Related papers: The integrals in Gradshteyn and Ryzhik. Part 11: T…
The auxiliary functions provide efficient computation of integrals arising at the self-consistent field (SCF) level for molecules using Slater-type bases. This applies both in relativistic and non-relativistic electronic structure theory.…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as…
We study the behavior of partially twisted multiple zeta-functions. We give new closed and explicit formulas for special values at non-positive integer points of such zeta-functions. Our method is based on a result of M. de Crisenoy on the…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
We discuss a method of calculating the zeta function of subshifts which have a presentation by a finite directed graph labeled by elements of the associated inverse semigroup. This class of subshifts is introduced as a class of property A…
In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and Laplace transform.
We evaluate some integrals over $[0,\infty)$ of quotients of powers of the hyperbolic functions $\sinh x$ and $\cosh x$ using a hypergeometric approach. Some of these results appear to be new but several verify the entries in the table of…
Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a…
In this paper, we investigate the complete monotonicity of some functions involving gamma function. Using the monotonic properties of these functions, we derived some inequalities involving gamma and beta functions. Such inequalities…
We describe methods to evaluate elementary logarithmic integrals. The integrand is the product of a rational function and a linear polynomial in ln x.
This is a review of some of the interesting properties of the Riemann Zeta Function.
In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These…
We calculate zeta functions for certain orders of rank $3$ defined by standard integral table algebras and integral fusion rings that have irrational-valued irreducible characters. The calculations are obtained from explicit calculations of…
As well known, the study of Riemanns zeta function {\zeta}(s) involves the related entire function {\xi}(s). A close relative of {\zeta}(s) is the alternating zeta function {\eta}(s). Similar to {\zeta}(s), also {\eta}(s) has a…
A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…
Two representations of the Bessel zeta function are investigated. An incomplete representation is constructed using contour integration and an integral representation due to Hawkins is fully evaluated (analytically continued) to produce two…
Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential…
We give the definition, main properties and integral expressions of the auxiliary function of Riemann $\mathop{\mathcal R }(s)$. For example we prove $$\pi^{-s/2}\Gamma(s/2)\mathop{\mathcal R }(s)=-\frac{e^{-\pi i s/4}}{…
A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As…