This article extends classical one variable results about Euler products defined by integral valued polynomial or analytic functions to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and also give a criterion, a la Estermann-Dahlquist, for the existence of a meromorphic extension to $\C^n.$ Among applications we deduce analytic properties of height zeta functions for toric varieties over $\Q$ and group zeta functions.
Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivati…
We consider the Dirichlet series associated to the number of representations of an integer as the sum of primes. Assuming the Riemann hypothesis on the distribution of the zeros of the Riemann zeta function we obtain the domain of meromorphic continuation of this series.
We extend Lax-Phillips' theorem on discreteness of pseudo-cuspforms, in the style of Colin de Verdi{è}re's use of the Friedrichs self-adjoint extension of a restriction of the Laplace-Beltrami operator, as opposed to the use of semigroup methods. We use this to prove meromorphic continuation of Eisenstein series in several $\mathbb{Q}$-rank one cases, again following Colin de Verdi{è}re, as opposed to the semigroup-oriented viewpoint of Lax-Phillips and W. Mueller.
We consider the one-parameter family of hypersurfaces in $\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5λX_1 X_2... X_5, (writing $λ$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane.
Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.
We consider modular functions (i.e., the Eisenstein series and Hecke-Maass forms) for the group PSL(2,Z). We fix a quadratic number field E. This gives rise to twisted (by a Hecke character of the field E) periods of a modular function along the torus corresponding to E. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.
We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.
For a positive irrational number $α,$ we study the ordinary Dirichlet series $ζ_α(s) = \sum\limits_{n\geq1} \lfloorαn\rfloor^{-s}$ and $S_α(s) = \sum\limits_{n\geq1} (\left\lceilαn\right\rceil - \left\lceil α(n-1)\right\rceil){n^{-s}}.$ We prove relations between them and $J_{\boldsymbolα}(s)=\sum\limits_{n\geq1}\left(\lbraceαn\rbrace-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\boldsymbolα},$ we show that $ζ_α$ and $S_α$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$ Based on the …
We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of "decalage" that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence {\it relations} between the values at $T-$tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.
We initiate the study of Selberg zeta functions $Z_{Γ,χ}$ for geometrically finite Fuchsian groups $Γ$ and finite-dimensional representations $χ$ with non-expanding cusp monodromy. We show that for all choices of $(Γ,χ)$, the Selberg zeta function $Z_{Γ,χ}$ converges on some half-plane in $\mathbb{C}$. In addition, under the assumption that $Γ$ admits a strict transfer operator approach, we show that $Z_{Γ,χ}$ extends meromorphically to all of $\mathbb{C}$.
We prove two theorems. Theorem 1 gives the meromorphic continuation of the multiple zeta function to the whole space. In Theorem 2, we prove asymptotic behavior near the non-positive integers.
This is an expository paper on the meromorphic continuation of zeta functions with Euler products (for example zeta functions of groups and height zeta functions) or without (for example the Goldbach zeta function). As an application we show how a natural boundary of analytic continuation can give asymptotic results.
We define, and obtain the meromorphic continuation of, shifted Rankin-Selberg convolutions in one and two variables. As sample applications, this continuation is used to obtain estimates for single and double shifted sums and a Burgess-type bound for $L$-series associated to modular forms of arbitrary central character. Further applications are furnished by subsequent works by the authors and their colleagues.
We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.
We prove that the meromorphic continuations of the Ruelle and Selberg zeta functions considered by Bunke and Olbrich are of finite order not larger than the dimension of the underlaying compact, odd-dimensional, locally symmetric space.
Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In t…
In this work we provide a meromorphic continuation in three complex variables of two types of triple shifted convolution sums of Fourier coefficients of holomorphic cusp forms. The foundations of this construction are based in the continuation of the spectral expansion of a special truncated Poincaré series recently developed by Jeffrey Hoffstein. As a result we are able to produce previously unstudied and nontrivial asymptotics of truncated shifted sums which we expect to correspond to off-diagonal terms in the third moment of automorphic L-functions.
In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalising an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.
In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork's unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p.
