Asymptotic coefficients of the Bettin–Conrey reciprocity function

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Numbers

$\ell$

$\widetilde C_\ell$

3.477368133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411

8.466570668896115525965796920167136743659285824356381609169052757289367087940845871222602368370547159

20.62455540221535487540737478097638825725786071939035940062336371568195699871289338172119097340702075

53.74544910044759888121191505768655754945466389246181651701545169792200455597278487467424394825459064

148.3993694250711456536662913673722773931503432308708453650922215836752850654409422270059991066572730

430.4231054127570447389859225912491353162941064455833785382683482579767272350808795243629682857464310

1300.528896155676913259031425430795192869567979608207433114963443603493372202027233960526695331670035

4084.702603319348064396973263172021423792202833413368477117213007006143764036348886426298558829342701

13293.87602901434282810910933950004989422169119709253015300943485819172122319455290641439441143488746

10:

44679.44445209355772202151729076415592784777366094388337241163127148471327020939933505176269995285359

11:

154711.2169125178896279935764319258982361327724849193421260893840256934760728257596422096749424337974

12:

550950.1892790863003734793379553186685533345664895110014949876656807366560527671178178210349283030236

13:

2014292.390037458904303570790744837511514129652097111833641076603000810792052265077248325959223000393

14:

7548156.443618963420729747707518076240501813145266000138113334260037633197058925005075289285789770288

15:

28953637.49851269565232468653163890691456340723018711895442181134468660027625981612018500743160073558

16:

113550218.1224931476893367063291745613407340033277293306631435504351991532735743947597796248816199629

17:

454812713.9963562585927651184795215044979424447321721646921713557533118152282028429331921124937791627

18:

1858649862.638561007467522858484107714389358932885556819128363440702663438632668883408514698545907307

19:

7742993654.086654068961581876486088150030230502644309329065339655048965627303508217975089800253017406

20:

32855445734.74011922653820181384906720354065183389230972231670589918596592913548060939845457182545094

21:

141899340885.9982402256685738662511454692305566454886461613069738609535328052649273586788792751565315

22:

623331813876.5695081826631537905174777021232455954376422039711382547237413401524931633588197728446627

23:

2783314682340.247329209854019159573476682449403948691616371003593750589670817271772958964916428844753

24:

12625332375848.08997282124726601980946507146091501139735761526089419643320725681631429027410957466649

25:

58148886886501.98885133905752773826077426698932288140499583899964361735370708541943260564652160592911

26:

271780340890796.0337022301113780681447257235151981618922808559934657294557935347277176787426879995056

27:

1288525638734749.375538327922389603338069598803305555124024185197643510984431188733139847940156216280

28:

6193577858452513.525393628738120364066845509780357968397213339035349858225033881148371318762576456736

29:

30173680777003622.09927401777113324111994255802124520988986361751001643397819353653658937578683712294

30:

148909224137996263.9462502549438224555934881318178146362796957404704451477286680522303602477268756657

31:

744303845469545376.3728731515676358669095008092894511594367807055071225761622033335394449329825861190

32:

3765610361193107083.374741343702377095482659163441676273530360189280304553841140083638301012457931438

33:

19285297364181214648.24198121516079893522437520805642951658317969276678658132675249533346125995781177

34:

99884034757512948278.93854723864223437824810970578798182055799200089320600205543019834618797254404866

35:

523567012397450607462.1155884654820419005590181010321245427845439563383477789112422431932312711830577

36:

2772105446088976323895.735462023436820673408253590968702478012700117898623148009044193461194905333734

37:

14861643275604085160518.16293870716824599369257127156550701965499116183428212086398632927948244085409

38:

80296160812599916478776.39511182537044132154508456613064531684420965748182271215685074757503459724053

39:

440403107151006672551592.3155517175233312661024662918478242348969124483609211321392645783766557543137

40:

2420189418114272537346433.647953238471326053751497789821818066660628072176707667139568697830366027964

41:

13624362515349780314314670.56881308710199872849986518204995987092022810287269625905178541900289249958

42:

75499374223705621938239922.55963317664569200722318136717983004753501073987817837879895083848821032225

43:

442143461084679414955675582.3477073706522547440893146121386725538023562303191241083885539555594354209

44:

2401405413844418006756005461.866935343296673230509471889823460743227695075745942368774406727193424503

