Related papers: Explicit Analytic Continuation of Euler Products

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…

Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…

In a previous paper the authors elaborated notions and technique which could be applied to compute such invariants of polynomials as Euler characteristics of fibres and zeta-functions of monodromy transformations associated with a…

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear…

Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note…

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An…

Topological descriptors have been increasingly utilized for capturing multiscale structural information in relational data. In this work, we consider various filtrations on the (box) product of graphs and the effect on their outputs on the…

We introduce polyhedral products in an $\infty$-categorical setting. We generalize a splitting result by Bahri, Bendersky, Cohen, and Gitler that determines the stable homotopy type of the a polyhedral product. We also introduce a motivic…

When it comes to partial numerical verification of the Riemann Hypothesis, one crucial part is to verify the completeness of a list of pre-computed zeros. Turing developed such a method, based on an explicit version of a theorem of…

Lie algebraic techniques are powerful and widely-used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite…

Given a finite simplicial complex L and a collection of pairs of spaces indexed by its vertex set, one can define their polyhedral product. We record a simple formula for its Euler characteristic. In special cases the formula simplifies…

This short note for non-experts means to demystify the tasks of evaluating the Riemann Zeta Function at non-positive integers and at even natural numbers, both initially performed by Leonhard Euler. Treading in the footsteps of G. H. Hardy…

Classically, Euler developed the theory of the Riemann zeta - function using as his starting point the exponential and partial fraction forms of cot(z) . In this paper we wish to develop the theory of $L$-functions of elliptic curves…

We establish an integral representation for the Dirichlet generating function of the coefficients of Euler's pentagonal number theorem. The Bromwich-type integral enables analytic continuation to the entire complex plane, filling a gap in…

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers,…

Let $\Lambda = \{\lambda_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{\lambda_{k} \in \Lambda : | \lambda_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an…

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…

All the $F:$N$\rightarrow $C having Ramanujan expansion $F(a)=\sum_{q=1}^{\infty}G(q)c_q(a)$ (here $c_q(a)$ is the Ramanujan sum) pointwise converging in $a\in $N, with $G:$N$\rightarrow $C a multiplicative function, may be factored into…

We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions,…

The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite…