Taylor coefficients of the Bettin–Conrey reciprocity function

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Numbers

$n$

$g_n$

-1

comment: $g_0=-1$, since $c_0(1)=0$.

1/2

comment: $g_1=1/2$, by $g(x)=xg(1/x)$ at $x=1$.

0.7149780222827421454908050555486750630729833004022661459118527431233358234677336246112096335399195684

0.3574890111413710727454025277743375315364916502011330729559263715616679117338668123056048167699597842

0.2520007955606285266443356146715930046848415867398846124437279571987265373214022225351441743738464564

0.1992566877702572535938021581202207412590165550092603821876287500172558501151699276499138531757897925

0.1660229037125837141458075273884724300217387529781478055726878910830921986980366663500614954434162347

0.1425442815212590414990823095665360948787345657967910557783262062725820651754529218428981194438842300

0.1248882394805122885658105319151631922985696226620479766456130893139235500362105444628822893706886736

0.1110884867324579818020842080647294505155267381723474900950832370712157035247054593772432873205770132

10:

0.1000074040344880910969273543360640795173404901073217664517765575694304861746393767831443713692366932

11:

0.09092170795163330805582998839522303016847133783720460024298556441397631366436689939271578758182515640

12:

0.08334316210799406004490296889460069179241167771455380271680859721977234439524981215085588557390195905

13:

0.07692893281245601721295541529348574096283887077011239810142643182578562328083422709413167629189482851

14:

0.07143139607754558308368824371873934251446638758287104509123320794791159516934062831233719520446802103

15:

0.06666767033485537693294576084553452666374401173723107094639451555102630308281036696138076819934182968

16:

0.06250009596794285412640331709823828984519826397200101001718520474376786898913701052361954771861984286

17:

0.05882327222315363750538001982147719306630862807359309259878320159782605596509674841370050289915225371

18:

0.05555523021028208784661854188117980069957373750216727811831014768360036875018193147117255983346072153

19:

0.05263130478071211889095057856026558718830097070766100070396694880912797876188676060273734424124888709

20:

0.04999980877453761043244295078753350055619924506806632980601187403301289759893366366329776181614577612

21:

0.04761893214608105622009576179669057145223466786135626344514954555031261545726571654359994788701472456

22:

0.04545448596891469828168427191240850903363966455772141197477744911322639634052194369516540396492600501

23:

0.04347823736679204390052220463033248744255704343655891851904069604111484477018715963633665231424596842

24:

0.04166666344272773262569853785409391702248954114420037512612300750541686470106564071188143671315301269

25:

0.04000000638525696443877524794433766125936588966738888483188914710968104208675318473764219275186724411

26:

0.03846154801475915548301269888802271246494685737113734283911083219552843730889876068100507966380110745

27:

0.03703704634646745820918493776296261032204316872381392978880324924649954549409435878484634458398902824

28:

0.03571429330462607464860771115725356088079992400019092196442340814017510813870194688730808785378060388

29:

0.03448276412628958467059185061571657615564903654452637591002328564094167525328172796703990992120753798

30:

0.03333333694205125995580836739024976607016999057270088546996678237758874446042016376354169533238970481

31:

0.03225806663050639724345464183752898695592072705382655959037520842970109875818413315664787117380010305

32:

0.03125000105189318956271237238452379038299556927013221855034823342571944604155392403697262498993107546

33:

0.03030303066668311768040734087814911595582848762080544218369679717676291124439717099961009568344107768

34:

0.02941176466861523576850339554380295785751786087216055998128613904239740990094102796231085482470233227

35:

0.02857142833451205395964285764411937971731309481601628727423303550332119501625629235335787323676736983

36:

0.02777777747054554944785863166957480375872242524276234927020771865185594586910070020317035198896947478

37:

0.02702702672465306697089865832978379740262273014850704694195047355534114094189578814941532220153998154

38:

0.02631578921381523426024787464716484364007820746898491498986921911238983634952744473848779407811339363

39:

0.02564102543731318776756380302690199967923059504742678206346498924380045551163193885420644086496223588

40:

0.02499999985213823495513400194589098600810524102395644231938264744463766531713441740829171085493626231

41:

