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{"Title": "Asymptotic coefficients of the Bettin\u2013Conrey reciprocity function", "Definition": "The numbers $\\widetilde C_\\ell$ are the signed coefficients specified by CITE{formula-coefficient} in the fixed-truncation asymptotic expansion CITE{formula-asymptotic} of $g_n-1/n$ CITE{AM}.", "Parameters": {"ell": {"type": "Z", "display": "$\\ell$", "title": "asymptotic correction order", "constraints": "$\\ell\\geq0$"}}, "References": {"AM": {"bib": "H. Akatsuka and Y. Murakami, An asymptotic property on a reciprocity law for the Bettin\u2013Conrey cotangent sum, International Journal of Number Theory 21 (2025), 2233\u20132253. Equation numbers refer to the 2024 preprint, version 1.", "arxiv": "2402.14216v1", "doi": "10.1142/S179304212550109X"}}, "Tags": ["number theory", "special values", "sequence"], "Links": {"DLMF": {"title": "DLMF 10.17.1: Hankel asymptotic coefficient normalization", "url": "https://dlmf.nist.gov/10.17.E1"}}, "Data properties": {"type": "R", "rigour": "proven", "complete": "no", "complete-note": "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", "rigour details": "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.\n\nA 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 CITE{AM}, 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\u2019s differential equation also hold.\n\nThe ingredient checks compare 62 HREF{Bernoulli_numbers}[Bernoulli numbers] and 1,071 specializations of the HREF{Hankel_asymptotic_coefficient_polynomials}[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 CITE{AM}, Corollary 5.6."}, "Display properties": {"number-header": "$\\widetilde C_\\ell$"}, "Comments": {"reciprocity-function": "For coprime positive integers $h,k$, let $c_0(h/k)=-\\sum_{a=1}^{k-1}(a/k)\\cot(\\pi ah/k)$ be the HREF{Bettin_Conrey_cotangent_sums}[Bettin\u2013Conrey 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 $|z|<1$ CITE{AM}, equations (1.1)\u2013(1.2). The normalized Taylor coefficients $g_n$ are recorded in HREF{T692}[the companion Taylor-coefficient table].", "conventions": "The constant $\\pi$ is positive. The amplitude is $2^{9/4}\\pi^{3/4}$, including the correction recorded by CITE{AM} 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.", "signs": "The coefficients retain their signs. The first negative coefficient is $\\widetilde C_{50}$ CITE{AM}, Remark 1.2(3).", "exactness": "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.", "tricomi": "The derivation in CITE{AM}, \u00a7\u00a74\u20135, uses $U(n,0,2\\pi i)$, where $U$ is HREF{Values_of_Tricomi's_confluent_hypergeometric_function}[Tricomi\u2019s confluent hypergeometric function] on its principal branch. That companion table contains positive real arguments, not these complex arguments."}, "Formulas": {"formula-asymptotic": "$\\displaystyle g_n-\\frac1n=2^{9/4}\\pi^{3/4}e^{-2\\sqrt{\\pi n}}\\sum_{\\ell=0}^{L}(2\\pi)^{-\\ell/2}\\widetilde C_\\ell n^{-\\ell/2-3/4}\\sin\\!\\left(2\\sqrt{\\pi n}+\\frac{\\pi\\ell}{4}+\\frac{3\\pi}{8}\\right)+O_L\\!\\left(n^{-L/2-5/4}e^{-2\\sqrt{\\pi n}}\\right)$ as $n\\to\\infty$ for every fixed integer $L\\geq1$, where $g_n$ is the HREF{T692}[normalized Taylor coefficient] CITE{AM}, Theorem 1.1.", "formula-generating": "$\\displaystyle \\lambda(u)=\\frac1{e^u-1}-\\frac1u+\\frac12=\\sum_{r\\geq1}\\frac{B_{r+1}}{(r+1)!}u^r$, 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 HREF{Bernoulli_numbers}[Bernoulli numbers] in the convention $u/(e^u-1)=\\sum_{r\\geq0}B_ru^r/r!$ CITE{AM}, equation (5.1).", "formula-hankel": "$\\displaystyle H(\\nu,j)=\\frac1{4^j j!}\\prod_{r=1}^{j}\\left(4\\nu^2-(2r-1)^2\\right)=2^j a_j(\\nu)$ for $j\\geq0$, with the empty product equal to 1, where $a_j$ are the HREF{Hankel_asymptotic_coefficient_polynomials}[Hankel asymptotic coefficient polynomials] CITE{DLMF}.", "formula-coefficient": "$\\displaystyle \\widetilde C_\\ell=\\sum_{k=0}^{\\ell}\\frac{H(k+1,\\ell-k)}{4^{\\ell-k}}(2\\pi i)^kP_k(2\\pi i)$ CITE{AM}, equation (5.2). Here $i^2=-1$, and the parity of $P_k$ makes every summand real.", "formula-first": "$\\widetilde C_0=1$, $\\widetilde C_1=\\pi^2/3+3/16$, and $\\widetilde C_2=\\pi^4/18+5\\pi^2/16-15/512$ CITE{AM}, Remark 1.2(2)."}, "Similar tables": [{"table": "HREF{T692}[Taylor coefficients of the Bettin\u2013Conrey reciprocity function]", "relation": "The sequence $g_n$ has the large-order expansion whose signed correction coefficients are recorded here."}, {"table": "HREF{Bettin_Conrey_cotangent_sums}[Bettin\u2013Conrey cotangent sums]", "relation": "The reciprocity defect of these sums defines the function $g$."}, {"table": "HREF{Hankel_asymptotic_coefficient_polynomials}[Hankel asymptotic coefficient polynomials]", "relation": "The conversion $H(\\nu,j)=2^j a_j(\\nu)$ supplies the factors in the finite coefficient formula."}, {"table": "HREF{Bernoulli_numbers}[Bernoulli numbers]", "relation": "The numbers $B_{r+1}$ determine the auxiliary series $\\lambda(u)$."}, {"table": "HREF{Values_of_Tricomi's_confluent_hypergeometric_function}[Values of Tricomi\u2019s confluent hypergeometric function]", "relation": "The asymptotic derivation uses the same function at $U(n,0,2\\pi i)$; this companion\u2019s stored arguments are positive real numbers."}], "Keywords": ["Bettin-Conrey", "cotangent reciprocity", "asymptotic coefficients", "large-order Taylor asymptotics", "signed correction coefficients"], "Programs": {"program-sage": {"language": "Sage", "code": "# With the attached generate.py beside this script:\nfrom generate import coefficient_polynomials, evaluate\ncoefficients = coefficient_polynomials(60)\nvalues = [evaluate(p, digits=100) for p in coefficients]\n"}}, "Numbers": {"0": "1", "1": "3.477368133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411", "2": "8.466570668896115525965796920167136743659285824356381609169052757289367087940845871222602368370547159", "3": 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