45:

15383249304479079527724018550.03907386987653724135415691400097378335214205630680414249163464410063337

46:

73500107709290030417395855217.04065358640347575446986127677575776881318592271189487670862841707442430

47:

619270434640016412583701706477.6365263822769754773366509718777065236921348456041482203849601658775219

48:

1521981080977051377001973972071.483551476840219930800748001958527999102201699018932516506366436987576

49:

34737706519732800277796420945711.38575549886738457477285051356355889512001645898037907619611549765935

50:

-100286153373538137863312130981931.0120879855347380050457102755468692034400035370578845773392609999866

51:

3225662493665835752913092908344438.105103618209680498986085411985044407210959210125810096509373932934

52:

-28601408388108147208377340268357527.65550323325499535995425928954977761424826268599577566534674089879

53:

446094548445548735645240818174104920.6447813023298131887670728303940652083788829732750332352850387885

54:

-5395014888101216406911215052991459388.596971657664745005815479801506215471891191373438960622547164153

55:

75473957051718362636838367009878046257.96396601490095969233862219527344354082310405413142111189334906

56:

-1012736079200431411783196618760957423749.215149613627881630921224257859995452364430408331902863161561

57:

14230138204159051533962748373277145664264.22190077634650143940091453652460673166429318255619796879164

58:

-200962855258061643506720311639452263097375.2237372944812808685588981875312737350267348398344215633568

59:

2905305685729364594133346961195491595843733.608526086079498204098340658418823026404748529720791096551

60:

-42621254774635596885825721316314770301616524.56741792742976387562603423999746640242384285538516118718

Definition

The numbers $\widetilde C_\ell$ are the signed coefficients specified by (4) in the fixed-truncation asymptotic expansion (1) of $g_n-1/n$ [1].

Parameters

$\ell$

— asymptotic correction order ($\ell\geq0$)

Formulas

(1)

$\displaystyle g_n-\frac1n=2^{9/4}\pi^{3/4}e^{-2\sqrt{\pi n}}\sum_{\ell=0}^{L}(2\pi)^{-\ell/2}\widetilde C_\ell n^{-\ell/2-3/4}\sin\!\left(2\sqrt{\pi n}+\frac{\pi\ell}{4}+\frac{3\pi}{8}\right)+O_L\!\left(n^{-L/2-5/4}e^{-2\sqrt{\pi n}}\right)$ as $n\to\infty$ for every fixed integer $L\geq1$, where $g_n$ is the normalized Taylor coefficient [1], Theorem 1.1.

(2)

$\displaystyle \lambda(u)=\frac1{e^u-1}-\frac1u+\frac12=\sum_{r\geq1}\frac{B_{r+1}}{(r+1)!}u^r$, and $\displaystyle e^{-z\lambda(u)}=\sum_{k\geq0}P_k(z)u^k$ for $|u|<2\pi$. The singularity of $\lambda$ at zero is removable. Here $B_r$ are the Bernoulli numbers in the convention $u/(e^u-1)=\sum_{r\geq0}B_ru^r/r!$ [1], equation (5.1).

(3)

$\displaystyle H(\nu,j)=\frac1{4^j j!}\prod_{r=1}^{j}\left(4\nu^2-(2r-1)^2\right)=2^j a_j(\nu)$ for $j\geq0$, with the empty product equal to 1, where $a_j$ are the Hankel asymptotic coefficient polynomials [2].

(4)

$\displaystyle \widetilde C_\ell=\sum_{k=0}^{\ell}\frac{H(k+1,\ell-k)}{4^{\ell-k}}(2\pi i)^kP_k(2\pi i)$ [1], equation (5.2). Here $i^2=-1$, and the parity of $P_k$ makes every summand real.

(5)

$\widetilde C_0=1$, $\widetilde C_1=\pi^2/3+3/16$, and $\widetilde C_2=\pi^4/18+5\pi^2/16-15/512$ [1], Remark 1.2(2).