0.02439024380306588945320639213741978267760815519070436499452621588131104120747867262293078890489977921

42:

0.02380952374862278203303083558082190711431182801953090200894538732433595146138258034242043616036002334

43:

0.02325581392095696033514062958583795112837569683147364907916694438158280840439231625165023070258674281

44:

0.02272727271423723926679821968664030071815096805580644671012215916627863026962984209489823089170114318

45:

0.02222222222155322943446559398915330443539826157984499859582970470178132353543510047877009228245598091

46:

0.02173913044114162324127155071744226330131684172403350583490231357493328196540528434963747060688782198

47:

0.02127659575432350536548755840070836849393577909582542314518880001706710331049175869628501692200735481

48:

0.02083333334381286292209249576917439820059906839911412513475568857124083260406630172044965987403543690

49:

0.02040816327516204706624793008009286120926461918544824741872863741861174324449392485217907089720816950

50:

0.02000000000847681440543635422534270287613883263092658174752477842200009978478929715228256303571985376

51:

0.01960784314406993927170571684391785738498727292895708734860930017460444687827345855590339068561866951

52:

0.01923076923593360200652350074269574229241390078944453596974394183924364031846845611633866031249483920

53:

0.01886792453199020820754274019847186857771898176190979331745782699824705709508693336596497256484382067

54:

0.01851851852097948400605584312483363332952339820318015965357219434119522582714123621755394381497963453

55:

0.01818181818331710618677399251373081163770712179028818762422867218922672045021153101569423537080845815

56:

0.01785714285792787174626049581289829049341992469642872132411207729869981324267342557459336865806189057

57:

0.01754385964940763553437451748723014341143036038055902152455898239424504826957679298967087766795643324

58:

0.01724137931030252421413346367632894247668441594711205686274534898362814807150274517760182493685357607

59:

0.01694915254213630827783630950404684964243902984795082045295425391392013551077366408444484396859932496

60:

0.01666666666633269312855132138871179390517140545178668511740700873683684785158494162519621926235387012

61:

0.01639344262258619702004854115595520967557550411522242592447611880423648065384300914361195917056516245

62:

0.01612903225771212528683636232258906017619217989233145097038152088549431016343974168578818477872657456

63:

0.01587301587270058068547056239522571940531464389780390244831100250876945080250846354404914067116881968

64:

0.01562499999973373398055602434371995280677771166584361403272892756014684675780884255397993501708893029

65:

0.01538461538440128430929264861034978370538649495255612629680477877824371191172795797743378387763439470

66:

0.01515151515135080036179855274567610581053258067384143227534494878530164755947733451226464728536172131

67:

0.01492537313420818829538974542933745360756785486078041685967531170941722621894308826464515266957078043

68:

0.01470588235285817381767961037814623479056567516041222386074551731185481606227783864707377444421801507

69:

0.01449275362313525427299440701043375859553967110550201772135758051170450337749350572075756134845097877

70:

0.01428571428568408003723369011180417375556787588768580752785750945653895428502994420938839311964179952

71:

0.01408450704224016315648493352887823352025970551362269205920388943147584947112777325431098285809344805

72:

0.01388888888888726172806170220845283237853571573348088584237602330847614149756782450246860670802855382

73:

0.01369863013699229600509118392658510981965888615021920325713480267079904175236267447397163705156346514

74:

0.01351351351352396467049475749476308623087204537533942401360718690961870729013255057655274433501890142

75:

0.01333333333334590578138517941656193973297942495774725279037687706932931064112489641583690034725269335

76:

0.01315789473685515923556046888406615119083994030648304445015931731707410794971168013843466885571111891

77:

0.01298701298702544159698971800024370747592246623228038410350624965187256548775611842177143207366173436

78:

0.01282051282052402649583211879479877723632160258649899292505632839477142226686160627238188708854719998

79:

0.01265822784811089416615018139930046806073642051350231268728184989884228540302911269957376127801649233

80:

0.01250000000000794844465380759940764253086085628211048957612451347658792489099284208424451479429439905

81:

0.01234567901235199692673841932390167143022932126909959469969131363340800218494104191414824940768054678

82:

0.01219512195122434232986924653989956535629623087743310660868895526036624739097243448413049344988306625

83:

0.01204819277108787239626961454587863241567672184067864032503058579695000511437726084009335939112935239

84:

0.01190476190476435631358611802041854120102744248408305966801894330224585173181091421610949923228070374

85:

0.01176470588235451868449634224832381369568464384408506656048687354166260922740606915375177081309853217

86:

0.01162790697674508355704642703196795501982381575870529983525062223496837461690555102699452425697184951

87:

0.01149425287356360711537922068380999688927371740694281157323945465447509118173301971002303152444417363

88:

0.01136363636363638872857163503401325541765450410746566113382942349940519232695341597194731973009171949

89:

0.01123595505617955550243851020892035928757907889806312679028749400405752758651271343565364981176992279

90:

0.01111111111111074035877794987512435368181325657477658471990243072403311028275563143580191033863278719

91:

0.01098901098901053892846683486605421426679050245747392853503157522670320102719276078889097924392448517

92:

0.01086956521739082745794138702571661766392804549006286234902341003457833075198990881264038197499719857

93:

0.01075268817204254371748466816849085306966550532190304778802665638066203701541029979219688333803408877

94:

0.01063829787233999228259148996407985233333362909779552243165150001292392449398030636578217111688820979

95:

0.01052631578947329879155506390336838033137013679400819554403890474012048536296157885058144235178202426

96:

0.01041666666666633571340502384977918643221006280110694093401318536028304706226758884873029165960592285

97:

0.01030927835051518873945533440314006115265585287237504761559658982285358913874881319477241235824421504

98:

0.01020408163265283950155313965799492075140384959851551220802097744737589220404953284813666073140314267

99:

0.01010101010100992813159396287303982656345263266065810359121565398861881299312411945532516548045548525

100:

0.009999999999999870087588030804141595221292975812025365412360941634804981758528168355095733686330432291

Definition

The normalized Taylor coefficients $g_n$ of the Bettin–Conrey reciprocity function $g$ are defined by $g(1+z)=\pi^{-1}\sum_{n=0}^{\infty}(-1)^n g_nz^n$ for $|z|<1$ [1], equations (1.1)–(1.3).

Parameters

$n$

— Taylor order ($n\geq0$)

Formulas

(1)

$g_n=\frac1{n(n+1)}+2b_n+2\sum_{j=0}^{n-2}\binom{n-1}{j}b_{j+2}$ for $n\geq2$, where $b_k=B_k\zeta(k)/k$ for $k\geq2$ [1], equation (1.3). Here $B_k$ is the Bernoulli number and $\zeta$ is the Riemann zeta function.

(2)

$b_{2m}=\frac{(-1)^{m+1}2^{2m-1}B_{2m}^2}{(2m)(2m)!}\pi^{2m}$ for $m\geq1$, by Euler’s even-zeta identity [2].

(3)

$g_0=-1$, $g_1=1/2$, $g_2=1/6+\pi^2/18$, and $g_3=1/12+\pi^2/36$.

(4)

$g(x)=xg(1/x)$ for $x>0$, directly from the rational-argument definition and analytic continuation. Equating Taylor coefficients gives $g_n=\sum_{j=2}^{n}(-1)^j\binom{n-2}{j-2}g_j$ for $n\geq2$.

Comments

(5)

The function $g$ is the holomorphic continuation to $\mathbb C\setminus(-\infty,0]$ of $g(x)=xc_0(x)+c_0(1/x)-1/(\pi\operatorname{Den}(x))$ on positive rationals, where $c_0$ is the Bettin–Conrey cotangent sum and $\operatorname{Den}(x)$ is the positive reduced denominator of $x$.

(6)

For coprime positive integers $h,k$, the cotangent sum is $c_0(h/k)=-\sum_{a=1}^{k-1}(a/k)\cot(\pi ah/k)$, including the empty sum $c_0(h)=0$ when $k=1$. The constant $\pi$ is positive. Thus $g_n=(-1)^n\pi g^{(n)}(1)/n!$; the expansion centre $1$ is fixed by $x\mapsto1/x$.

(7)

The Bernoulli numbers use $u/(e^u-1)=\sum_{k\geq0}B_ku^k/k!$, so $B_1=-1/2$. In (1), $\zeta$ is the Riemann zeta function. Odd $b_k$ vanish for $k\geq3$; $b_1$ is not used.

