190368400190187537794800315781773934142098825010189751921
9
7:
-0.5271535360753661117781664521513872449364878125243437585610611095032542550035847017086968891789017766
9
8:
-3.011054187492305334650747252573366340220116505432997700785803557271525381383385535194952690540152748
10
1:
3.643576932471632464569919195411334859758318967925507544812317934436982083986798750836015329175575510
10
3:
-0.5695148261113217403638237553771720729775691587542035281271571166315305037445022686257975135400950928
10
7:
0.5695148261113217403638237553771720729775691587542035281271571166315305037445022686257975135400950928
10
9:
-3.643576932471632464569919195411334859758318967925507544812317934436982083986798750836015329175575510
11
1:
4.308159528295663201650532873081824984559069747163093293937080206079514547249569727484558151073621071
11
2:
0.9524520668736869598304176074543598747137664349864554768511220749832857876651836688713967704117258234
11
3:
0.8805173531469863493322560906238359328964280284006256522552676617314814444072148520795421039496665959
11
4:
1.268033496071480517119353404044809097627326932539554296293683172168225908543110534924403319329166318
11
5:
-2.187633520285043259156792023625423456441969727527464168576226818879451469925667598908263404812437511
11
6:
2.187633520285043259156792023625423456441969727527464168576226818879451469925667598908263404812437511
11
7:
-1.268033496071480517119353404044809097627326932539554296293683172168225908543110534924403319329166318
11
8:
-0.8805173531469863493322560906238359328964280284006256522552676617314814444072148520795421039496665959
11
9:
-0.9524520668736869598304176074543598747137664349864554768511220749832857876651836688713967704117258234
11
10:
-4.308159528295663201650532873081824984559069747163093293937080206079514547249569727484558151073621071
12
1:
5.001851166488378312872978048841815641139607588837216462747276748097236579979377831339433380871581276
12
5:
-0.001851166488378312872978048841815641139607588837216462747276748097236579979377831339433380871581275976
12
7:
0.001851166488378312872978048841815641139607588837216462747276748097236579979377831339433380871581275976
12
11:
-5.001851166488378312872978048841815641139607588837216462747276748097236579979377831339433380871581276
13
1:
5.722200844896039873671275568276463473241069464813122904850605681347663693912457794080745046102385727
13
2:
1.437018840129331213921185706698815847745478506918741546537680693492684025318776050177252751852197006
13
3:
-0.4240803878994121971305047272383040970705275287042340922874955085042413574460417454576329463369045396
13
4:
-1.603568296691409779879855728427281285361477355371055216888684918953351567477387304865958167096163181
13
5:
-0.3644694417893710707079086675397653300089213061882877615774289798652975952696990476781798291528096331
13
6:
-3.162212612165591003281206578310227429140133315627269633074819615374204899671106123412230711871884549
13
7:
3.162212612165591003281206578310227429140133315627269633074819615374204899671106123412230711871884549
13
8:
0.3644694417893710707079086675397653300089213061882877615774289798652975952696990476781798291528096331
13
9:
1.603568296691409779879855728427281285361477355371055216888684918953351567477387304865958167096163181
13
10:
0.4240803878994121971305047272383040970705275287042340922874955085042413574460417454576329463369045396
13
11:
-1.437018840129331213921185706698815847745478506918741546537680693492684025318776050177252751852197006
13
12:
-5.722200844896039873671275568276463473241069464813122904850605681347663693912457794080745046102385727
14
1:
6.467139831053407961645496223112784665388624444701313487153440280040072809367755454739478450253447238
14
3:
1.438451808133128363317046158274040435152785882900186994761846767287366382223354197877995565883504489
14
5:
2.382936711855689007826834311199483804525579378718676312023259053551637603914117629101089997895399138
14
9:
-2.382936711855689007826834311199483804525579378718676312023259053551637603914117629101089997895399138
14
11:
-1.438451808133128363317046158274040435152785882900186994761846767287366382223354197877995565883504489
14
13:
-6.467139831053407961645496223112784665388624444701313487153440280040072809367755454739478450253447238
15
1:
7.234898413360283157043378132833924530217047143757250643033490422713346446187927541870324642485476587
15
2:
1.971396196695157741834476453697606265132885721489324756349780629343941937882985574498156967727400668
15
4:
2.050668996893552652006216081402761239842400936879602679524596646804226863827186113684824499460535850
15
7:
-4.234969777880673989832990931195306408139353019040185227442427365592417119849816904023843397761280968
15
8:
4.234969777880673989832990931195306408139353019040185227442427365592417119849816904023843397761280968
15
11:
-2.050668996893552652006216081402761239842400936879602679524596646804226863827186113684824499460535850
