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Many special functions and analytic constants allow for a probabilistic representation in terms of inverse moments of [0, 1]-valued random variables. Under this assumption, we give fast computations of them with an explicit upper bound for the remainder term. One of the main features of the method is that the coefficients of the main term of the approximation always contain negative binomial probabilities which, in turn, can be precomputed and stored. Applications to the arctangent function, Dirichlet functions and their nth derivatives, and the Catalan, Gompertz, and Stieltjes constants are provided. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

We present a new asymptotic formula for the Stieltjes constants which is both simpler and more accurate than several others published in the literature (see e.g. [1-3]). More importantly, it is also a good starting point for a detailed analysis of some surprising regularities in these important constants. [ABSTRACT FROM AUTHOR]

We present a simple but efficient method of calculating Stieltjes constants at a very high level of precision, up to about 80 000 significant digits. This method is based on the hypergeometric-like expansion for the Riemann zeta function presented by one of the authors in 1997 [19]. 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 + ε), ζ(1 + 2ε), ζ(1 + 3ε), ... where e is some real parameter. (Practical choice of e is described in the main text.) Such values of zeta may be readily obtained using the PARI/GP program, which is especially suitable for this. [ABSTRACT FROM AUTHOR]

We discuss evaluating fractional Stieltjes constants γα(a), arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions ζα)(s,a). We give an upper bound for the absolute value of Cα(a)=γα(a)-logα(a)/a and an asymptotic formula ˜Cα(a) for Cα(a) that yields a good approximation even for most small values of α. We bound |˜Cα(a)| and based on this we conjecture a tighter bound for |Cα(a)|. [ABSTRACT FROM AUTHOR]

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]

The generalized Stieltjes constants γn(v) are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function ζ(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 γn(1) to 1000 digits in a minute for any n up to n = 10100. We also provide other interesting integral representations for γn(v), ζ(s), ζ(s, v), some polygamma functions, and the Lerch transcendent. [ABSTRACT FROM AUTHOR]

The Stieltjes constants γn(K) of a number field K are the coefficients of the Laurent expansion of the Dedekind zeta function ςK(s) at its pole s = 1. In this paper, we establish a similar expression of γn(K) as Stieltjes obtained in 1885 for γn(Q). We also study the signs of γn(K). [ABSTRACT FROM AUTHOR]

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 γ and the constant ln2π 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. [ABSTRACT FROM AUTHOR]

Abstract 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 Grünwald–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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the following generalized multiple fractional part integrals:andare considered for complex numbers, positive integersand positive numbera, wheredenotes the fractional part ofu. Furthermore,andcan be transformed into the calculation of definite fractional part integraland, 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. [ABSTRACT FROM PUBLISHER]

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 arguments, this 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 $$a \rightarrow 0$$ as well as a novel technique for evaluating integrals with integrands with $$\ln (-\ln x)$$ and rational factors. [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]

The article discusses the original formulation of the famous Riemann hypothesis. It explores the assertion about the positions of the zeros of the Riemann zeta functions which can be defined by the Dirichlet series. It also presents several formulas about the Riemann hypothesis which be can be defined and expanded in the Laurent series.

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 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. [ABSTRACT FROM AUTHOR]

In this paper, we provide the Euler-Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$, where f∈ C( a, b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f( x) has asymptotic expansions of the general forms [Equation not available: see fulltext.] where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The γ and δ are distinct, complex in general, and different from −1. They also satisfy [Equation not available: see fulltext.] The results we obtain in this work extend the results of a recent paper [A. Sidi, Numer. Math. 98:371-387, ], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$. They are expressed in very simple terms based only on the asymptotic expansions of f( x) as x→ a+ and x→ b−. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$, h=( b− a)/ n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$. Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$, one of these results reads [Equation not available: see fulltext.] where ζ( z) is the Riemann Zeta function and σ are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$, i=0,1,... . [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: 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. [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]