Search

Let f be an arithmetic function and let $${\mathcal {S}}^\#$$ S # denote the extended Selberg class. We denote by $${\mathcal {L}}(s) = \sum _{n = 1}^{\infty }\frac{f(n)}{n^s}$$ L ( s ) = ∑ n = 1 ∞ f ( n ) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of $${\mathcal {L}}(s)$$ L ( s ) , which belongs to $${\mathcal {S}}^\#$$ S # , are the coefficients of the Laurent expansion of $${\mathcal {L}}$$ L at its pole $$s=1$$ s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results.

We study the distribution of values of the Riemann zeta function $\zeta(s)$ on vertical lines $\Re s + i \mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $\zeta(s)$ has a Fourier expansion in the half-plane $\Re s \geq 1/2$ and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of $\zeta(s) - s/(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindel\"{o}f hypotheses on the growth of the Fourier coefficients., Comment: 21 pages

We discuss evaluating fractional Stieltjes constants $\gamma_{\alpha}(a)$, arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions $\zeta^{(\alpha)}(s,a)$. We give an upper bound for the absolute value of $C_\alpha(a)=\gamma_\alpha(a)-\log^\alpha(a)/a$ and an asymptotic formula $\widetilde{C}_{\alpha}(a)$ for $C_{\alpha}(a)$ that yields a good approximation even for most small values of $\alpha$. We bound $|\widetilde{C}_{\alpha}(a)|$ and based on this we conjecture a tighter bound for $|C_\alpha(a)|$.

We study the non-integral generalized Stieltjes constants γ α ( a ) arising from the Laurent series expansions of fractional derivatives of the Hurwitz zeta functions ζ ( α ) ( s , a ) , and we prove that if h a ( s ) ≔ ζ ( s , a ) − 1 ∕ ( s − 1 ) − 1 ∕ a s and C α ( a ) ≔ γ α ( a ) − log α ( a ) a , then C α ( a ) = ( − 1 ) − α h a ( α ) ( 1 ) , for all real α ≥ 0 , where h ( α ) ( x ) denotes the α -th Grunwald–Letnikov fractional derivative of the function h at x . This result confirms the conjecture of Kreminski (2003), originally stated in terms of the Weyl fractional derivatives.

In this article, new series for the first and second Stieltjes constants (also known as generalized Euler’s constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as the (reciprocal) logarithmic numbers, the Cauchy numbers of the first kind and the Bernoulli numbers of the second kind. In addition, two interesting series with rational terms for Euler’s constant $$\gamma $$ and the constant $$\ln 2\pi $$ are given, and yet another generalization of Euler’s constant is proposed and various formulas for the calculation of these constants are obtained. Finally, we mention in the paper that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

The connection between derivatives of $L(s,f)$ for periodic arithmetical functions $f$ at $s=1$ and generalized Stieltjes constants has been noted earlier. In this paper, we utilize this link to throw light on the arithmetic nature of $L'(1,f)$ and certain Stieltjes constants. In particular, if $p$ is an odd prime greater than $7$, then we deduce the transcendence of at least $(p-7)/2$ of the generalized Stieltjes constants, $\{ \gamma_1(a,p) : 1 \leq a < p \}$, conditional on a conjecture of S. Gun, M. R. Murty and P. Rath., Comment: 11 pages

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants $\gamma_k(\chi)$ for a principal character $\chi.$ In particular, we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant $\gamma_1(r/q)$ which was given by Blagouchine in 2015., Comment: 27 pages

In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series whose coefficients are obtained by a regularisation process, similar to the one used in defining the classical Stieltjes constants for the Riemann zeta function. We therefore call these coefficients {\it multiple Stieltjes constants}. The remaining part of the above mentioned Laurent type expansion is then expressed in terms of the multiple Stieltjes constants arising in smaller depths., Comment: This work was carried out in 2017-18, in Institut de Math\'ematiques de Jussieu, with support from IRSES Moduli and LIA

Let [Formula: see text] be a number field of a finite degree and let [Formula: see text] be the Rankin–Selberg [Formula: see text]-function associated to unitary cuspidal automorphic representations [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. The main result of the paper is an asymptotic formula for evaluation of coefficients appearing in the Laurent (Taylor) series expansion of the logarithmic derivative of the function [Formula: see text] at [Formula: see text]. As a corollary, we derive orthogonality and weighted orthogonality relations.