15
13:
-1.971396196695157741834476453697606265132885721489324756349780629343941937882985574498156967727400668
15
14:
-7.234898413360283157043378132833924530217047143757250643033490422713346446187927541870324642485476587
16
1:
8.023945254791337122116110206006549304429010254208466090813750401103604986928623512560134808056661530
16
3:
-0.2023066047141689893104624877021993099060344991886120193818430665287191789161446912472800727138510431
16
5:
-2.859160854206549184517217384540991624184722000696404312088562018491649092764572846648830210024417334
16
7:
-0.6329089947010430730906446908322430098496772472993262018929685508593249269198046428414153292539047611
16
9:
0.6329089947010430730906446908322430098496772472993262018929685508593249269198046428414153292539047611
16
11:
2.859160854206549184517217384540991624184722000696404312088562018491649092764572846648830210024417334
16
13:
0.2023066047141689893104624877021993099060344991886120193818430665287191789161446912472800727138510431
16
15:
-8.023945254791337122116110206006549304429010254208466090813750401103604986928623512560134808056661530
17
1:
8.832942148062134517391131145436599764728702181954002602827921330559523171786609283733862250385328257
17
2:
2.548759745566134831092713094459931477878079190781416929988494009457511784993572236916582332077457859
17
3:
2.066894143523475194931668079153691584275727953673145678940328293249021867744794081744039668663378074
17
4:
-1.772989858903526598017472313386371879583009222836140766031989652238321672889283108891567081376323517
17
5:
0.3223712602158516715391200048403137116226045597335227362931808124365314870676988092071807381115339668
17
6:
3.702840034695209683659679777519333625366834528549818344902251406912447026516523826177353196407068765
17
7:
1.281081635921007977474601963462698119183694280050872534506030158494313414816365938232938690362518104
17
8:
-5.392762080187484306114233018383600784327666230222538789172229106378978729856718520389444502315207281
17
9:
5.392762080187484306114233018383600784327666230222538789172229106378978729856718520389444502315207281
17
10:
-1.281081635921007977474601963462698119183694280050872534506030158494313414816365938232938690362518104
17
11:
-3.702840034695209683659679777519333625366834528549818344902251406912447026516523826177353196407068765
17
12:
-0.3223712602158516715391200048403137116226045597335227362931808124365314870676988092071807381115339668
17
13:
1.772989858903526598017472313386371879583009222836140766031989652238321672889283108891567081376323517
17
14:
-2.066894143523475194931668079153691584275727953673145678940328293249021867744794081744039668663378074
17
15:
-2.548759745566134831092713094459931477878079190781416929988494009457511784993572236916582332077457859
17
16:
-8.832942148062134517391131145436599764728702181954002602827921330559523171786609283733862250385328257
18
1:
9.660709564780801421192372866572626687019065206314903095833554215279277233752438595263591973613055518
18
5:
-0.1781543867584937821061950953230886995500444368680508822603948258385333806674498002859084511423960359
18
7:
-1.755960182884534500170485034868311007502685125401177713833516470342123202178821555837484886554958200
18
11:
1.755960182884534500170485034868311007502685125401177713833516470342123202178821555837484886554958200
18
13:
0.1781543867584937821061950953230886995500444368680508822603948258385333806674498002859084511423960359
18
17:
-9.660709564780801421192372866572626687019065206314903095833554215279277233752438595263591973613055518
19
1:
10.50619995476380841294725903024115720831291841284380222138078082457754355418725816911866015921677081
19
2:
3.163944987322444341689443552874531237809042393556777974831077966810883082041876207819862569280057833
19
3:
0.08068063034967431612727747858822037578596449870249623669535964640441246166937900673843235075331177079
19
4:
2.922683081171001926065931673725356646050844361694423039140956238307641140772298696321254077846538844
19
5:
3.561534568857377647583158320023649550398050790676992399476129427551396312267313283130650082617197391
19
6:
-4.294384703025416559768982956770910903699646658046097778685260368637719541119905760302622788858904945
19
7:
0.3263060007033016518292037385561524437687674753849413086688809510951604704820358332408607287796715113
19
8:
-1.206740563993190389379424179967999613150662831863189610857889096542324537221488608481535975571985774
19
9:
-6.625554905250471257982290970421037285249586396925531652422594418061493601896075529775884159287945249
19
10:
6.625554905250471257982290970421037285249586396925531652422594418061493601896075529775884159287945249
19
11:
1.206740563993190389379424179967999613150662831863189610857889096542324537221488608481535975571985774
19
12:
-0.3263060007033016518292037385561524437687674753849413086688809510951604704820358332408607287796715113
