On Validated Numeric of Values of Some Zeta and L-functions, and Applications
Nikolaj M. Glazunov
本文研究了黎曼zeta函数、狄利克雷L-函数、戴德金zeta函数和哈塞-韦尔L-函数的验证数值计算及其应用。
本文描述了黎曼zeta函数、狄利克雷L-函数、戴德金zeta函数和哈塞-韦尔L-函数的一些验证数值方面。
This paper describes some validated numerics aspects of Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions and Hasse-Weil L-functions.
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The distribution of the zeros of the Euler double zeta-function $ζ_2(s_1,s_2)$, in the case when $s_1=s_2$, is studied numerically. Some similarity to the distribution of the zeros of Hurwitz zeta-functions is observed.
The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.
The paper describes a method for calculating values of Riemann's Zeta function within the critical strip 0< σ <1 and on its boundary. The approach is based on the "Alternating Zeta function" η(s). The actual Riemann Zeta function ζ(s) is easily obtained from η(s). The obtained accuracy, within certain limits of the described method is discussed. A decent scientific calculator suffices to carry out the involved computations.
We derive an explicit expression for an inverse power series over the gaps values of numerical semigroups generated by two integers. It implies a set of new identities for the Hurwitz zeta function.
Simple and accurate approximations to the Riemann zeta function
本文提出了一种简单且准确的Riemannzi函数近似方法，结合Riemann-Siegel公式的主要求和部分和一个仅包含初等函数和预计算系数的余项近似，实现了在临界带和其他垂直带内的高精度计算，同时提供了导数的近似方法和大量数值证据证明其准确性。
The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of Riemann's zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large scale high accuracy numerical experiments. So far no theoretical explanation to the observed phenomena was found.
New computations of the Riemann zeta function on the critical line
本文提出了一种快速算法用于计算黎曼zi函数的高零点，通过改进的二次指数和计算方法提高了效率。
We study the behavior of zero-divisors of the double zeta-function $ζ_2(s_1,s_2)$. In our former paper \cite{MatSho14} we studied the case $s_1=s_2$, but in the present paper we consider the more general two variable situation. We carry out numerical computations in order to trace the behavior of zero-divisors. We observe that some divisors approach the points with $s_2=0$, while other divisors approach the points which are solutions of $ζ(s_2)=1$.
In this article, we introduce a recurrence formula which only involves two adjacent values of the Riemann zeta function at integer arguments. Based on the formula, an algorithm to evaluate $ζ$-values(i.e. the values of Riemann zeta function) at odd-integers from the two nearest $ζ$-values at even-integers is posed and proved. The behavior of the error bound is $O(10^{-n})$ approximately where $n$ is the argument. Our method is especially powerful for the calculation of Riemann zeta function at large argument, while for smaller ones it can also reach spectacular accuracies such as more than ten…
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.
Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 10^9 zeros near zero number 10^23. More than anything else, the variability illustrates the limits of what one can learn about the zeta fu…
In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, $G_q$ ($q>=3$) is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given. We give numerical support for the claim that the method works and can be used to compute the Selberg Zeta function for $G_q$ to any desired precision. We also present some numerical results obtained by implementing the algorithm.
I use the numerical values of the generalised Glaisher-Kinkelin-Bendersky (GKB) constants to give numerical values for the derivatives of the Hurwitz zeta function at negative integers, rather than the other way round. I point out that both Glaisher's numerical approach and Bendersky's recursion for the generalised gamma function were anticipated by Jeffery in 1862 who gave the value of the second constant as an example. I therefore propose that GKB become GKBJ.
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function $ζ(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
We review motivic aspects of multiple zeta values, and as an application, we give an exact-numerical algorithm to decompose any (motivic) multiple zeta value of given weight into a chosen basis up to that weight.
We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup $Γ_0(4)$ as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results.
We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favour of the conjecture.
Computation of anisotropic singular sums from high-order derivatives of Epstein zeta functions
本文定义并分析各向异性Epstein zeta函数，推导其稳定可计算表示，用于高效精确计算各向异性幂律核晶格求和，并开发数值算法，在多种晶格、指数和阶数下达到机器精度。
The Zetafast algorithm for computing zeta functions
该算法通过三个绝对收敛级数表达黎曼ζ函数，能以高精度计算ζ(s)及其导数，适用于多种ζ函数类型。
We prove that the sum of multiple zeta-star values over all indices inserted two 2's into the string $(\underbrace{3,1, ..., 3,1}_{2n})$ is evaluated to a rational multiple of powers of $π^2$. We also establish certain conjectures on evaluations of multiple zeta-star values observed by numerical experiments.
Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.
In this article, we give a proof of the link between the zeta function of two families of hypergeometric curves and the zeta function of a family of quintics that was observed numerically by Candelas, de la Ossa, and Rodriguez Villegas. The method we use is based on formulas of Koblitz and various Gauss sums identities; it does not give any geometric information on the link.
In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function
本文通过数值证据探讨交替ζ函数及其导数之间的近似多项式依赖关系，并提出相关猜想。
The numerical data of an embedded resolution determine the candidate poles of Igusa's p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.
As well known, the study of Riemanns zeta function ζ(s) involves the related entire function ξ(s). A close relative of ζ(s) is the alternating zeta function η(s). Similar to ζ(s), also η(s) has a corresponding entire function ξ_a (s). After establishing its definition and a related functional equation, formulas based on incomplete gamma functions are worked out, allowing to numerically evaluate ξ_a (s). Examples verifying the obtained formulas are included.
The paper deals with the analytic entire function Chi(s) closely related to Riemann Zeta Function Zeta(s). A formula is obtained for Chi(s) essentially within the so-called critical strip. This is achieved by applying Cauchy integral formula to an infinite loop encircling the critical strip. In the obtained formula a remarkable role is played by a special type of complex function known as Incomplete Gamma Function. Numerical examples verifying the obtained formula are included.
On a Generalized Moment Integral containing Riemann's Zeta Function: Analysis and Experiment
本文研究了与黎曼ζ函数广义动量均值相关的积分Z(σ,r)，通过分析和数值方法验证了其导数函数与狄拉克梳函数的关系，并发现被积函数的周期性特征。
Analytical continuation of prime zeta function for $\Re(s)>\frac{1}{2}$ assuming (RH)
本文在RH假设下推导出素数ζ函数在Re(s)>1/2区域的解析延拓表达式，并通过数值验证和图表展示结果。
In [Ok] Okounkov studies a specific $q$-analogue of multiple zeta values and makes some conjectures on their algebraic structure. In this note we compare Okounkovs $q$-analogues to the generating function for multiple divisor sums defined in [BK1]. We also state a conjecture on their dimensions that complements Okounkovs conjectural formula and present some numerical evidences for it.
We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $Ψ$, a group cocycle for $GL_{n} (\Z )$; the special values are computed as periods of $Ψ$, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.
We describe a method to accelerate the numerical computation of the coefficients of the polynomials $P_k(x)$ that appear in the conjectured asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out our method to compute the moment polynomials for $k \leq 13$, and used these to experimentally test conjectures for the moments up to height $10^8$.
We consider sums of the form $\sum ϕ(γ)$, where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
A q-analogue of the Riemann zeta function was studied in [Kaneko et al. 03] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of the q-analogue left open in [Kaneko et al. 03]. We also examine a "crystal" limit (i.e. q->0) behavior of the q-analogue. The q-trajectories of the trivial and essential zeros of the Riemann zeta function are investigated numerically when q moves in (0,1]. Moreover, conjectures for the crystal limit behavior of zeros o…
Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and /psi_(1)(z)*/psi_(1)(-z). These series were used to find summation rules which include generalized harmonic numbers of first, second and third powers and values of the Riemann zeta-functions at integers / Bernoulli numbers, for example 2*Sum_(k-1)^(infinity)(H_(k)^((2))/k^3)=6*/zeta(2)*/zeta(3)-9*/zeta(5). Some of these rules were tested numerically.
Explicit conditional bounds for the residue of a Dedekind zeta-function at $s=1$
本文通过显式数值方法给出了Dedekind ζ函数在s=1处残差的条件界。
On the series expansion of the prime zeta function about $s=1$ and its coefficients
本文推导了素数ζ函数在s=1处的级数展开，并证明了其系数的通用公式，类似于黎曼ζ函数的Stieltjes系数。结果可视为对梅特恩斯定理的推广，并通过数值验证计算了高精度公式。
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.
We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions $V_z: s \mapsto ζ(s) \exp (zs)$. We observe convergence of $V_z$ fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.
We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function $ζ(s,a)$ for $s, a \in \mathbb{C}$, along with an arbitrary number of derivatives with respect to $s$, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.
This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we hav…
On the series expansion of the secondary zeta function about $s=1$ and its coefficients
本文研究二次ζ函数在s=1处的展开系数，推导新公式并验证其精度，同时应用Brent定理提升收敛性。
The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
We compute the complete set of candidates for the zeta function of a K3 surface over F_2 consistent with the Weil conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over F_2. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda-Tate theorem for transcendental zeta functions of K3 surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.
