Related papers: Essential singularities of Euler products

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…

In the present work we show that the Dirichlet series with the Euler product having analytical continuation to the critical strip without singularities, in some natural conditions, can be approximated by partial products of Euler type in…

Many Dirichlet series of number theoretic interest can be written as a product of generating series $\zeta_{\,d,a}(s)=\prod\limits_{p\equiv a\pmod{d}}(1-p^{-s})^{-1}$, with $p$ ranging over all the primes in the primitive residue class…

We focus on a well-known convergence phenomenon, the fact that the $\zeta$ zeros are the universal singularities of certain Euler products.

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of…

Finite Euler product is known to be one of the classical zeta functions in number theory. In [1], [2] and [3], we have introduced some multivariable zeta functions and studied their definable probability distributions on R^d. They include…

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of…

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re…

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\sigma_{\alpha}(n):=\sum_{d|n}d^{\alpha}$. We obtain an exact identity relating the Dirichlet series…

We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothesis, which is named "the Deep Riemann Hypothesis" (DRH), is examined. We…

In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the…

In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for…

We study zeta functions enumerating subalgebras or ideals of Lie algebras over finite field of prime order $\mathbb{F}_p$. We first develop a general blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras, and…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…

In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets…

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the…

We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a…

We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes…

We study a class of approximations to the Riemann zeta function introduced earlier by the second author on the basis of Euler product. This allows us to justify Euler Product Sieve for generation of prime numbers. Also we show that Bounded…