We study the meromorphic continuation and the spectral expansion of the oppposite sign Kloosterman sum zeta function, $$(2π\sqrt{mn})^{2s-1}\sum_{\ell=1}^\infty \frac{S(m,-n,\ell)}{\ell^{2s}}$$ for $m,n$ positive integers, to all $s \in \mathbb{C}$. There are poles of the function corresponding to zeros of the Riemann zeta function and the spectral parameters of Maass forms. The analytic properties of this function are rather delicate. It turns out that the spectral expansion of the zeta function converges only in a left half-plane, disjoint from the region of absolute convergence of the Diric…
We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1/d$ of the L-functions of any degree $d \geq 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
We review some topics in the analytic theory of Eisenstein series, including meromorphic continuation, $L^2$-spectral expansion and Fourier coefficients. We also discuss some open problems.
Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case…
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 plane, giving explicit formulas for its pole set and residues, as well as for its finite values at negative integers, which are shown to be rational numbers. To illustrate the results, we focus on some concrete examples which have also been studied previously by other authors.
Let $π=π_1 \otimes π_2 \otimes π_3$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_3^3(\mathbb{A}_F)$ where $F$ is a number field. Assume that $π$ is everywhere tempered. Under suitable local hypotheses, for a sufficiently large finite set of places $S$ of $F$ we prove that the triple product $L$-function $L^S(s,π,\otimes^3)$ admits a meromorphic continuation to $\mathrm{Re}(s) >\tfrac{1}{2}$. We also give some information about the possible poles.
First, we recount a history of how certain methods using natural self-adjoint operators have, thus far, failed to prove the Riemann Hypothesis. In Section 2, we set the analytical context necessary to have genuine proofs in later sections, rather than attractive heuristics. In Section 3, we recall the utility of designed pseudo-Laplacians by reproving meromorphic continuation of certain Eisenstein series and proving a spacing result for zeros of $ζ_k(s)$ for $k$ a complex quadratic field with negative determinant. In Section 4, we construct an automorphic Hamiltonian which has purely discrete …
Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced.In this paper, similarly we consider generalized multiple zeta-functions and $ L $-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases.We prove the meromorphic continuation to…
In this article we introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irra…
The aim of this paper is to define some new number-theoretic functions including necklaces polynomials and the numbers of special words such as Lyndon words. By using Dirichlet convolution formula with well-known number-theoretic functions, we derive some new identities and relations associated with Dirichlet series, Lambert series, and also the family of zeta functions including the Riemann zeta functions and polylogarithm functions. By using analytic (meromorphic) continuation of zeta functions, we also derive identities and formulas including Bernoulli numbers and Apostol-Bernoulli numbers.…
This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series.…
We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence $2$.
In this article we discuss the limit $p$ approaches to one of tree-level $p$-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that $p$-adic strings are related to the ordinary strings in the $p \to 1$ limit. Previously, we established that $p$-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa's local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to …
The aim of this paper is to describe explicitly the poles of the meromorphic continuation of the Igusa local zeta function associated to several polynomials. Using resolution of singularities is possible to express the Igusa's local zeta function as a finite sum of p-adic monomial integrals. We compute these monomial integrals using techniques of toroidal geometry. In this way, we obtain an explicit list for the candidates to poles of the local zeta function associated to several polynomials.
We consider zeta functions: $Z(f ;P ;s)=\sum_{\m \in \N^{n}} f(m_1,..., m_n) P(m_1,..., m_n)^{-s/d}$ where $P \in \R [X_1,..., X_n]$ has degree $d$ and $f$ is a function arithmetic in origin, e.g. a multiplicative function. In this paper, I study the meromorphic continuation of such series beyond an a priori domain of absolute convergence when $f$ and $P$ satisfy properties one typically meets in applications. As a result, I prove an explicit asymptotic for a general class of lattice point problems subject to arithmetic constraints.
Two dimensional adelic objects were introduced by I. Fesenko in his study of the Hasse zeta function associated to a regular model $\mathcal E$ of the elliptic curve $E$. The Hasse-Weil $L$-function $L(E,s)$ of $E$ appears in the denominator of the Hasse zeta function of $\mathcal E$. The two dimensional adelic analysis predicts that the integrand $h$ of the boundary term of the two dimensional zeta integral attached to $\mathcal E$ is mean-periodic. The mean-periodicity of $h$ implies the meromorphic continuation and the functional equation of $L(E,s)$. On the other hand, if $E$ is modular, s…
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $θ_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $ζ_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $ζ_{U}\left( s,z\right)$ at all poles in the half-plane …
This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. This correspond…
Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $ζ^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $ζ^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the o…