We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 \cite{Maslanka 1}. The crucial ingredient in this method is a sequence of high-precision numerical values of the Riemann zeta function computed in equally spaced real arguments, i.e. $ζ(1+\varepsilon),ζ(1+2\varepsilon),ζ(1+3\varepsilon),...$ where $\varepsilon$ is some real parameter. (Practical choice of $\varepsilon$ is…
We exhibit very small eigenvalues of the quadratic form associated to the Weil explicit formulas restricted to test functions whose support is within a fixed interval with upper bound S. We show both numerically and conceptually that the associated eigenvectors are obtained by a simple arithmetic operation of finite sum using prolate spheroidal wave functions associated to the scale S. Then we use these functions to condition the canonical spectral triple of the circle of length L=2 Log(S) in such a way that they belong to the kernel of the perturbed Dirac operator. We give numerical evidence …
The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.
On the distribution of the zeros of the Euler double zeta-function $ζ_2(s,s)$
本文研究Euler双zeta函数在对角线情形$\zeta_2(s,s)$的零点分布，用严格的理论证明改进了先前数值观察结果。
We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point of appropriate integrands. We provide simplified derivations of the RS formulas, containing finite series sums and residual integrals, for the Lerch and the Dirichlet $L$-functions. The MDE method allows us to remove the contribution of the singularities near the contour of integration for the residual integrals. The method allows us to evaluate the residua…
We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two multiplicative relations, which arise naturally from comparison of the MZVs with an underlying algebraic structure. We generalise these concepts by introducing the parameter $q$ in such a way that as $q\to 1^-$ we return to the ordinary MZVs. Our special interest lies in two $q$-models recently introduced by H. Bachmann. He further conjectures that the $\mathbb{Q}$-s…
Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain $2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree $k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first $k$ coefficients of the moment polynomial are derived. Nu…
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
Optimal Cosine Polynomials for Riemann Zeta Zero-Free Region
本文用数值方法确定次数为 7 和 8 的最优余弦多项式，并给出黎曼 zeta 函数相应的无零点区域，推进了 Landau 构造下最大无零点区域问题的研究。
In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …
On the series expansion of the secondary zeta function
本文研究次级 zeta 函数 $Z(s)$，通过基于素数 zeta 函数的新显式公式建立非平凡零点与素数的联系，并给出新级数展开、高精度系数计算及解析延拓算法。
Using a generalized Littlewood theorem concerning integrals of the logarithm of analytical functions, we have established a few equalities involving integrals of the logarithm of the Riemann Zeta-function and have rigorously proven that they are equivalent to the Riemann hypothesis. Separate consideration for imaginary and real parts of these equalities, which deal correspondingly with the integrals of the logarithm of the module of the Riemann function and with the integrals of its argument is given. Preliminary results of the numerical research performed using these equalities to test the Ri…
We introduce a $q$-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple $q$-zeta values satisfy a $q$-stuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple $q$-zeta values can be viewed as special values of the multiple $q$-polylogarithm, which admits a multiple Jackson $q$-integral representation whose limiting case is the Drinfel'd simplex integral for the ordinary multiple polylogarithm when $q=1$. The multiple Jackson $q$-integral r…
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple…
We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function zeta(s), this is expected to be an accurate description for the horizontal distribution of the zeros of zeta'(s) to the right of the critical line. We show that as N --> infinity the fraction of roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in …
In the present series of papers, we study the behavior of the r-fold zeta-function of Euler-Zagier type with identical arguments on the real line. In this first part, we consider the behavior on the interval [0,1]. Our basic tool is an "infinite" version of Newton's classical identities. We carry out numerical computations, and draw graphs for real s in [0,1], for several small values of r. Those graphs suggest various properties of the r-fold zeta-function, some of which we prove rigorously. For example, we show that the r-fold zeta-function has r asymptotes, and determine the asymptotic beha…
In this article, we develop a formula for an inverse Riemann zeta function such that for $w=ζ(s)$ we have $s=ζ^{-1}(w)$ for real and complex domains $s$ and $w$. The presented work is based on extending the analytical recurrence formulas for trivial and non-trivial zeros as to solve an equation $ζ(s)-w=0$ for a given $w$-domain using logarithmic differentiation and zeta recursive root extraction methods. We further explore formulas for trivial and non-trivial zeros of the Riemann zeta function in greater detail, and next, we also explore an expansion of the inverse zeta function by an attracto…
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