We introduce a probabilistic perspective to the problem of accelerating the convergence of a wide class of series, paying special attention to the computation of the coefficients, preferably in a recursive way. This approach is mainly based on a differentiation formula for the negative binomial process which extends the classical Euler’s transformation. We illustrate the method by providing fast computations of the logarithm and the alternating zeta functions, as well as various real constants expressed as sums of series, such as Catalan, Stieltjes, and Euler–Mascheroni constants.

Let E be Galois extension of Q of finite degree and let π and π' be two irreducible automorphic unitary cuspidal representations of GLm(EA) and GLm'(EA), respectively. We prove an asymptotic formula for computation of coefficients γπ,π'(k) in the Laurent (Taylor) series expansion around s=1 of the logarithmic derivative of the Rankin-Selberg L-function L(s, π × π') under assumption that at least one of representations π, π' is self-contragredient and show that coefficients γπ,π'(k) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π' is self-contragredient by a weaker one.

In this paper, the following generalized multiple fractional part integrals: In,μα1,α2,…,αn(a)=∫01∫01⋯∫01x1α1x2α2⋯xnαn1ax1x2⋯xnμ×dx1dx2⋯dxn, and k1,k2,…,knIn,μα1,α2,…,αn(a)=∫01∫01⋯∫011ax1x2⋯xnμ×∏i=1nxiαilnki⁡1xidx1dx2⋯dxn, are considered for complex numbers μ,αi (i=1,2,3,…,n), positive integers ki (i=1,2,…,n) and positive number a, where {u} denotes the fractional part of u. Furthermore, In,μα1,α2,…,αn(a) and k1,k2,…,knIn,μα1,α2,…,αn(a) can be transformed into the calculation of definite fractional part integral Iμα(a) and kIμα(a), which overcomes the shortcomings of low speed and poor accuracy of the multiple case. Moreover, some identities and recursive formulas of the above integrals are obtained.

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π − 2 with rational coefficients and converges slightly better than Euler's series ∑ n − 2 . The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A , the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

International audience; The generalized Stieltjes constants $\gamma_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order~$n$. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute $\gamma_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma_n(v)$, $\zeta(s)$, $\zeta(s,v)$, some polygamma functions and the Lerch transcendent.

We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm, Dirichlet $L$- and Clausen functions. These results then enable a variety of Addison-type series representations of functions. Moreover, we obtain integral and Addison-type series for a variety of mathematical constants., 36 pages, no figures

Thus, the Riemann hypothesis is an assertion concerning the infinite sequence of coefficients γ0, γ1, . . . known as Stieltjes constants (γ0 = γ = 0.577215 . . . is the Euler constant). As is well known, to prove the Riemann hypothesis, it would be sufficient to establish the validity of an appropriate infinite system of polynomial inequalities P1(γ0, . . . , γm1) > 0, . . . , Pn(γ0, . . . , γmn) > 0, . . . , (2) each of which contains only finitely many Stieltjes constants (and possibly also some classical constants) and hence admits a numerical verification. Polynomials Pn giving this reformulation of the Riemann hypothesis can be chosen in many ways. In one of the most well-known ways described in [1] (see also [2]), one has

We derive telescoping majorants and minorants for some classes of series and give applications of these results. P ∞ k=n k −s , s > 1. The present article is dedicated to the question which se- ries can be treated in a similar way. In the following we shall show that this is the case for a big class of series P ∞=n f(k). They have to satisfy only certain mild smoothness conditions on the function f. The proofs are based on the comparison of f(k) and Z k+1−c k−c f(x) dx, c 2 (0,1), that may be regarded as special cases of theorems from the theory of numer- ical integration. We will demonstrate the usefulness of this method by some applications. Among them there will be a generalization of the Stieltjes constants and an elementary proof of Stirling's formula.

The Stieltjes constants $\gamma_{n}(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $\zeta_{K}(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $\gamma_{n}(K)$ as Stieltjes obtained in 1885 for $\gamma_{n}(\mathbf{Q})$. We also study the signs of $\gamma_{n}(K)$.

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function $x\mapsto \ln \ln(1/x)$.

