Related papers: The Generating Function for the Dirichlet Series $…

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these…

For a two parameter family of Bernoulli numbers $B_{n, p}$ the exponential generating function is derived by elementary methods.

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of the appropriate form, then $F(s)=L_f(s)$ for some…

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…

Using a general $q$-series expansion, we derive some nontrivial $q$-formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some general formulas for sums of any number of squares are given. A…

We introduce a new set of prime numbers functions including an exact Generating Function and a Discriminating Function of Prime Numbers neither based on prime number tables nor on algorithms. Instead these functions are defined in terms of…

Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $\lambda_{A}(n) =(-1)^{\Omega_A(n)}$ where $\Omega_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville…

We introduce two convergent series expansions (direct and recursive) in terms of Bessel functions and representations of sums $r_N(m)$ of squares for $N$-dimensional Madelung constants, $M_N(s)$, where $s$ is the exponent of the Madelung…

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $\eta_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $\eta(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by…

We consider a Hankel determinant formula for generic solutions of the Painlev\'e IV equation. We show that the generating functions for the entries of the Hankel determinants are related to the asymptotic solution at infinity of the…

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann-Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine…

In this paper we use a contour integral method to derive a generating function in the form of a double series involving the product of two Chebyshev polynomials over generalized independent indices expressed in terms of the incomplete gamma…

We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point…

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit…

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit…

For integer $n\geqslant 1$ and real number $z\geqslant 1$, define $M(n,z):=\sum_{d|n,\,d\leqslant z}\mu(d)$ where $\mu$ denotes the M\"obius function. Put ${\cal L}(y):=\exp\left\{(\log y)^{3/5}/(\log_2y)^{1/5}\right\}$ $(y\geqslant 3)$. We…

We prove joint universality theorems on the half plane of absolute convergence for general classes of Dirichlet series with an Euler-product, where in addition to vertical shifts we also allow scaling. This generalizes our recent joint…

We give a short generating function proof of the Almkvist-Meurman theorem: For integers $h$ and $k\ne0$, define the numbers $M_n(h,k)$ by $kx(e^{hx}-1)/(e^{kx}-1)=\sum_{n=0}^\infty M_n(h,k) x^n/n!$. Equivalently, $M_n(h,k) = k^n(B_n(h/k) -…

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that…

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…