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We study the distribution of values of the Riemann zeta function ζ (s) on vertical lines ℜ s + i R , by using the theory of Hilbert space. We show among other things, that, ζ (s) has a Fourier expansion in the half-plane ℜ s ≥ 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 ζ (s) − s / (s − 1). Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients. For a video summary of this paper, please visit https://youtu.be/wI5fIJMeqp4. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 representations and Addison-type series for a variety of mathematical constants. [ABSTRACT FROM AUTHOR]

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the Γ-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of γ 1 ( 1 / 2 ) , γ 1 ( 1 / 3 ) , γ 1 ( 2 / 3 ) , γ 1 ( 1 / 4 ) , γ 1 ( 3 / 4 ) , γ 1 ( 1 / 6 ) , γ 1 ( 5 / 6 ) , which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulae involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the Γ-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form. [ABSTRACT FROM AUTHOR]

Abstract: The Stieltjes constants appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function about its only pole at . We generalize a technique of Addison for the Euler constant to show its application to finding series representations for these constants. Other generalizations of representations of γ are given. [Copyright &y& Elsevier]

Abstract: The Stieltjes constants are the expansion coefficients in the Laurent series for the Hurwitz zeta function about . We present new asymptotic, summatory, and other exact expressions for these and related constants. [Copyright &y& Elsevier]