The Laurent-Stieltjes constants $\gamma_{n}(\chi)$ are, up to a trivial coefficient, the coefficients of the Laurent expansion of the usual Dirichlet $L$-series: when $\chi$ is non-principal, $(-1)^{n}\gamma_{n}(\chi)$ is simply the value of the $n$-th derivative of $L(s,\chi)$ at $s=1$. In this paper, we give an approximation of the Dirichlet $L$-functions in the neighborhood of $s=1$ by a short Taylor polynomial. We also prove that the Riemann zeta function $\zeta(s)$ has no zeros in the region $|s-1|\leq 2.2093$, with $0\leq \Re{(s)}\leq 1$. This work is a continuation of [24].

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz zeta function. As a concrete application, a superzeta function enters an integral representation for the KeiperLi coefficients, whose large-order behavior thereby becomes computable by the method of steepest descents; then the dominant saddle-point entirely depends on the Riemann Hypothesis being true or not, and the outcome is a sharp exponential-asymptotic criterion for the Riemann Hypothesis that only refers to the large-order KeiperLi coefficients. As a new result, that criterion, then Li' s criterion, are transposed to a novel sequence of Riemann-zeta expansion coefficients based at the point 1/2 (vs 1 for KeiperLi)., "Exponential Analysis of Differential Equations and Related Topics". October 15～18, 2013. edited by Yoshitsugu Takei. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This article is devoted to a family of logarithmic integrals recently treated in mathematical literature, as well as to some closely related results. First, it is shown that the problem is much older than usually reported. In particular, the so-called Vardi’s integral, which is a particular case of the considered family of integrals, was first evaluated by Carl Malmsten and colleagues in 1842. Then, it is shown that under some conditions, the contour integration method may be successfully used for the evaluation of these integrals (they are called Malmsten’s integrals). Unlike most modern methods, the proposed one does not require “heavy” special functions and is based solely on the Euler’s Γ-function. A straightforward extension to an arctangent family of integrals is treated as well. Some integrals containing polygamma functions are also evaluated by a slight modification of the proposed method. Malmsten’s integrals usually depend on several parameters including discrete ones. It is shown that Malmsten’s integrals of a discrete real parameter may be represented by a kind of finite Fourier series whose coefficients are given in terms of the Γ-function and its logarithmic derivatives. By studying such orthogonal expansions, several interesting theorems concerning the values of the Γ-function at rational arguments are proven. In contrast, Malmsten’s integrals of a continuous complex parameter are found to be connected with the generalized Stieltjes constants. This connection reveals to be useful for the determination of the first generalized Stieltjes constant at seven rational arguments in the range (0,1) by means of elementary functions, the Euler’s constant γ, the first Stieltjes constant γ 1 and the Γ-function. However, it is not known if any first generalized Stieltjes constant at rational argument may be expressed in the same way. Useful in this regard, the multiplication theorem, the recurrence relationship and the reflection formula for the Stieltjes constants are provided as well. A part of the manuscript is devoted to certain logarithmic and trigonometric series related to Malmsten’s integrals. It is shown that comparatively simple logarithmico–trigonometric series may be evaluated either via the Γ-function and its logarithmic derivatives, or via the derivatives of the Hurwitz ζ-function, or via the antiderivative of the first generalized Stieltjes constant. In passing, it is found that the authorship of the Fourier series expansion for the logarithm of the Γ-function is attributed to Ernst Kummer erroneously: Malmsten and colleagues derived this expansion already in 1842, while Kummer obtained it only in 1847. Interestingly, a similar Fourier series with the cosine instead of the sine leads to the second-order derivatives of the Hurwitz ζ-function and to the antiderivatives of the first generalized Stieltjes constant. Finally, several errors and misprints related to logarithmic and arctangent integrals were found in the famous Gradshteyn & Ryzhik’s table of integrals as well as in the Prudnikov et al. tables.

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form., Comment: 17 pages, no figures

We prove that if $f:(0,\infty)\to\mathbb{R}$ is a completely monotonic function then the generalized Euler sequence $$(a_n)_{n\ge1},\quad a_n=f(1)+\cdots+f(n)-\int _1^n f(t)\,dt $$ is completely monotonic, and under appropriate conditions on f we obtain an explicit formula for its limit. Some particular cases of Stieltjes constants are studied. We also give representations for the sum of some generalized harmonic series.