Comments

(6)

For coprime positive integers $h,k$, let $c_0(h/k)=-\sum_{a=1}^{k-1}(a/k)\cot(\pi ah/k)$ be the Bettin–Conrey cotangent sum, with the empty sum equal to zero. On positive rationals, $g(x)=xc_0(x)+c_0(1/x)-1/(\pi\operatorname{Den}(x))$, where $\operatorname{Den}(x)$ is the positive reduced denominator. The function extends holomorphically to $\mathbb C\setminus(-\infty,0]$, and $g(1+z)=\pi^{-1}\sum_{n\geq0}(-1)^n g_nz^n$ for $|z|<1$ [1], equations (1.1)–(1.2). The normalized Taylor coefficients $g_n$ are recorded in the companion Taylor-coefficient table.

(7)

The constant $\pi$ is positive. The amplitude is $2^{9/4}\pi^{3/4}$, including the correction recorded by [1] to the earlier leading-term formula. The expansion is interpreted with the truncation order $L$ fixed while $n$ tends to infinity; it is not a claim of convergence as $L$ tends to infinity.

(8)

The coefficients retain their signs. The first negative coefficient is $\widetilde C_{50}$ [1], Remark 1.2(3).

(9)

Writing $\widetilde C_\ell=C_\ell(\pi^2)$, the polynomial $C_\ell(X)\in\mathbb Q[X]$ has degree $\ell$ and leading coefficient $1/(3^\ell\ell!)$. Therefore $\widetilde C_0=1$ is exact, whereas $\widetilde C_\ell$ is transcendental for $\ell\geq1$. The auxiliary exact polynomials are included in the reproducibility files.

(10)

The derivation in [1], §§4–5, uses $U(n,0,2\pi i)$, where $U$ is Tricomi’s confluent hypergeometric function on its principal branch. That companion table contains positive real arguments, not these complex arguments.

Programs

(P1)

Sage

# With the attached generate.py beside this script:
from generate import coefficient_polynomials, evaluate
ell = 61
print(evaluate(coefficient_polynomials(ell)[ell], digits=100))

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 10.17.1: Hankel asymptotic coefficient normalization

Similar tables

Taylor coefficients of the Bettin–Conrey reciprocity function — The sequence $g_n$ has the large-order expansion whose signed correction coefficients are recorded here.

Bettin–Conrey cotangent sums — The reciprocity defect of these sums defines the function $g$.

Hankel asymptotic coefficient polynomials — The conversion $H(\nu,j)=2^j a_j(\nu)$ supplies the factors in the finite coefficient formula.

Bernoulli numbers — The numbers $B_{r+1}$ determine the auxiliary series $\lambda(u)$.

Values of Tricomi’s confluent hypergeometric function — The asymptotic derivation uses the same function at $U(n,0,2\pi i)$; this companion’s stored arguments are positive real numbers.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every correction order $0\leq\ell\leq60$ in increasing order, including all orders before the first negative coefficient at 50 and ten further orders. This consecutive prefix supports successive truncations and comparisons of the sign transition. Sixty is a practical, noncanonical cutoff; the centre, order-zero cotangent sum and normalization are fixed parts of the sequence, and there are no sampled continuous parameters. The attached argument-selection note explains the coverage)

How they were obtained:

Each value is evaluated as an exact rational polynomial in $\pi^2$, using Arb real balls at 525 working bits. The smallest retained relative accuracy is 517 bits; the widest cancellation ratio corresponds to less than 0.50 decimal digits. Every resulting enclosure lies strictly inside its written 100-significant-digit interval. The leading coefficient $\widetilde C_0=1$ is stored exactly.

A separate reciprocal-series and exponential-convolution construction agrees with all 61 exact polynomials. Its 700-bit evaluations retain at least 692 relative accuracy bits and lie within the same written intervals. The six printed coefficients in [1], Remark 1.2(2), agree exactly. The degree, leading term, constant term and signs were checked across the whole selection. All 60 adjacent formal identities from Kummer’s differential equation also hold.

The ingredient checks compare 62 Bernoulli numbers and 1,071 specializations of the Hankel coefficient polynomials. Direct 1,200-bit complex Tricomi evaluations check the amplitude, phase and truncation behavior at $n=25,100,400,1600$. Known-answer controls and deliberately changed coefficients test the checking methods. These experiments do not provide a general remainder bound for a truncated asymptotic expansion; the coefficient enclosures come from the finite formula. The attached generator, independent checks, derivation, argument-selection note and numerical evidence describe the methods and the localized typographical inconsistency in the statement of [1], Corollary 5.6.