Related papers: Local zeta functions and Newton polyhedra

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef-Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing…

In this paper we study (h,q)-zeta functions associated with (h,q)-Bernoulli numbers and polynomials.

In this paper, we study the arithmetic zeta function $$\mathscr{Z}_{\mathcal{X}}(s) = \prod_p \prod_{\substack{x \in \mathcal{X}_p \\ \text{closed}}} \Big( \frac{1}{1-|\kappa(x)|^{-s}} \Big)^{\mathfrak{m}_{p}(x)}$$ associated to a scheme…

We prove for abelian varieties a global form of Denef and Loeser's motivic monodromy conjecture, in arbitrary characteristic. More precisely, we prove that for every tamely ramified abelian variety $A$ over a complete discretely valued…

In the present paper, we construct an algorithm for the evaluation of real Riemann zeta function $\zeta(s)$ for all real $s$, $s>1$, in polynomial time and linear space on Turing machines in Ko-Friedman model. The algorithms is based on a…

In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a…

In this paper, we give the values of a certain kind of $q$-multiple zeta functions at roots of unity. Various multiple zeta values have been proposed and studied by many researchers, but these multiple zeta values naturally arise from…

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the…

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in…

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to…

We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational…

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary…

We prove a 1999 conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of Denef and Loeser's motivic zeta function associated with a germ of a regular function on a smooth…

In this paper we obtain a closed form expression of the zeta function $Z(X_\Gamma, u)$ of a finite quotient $X_\Gamma = \Gamma \backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$.…

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For…

In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some…

We consider some nonlocal and nonpolynomial scalar field models originated from p-adic string theory. Infinite number of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its…

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of…