(8)

For every $n\geq2$, (1) and (2) express $g_n$ as a nonconstant rational polynomial in $\pi^2$ of degree $\lfloor n/2\rfloor$. Consequently these coefficients are transcendental, whereas $g_0$ and $g_1$ are rational.

(9)

The coefficients satisfy $g_n\sim1/n$ as $n\to\infty$. The difference $g_n-1/n$ has an exponentially small oscillatory asymptotic expansion [1], Theorem 1.1; its correction coefficients form a different sequence.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from math import comb
from sage.arith.misc import bernoulli
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

# Evaluate one normalized coefficient, with enough precision for n <= 100.
n = 20
R = RealBallField(numberdb.bits(100, losing=384))
def b(k):
    return R(0) if k % 2 else R(bernoulli(k)) * R(k).zeta() / k
if n == 0:
    answer = QQ(-1)
elif n == 1:
    answer = QQ(1)/2
else:
    answer = R(QQ(1)/(n*(n+1))) + 2*b(n)
    answer += 2*sum((comb(n-1,j)*b(j+2) for j in range(n-1)), R(0))
print(answer)

References

[1]

H. Akatsuka and Y. Murakami, An asymptotic property on a reciprocity law for the Bettin–Conrey cotangent sum, International Journal of Number Theory 21 (2025), 2233–2253. Equation numbers refer to the 2024 preprint, version 1. (arXiv) (doi)

Links

[2]

DLMF 25.6.2: Zeta values at positive even integers

Similar tables

Bettin–Conrey cotangent sums — The sums $c_0(x)$ and $c_0(1/x)$ determine the reciprocity function whose local expansion is recorded here.

Bernoulli numbers — The exact numbers $B_k$ supply the rational factors in (1) and (2).

Values of the Riemann zeta function at rational numbers — The even positive zeta values supply the transcendental factors in $b_{2m}=B_{2m}\zeta(2m)/(2m)$.

Asymptotic coefficients of the Bettin–Conrey reciprocity function — The signed coefficients $\widetilde C_\ell$ give the successive corrections in the fixed-truncation large-order expansion of $g_n-1/n$, with amplitude $2^{9/4}\pi^{3/4}$ and phase $2\sqrt{\pi n}+\pi\ell/4+3\pi/8$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every Taylor order $0\leq n\leq100$ in increasing order, including both parities and the two exact initial coefficients. This consecutive prefix supports local reconstruction, reciprocal-symmetry identities and comparison of the residual $g_n-1/n$ across orders. The noncanonical cutoff balances this coverage against the factorial growth and cancellation of the finite formula; the centre and normalization are fixed parts of the sequence, not sampled parameters. The attached argument-selection note gives the rationale)

How they were obtained:

For $n\geq2$, each coefficient is evaluated as an exact rational polynomial in $\pi^2$ using Arb real balls at 717 working bits. Exact coefficient arithmetic and ball propagation enclose the result. The largest measured cancellation is about 82.36 decimal digits, at $n=100$; the smallest retained relative accuracy is 441 bits, sufficient for the 100 significant digits written. The values $g_0=-1$ and $g_1=1/2$ are exact.

An independent Bernoulli recurrence and direct Arb zeta evaluation at 1000 bits agree on every coefficient, retaining at least 723 relative accuracy bits for $n\geq2$. Both enclosures lie strictly inside each written last-place interval. A separate certified computation of the corrected contour representation [1], equations (3.1)–(3.5), agrees on all 101 coefficients with radii less than $6.02\times10^{-37}$. The explicit bound $(12T+14)e^{-3T/2}$ for the two omitted tails is included with $T=60$; the attached contour-bound note derives it.

The checks verify the first four closed forms and every reciprocal coefficient identity through order 100, both in exact rational-polynomial arithmetic and on the written intervals. Six Taylor reconstructions are compared with the finite cotangent definition using an explicit series remainder. Ingredient values agree with 101 Bernoulli numbers, 20 even positive zeta values and 31 cotangent sums. Known-answer and deliberate-error controls precede the comparisons. The generator, checks, argument-selection note and numerical evidence are attached.