19
13:
4.294384703025416559768982956770910903699646658046097778685260368637719541119905760302622788858904945
19
14:
-3.561534568857377647583158320023649550398050790676992399476129427551396312267313283130650082617197391
19
15:
-2.922683081171001926065931673725356646050844361694423039140956238307641140772298696321254077846538844
19
16:
-0.08068063034967431612727747858822037578596449870249623669535964640441246166937900673843235075331177079
19
17:
-3.163944987322444341689443552874531237809042393556777974831077966810883082041876207819862569280057833
19
18:
-10.50619995476380841294725903024115720831291841284380222138078082457754355418725816911866015921677081
20
1:
11.36847672841355893380268569669566268969135435916487632008546395649621432279675451979675013215328045
20
3:
2.752781736422445000142780757655111467993860617425897719799027935916165545609617633557655832323539356
20
7:
5.191490173576713785493913917269993473768612821020205177284561045725918156941601964100001801191589744
20
9:
-1.424204818414400148165991021770557747928880920718773423001874974854130620245534922139092498638151345
20
11:
1.424204818414400148165991021770557747928880920718773423001874974854130620245534922139092498638151345
20
13:
-5.191490173576713785493913917269993473768612821020205177284561045725918156941601964100001801191589744
20
17:
-2.752781736422445000142780757655111467993860617425897719799027935916165545609617633557655832323539356
20
19:
-11.36847672841355893380268569669566268969135435916487632008546395649621432279675451979675013215328045
21
1:
12.24669748415600370295164670789604738096669773231458266018799465942833947147248334437215270887508678
21
2:
3.812903628802104665157833299653318758293784836574094313226620614566661249422796757006777524835698872
21
4:
-1.864455037783661831948927681938513848179614349415863813595707737239602436726922391624632929121448265
21
5:
-3.423864378140236846014508522740726051411814283783658931267987700300584753708790531748160048204829067
21
8:
0.9130681093747205147050540743923030088645023860795970747699643869125873967388718700399793798770601970
21
10:
-7.925434547956502008136042613648722905745406941488432637826484968042117797686816166061492132074209459
21
11:
7.925434547956502008136042613648722905745406941488432637826484968042117797686816166061492132074209459
21
13:
-0.9130681093747205147050540743923030088645023860795970747699643869125873967388718700399793798770601970
21
16:
3.423864378140236846014508522740726051411814283783658931267987700300584753708790531748160048204829067
21
17:
1.864455037783661831948927681938513848179614349415863813595707737239602436726922391624632929121448265
21
19:
-3.812903628802104665157833299653318758293784836574094313226620614566661249422796757006777524835698872
21
20:
-12.24669748415600370295164670789604738096669773231458266018799465942833947147248334437215270887508678
22
1:
13.14010045915023562778450425379315673236677639115199805346984985547203514374224397328717565865859513
22
3:
0.4148358680893393961347937192893059163290995910871285681623001813352487463206347454260449255029576800
22
5:
0.7463023285981397160061537422857920762466886126169555658245207983727046592036839114156047868471586941
22
7:
-5.880716623706207807311508189972548381588397351973013012834145937152580735321688174985300714804272642
22
9:
3.034418252943876396567569251354003680958657872296379979119958250482823213675056121847777708085050414
22
13:
-3.034418252943876396567569251354003680958657872296379979119958250482823213675056121847777708085050414
22
15:
5.880716623706207807311508189972548381588397351973013012834145937152580735321688174985300714804272642
22
17:
-0.7463023285981397160061537422857920762466886126169555658245207983727046592036839114156047868471586941
22
19:
-0.4148358680893393961347937192893059163290995910871285681623001813352487463206347454260449255029576800
22
21:
-13.14010045915023562778450425379315673236677639115199805346984985547203514374224397328717565865859513
23
1:
14.04799346344219282480948240631876505200249787483728174646969547319322062132248776648059875990628129
23
2:
4.492375094052188202526794034901403968416038382494105719254142238688425181058525324347343912398223822
23
3:
3.487193700729839512471016139670688899302106515866285550027585850186929931232404158056529869729391374
23
4:
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Definition
The Bettin–Conrey cotangent sum is $c_0(h/k)=-\sum_{m=1}^{k-1}(m/k)\cot(\pi m h/k)$ for coprime integers $k\geq2$ and $1\leq h<k$, with the positive real constant $\pi$ [1].
Parameters
$k$
—   Denominator ($k\geq2$)
$h$
—   Reduced numerator ($1\leq h<k$, $\gcd(h,k)=1$)
Formulas
(1)
$c_0((h+k)/k)=c_0(h/k)$ and $c_0((k-h)/k)=-c_0(h/k)$, where the same finite-sum definition extends to every integer numerator coprime to $k$.
(2)
$V(h/k):=\sum_{m=1}^{k-1}\{mh/k\}\cot(\pi m/k)=-c_0(\bar h/k)$, where $1\leq\bar h<k$ and $h\bar h\equiv1\pmod k$ [1], §2, p. 225, definition of $V$ preceding equation (13).
(3)
$c_0(h/k)=\frac{1}{\pi k}\sum_{m=1}^{k-1}m\bigl(\psi(r_m/k)-\psi(1-r_m/k)\bigr)$, where $r_m$ is the least positive residue of $mh$ modulo $k$ and $\psi$ is the digamma function. This follows from the reflection formula $\psi(1-x)-\psi(x)=\pi\cot(\pi x)$ [3].
(4)
$D(0,h/k)=\frac14+\frac{i}{2}c_0(h/k)=\sum_{a=1}^{k}\sum_{b=1}^{k}e^{2\pi i hab/k}(\frac12-\frac ak)(\frac12-\frac bk)$, where $D(s,h/k)=\sum_{n\ge1}d(n)e^{2\pi ihn/k}n^{-s}$ for $\Re s>1$, $d(n)$ is the number of positive divisors of $n$, and $D(0,h/k)$ denotes its analytic continuation [1], equation (12). The finite expression follows by grouping both divisor variables modulo $k$ and using $\zeta(0,x)=\frac12-x$ for the Hurwitz zeta function [4].
(5)
$c_0(1/2)=0$, $c_0(1/3)=1/(3\sqrt3)$, $c_0(1/4)=1/2$, and $c_0(1/6)=7/(3\sqrt3)$, with positive square roots.
Comments
(6)
This is the order-zero cotangent sum. The Vasyunin sum, also spelled Vassiounine sum [2], uses a modular-inverse numerator and the opposite sign, as specified in (2). Here $\{x\}=x-\lfloor x\rfloor$ is the ordinary fractional part.
(7)
These sums enter the Gram matrices of integer dilates of the fractional-part function in the Nyman–Beurling approach to the Riemann hypothesis [1], §2, pp. 225–226, equation (13) and its discussion. The finite sums themselves require no assumption about the Riemann hypothesis.
(8)
Each value is real algebraic: $\cot(\pi r/k)=i(\zeta_k^r+1)/(\zeta_k^r-1)$, where $\zeta_k=e^{2\pi i/k}$ and $i^2=-1$ with positive imaginary part. Thus $c_0(h/k)$ lies in the real subfield of $\mathbb Q(\zeta_{\operatorname{lcm}(4,k)})$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=96))
k, h = 7, 2
c0 = -sum((R(QQ(m)/k) * (R.pi()*R(QQ((m*h) % k)/k)).cot()
           for m in range(1, k)), R(0))
print(c0)  # c_0(2/7)
References
[1]
S. Bettin and B. Conrey, Period functions and cotangent sums, Algebra & Number Theory 7 (2013), no. 1, 215–242. (arXiv) (doi)
[2]
L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire », 2003, §4. (arXiv)
Links
Similar tables
Values of the digamma function at rational numbers —   The values $\psi(r_m/k)$ and $\psi(1-r_m/k)$ are the ingredients of (3).
Values of Dirichlet L-functions at positive integers —   For odd characters, the formulas for $L(1,\chi)$ use cotangents weighted by character values; $c_0$ uses linear weights.
Atiyah–Patodi–Singer eta invariants of lens spaces —   The lens-space invariants have finite sums of products of two cotangents, related to Dedekind sums.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds every reduced residue $1\leq h<k$ for $2\leq k\leq30$, ordered by $k$ and then $h$. Full residue systems retain modular inversion and both reflection signs, and include prime, prime-power and composite moduli. This covers the nontrivial cotangent sums needed for ratios of the first 30 integer dilates in the Nyman–Beurling Gram matrices. Thirty is a practical, noncanonical initial cutoff for the shared finite approximation problem; the argument-selection attachment explains this choice)
How they were obtained:

Exact arithmetic in cyclotomic fields identifies rational answers, which are stored exactly. The remaining 274 entries are evaluated using Arb real balls with 96 extra working bits. Every ball is finite and contains the exact finite sum; the smallest measured relative accuracy is 411 bits and the largest absolute radius is less than $1.09\times10^{-126}$. These bounds justify the 100 significant digits stored.

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Every selected value agrees with an independent exact evaluation of (4) using rational coefficients and roots of unity, and with a 550-bit digamma evaluation of (3). Zero and nonzero controls precede the comparisons. Periodicity, reflection, and the modular-inverse Vasyunin conversion are checked throughout the selection. The generator, checks and numerical evidence are attached.



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