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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 $|C_\alpha(a)|$.

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

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 approximate each generalized Stieltjes constant γ n ( a ) by means of a finite sum involving Bernoulli numbers. Such an approximation has a quasi-geometric rate of convergence, which improves as Re( a ) increases. A more detailed analysis, including numerical computations, is carried out for the constants γ 0 (1) and γ 1 (1). The key point in the proof is a probabilistic representation of the aforementioned constants, obtained as a consequence of a differential calculus concerning the gamma process.

We present expressions in terms of a double infinite series for the Stieltjes constants $\gamma_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case $\gamma_k(1)=\gamma$ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function., Comment: 12 pages, no figures, updated and typos corrected; to appear in Analysis

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γ k ( a ) \gamma _k(a) . These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials L n α L_n^\alpha to investigate a certain sum S γ ( n ) S_\gamma (n) involving the constants γ k ( 1 ) \gamma _k(1) that appears in application of the Li criterion for the Riemann hypothesis. We confirm the sublinear growth of S γ ( n ) + n S_\gamma (n)+n , which is consistent with the validity of the Riemann hypothesis.

The Stieltjes constants $\gamma_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler constant $\gamma=\gamma_0(1)$ to show its application to finding series representations for these constants. Other generalizations of representations of $\gamma$ are given., Comment: 21 pages, no figures

A simple class of algorithms for the efficient computation of the Hurwitz zeta and related special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical examples are briefly described. The algorithms are easy to implement and compete with Euler–Maclaurin summation-based methods.

The Stieltjes constants have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function about . We obtain novel single and double summatory relations for , including single summation relations for and , where a and b are real and p and q are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants . Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present the evaluation of $\gamma_1(a)$ and $\gamma_2(a)$ at rational argument, being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $\gamma_0(a)$, $\gamma_1(a)$, and $\gamma_2(a)$, and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of $\gamma_k(a)$ as $a \to 0$ as well as a novel technique for evaluating integrals with integrands with $\ln(-\ln x)$ and rational factors., Comment: 30 pages, no figures

The Stieltjes constants \gamma_k(a) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \zeta(s,a) about s=1. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for \gamma_k=\gamma_k(1). Some extensions are briefly described, as well as the relevance to expansions of Dirichlet L functions., Comment: 37 pages, no figures New material added at end, including Corollary 6

The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and computational., Comment: 39 pages, no figures Prop. 8 strengthened

The Stieltjes constants are the expansion coefficients in the Laurent series for the Hurwitz zeta function about s=1. We present new asymptotic, summatory, and other exact expressions for these and related constants., Comment: to appear in J. Math. Anal. Appl., 17 pages, no figures