Rankin selberg zeta function pdf

On riesz mean for the coefficients of twisted rankinselberg lfunctions ichihara, yumiko, journal of the mathematical society of japan, 2003. Standard zerofree regions for rankinselberg lfunctions. The simplest rankinselberg integral from rankin 1939 and selberg 1940 is hfe s. Rankin lfunctions and the birch and swinnertondyer. Nonvanishing of rankinselberg l functions for hilbert modular forms shengchi liu and riad masri abstract. Archimedean rankinselberg integrals columbia university. Euler systems for rankinselberg convolutions of modular forms pages 653771 from volume 180 2014, issue 2 by antonio lei, david loeffler, sarah livia zerbes abstract. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. A rankin selberg integral using the automorphic minimal representation of so7 by bump, friedberg and ginzburg.

Quantum chaos, random matrix theory, and the riemann. After the war his accomplishments became known, including a proof that a positive proportion of the zeros of the riemann zeta function lie on the line. Rankin selberg lfunctions kohji matsumoto graduate school of mathematics, nagoya university, chikusaku, nagoya 4648602, japan. The selberg zetafunction was introduced by atle selberg 1956. Test vectors for rankinselberg lfunctions university of. An introduction jens marklof school of mathematics, university of bristol, bristol bs8 1tw, u. We give a simple proof of a standard zerofree region in the taspect for the rankinselberg lfunction ls. Pdf euler products of selberg zeta functions in the critical strip. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in my webpage. Zagier in tills paper we will consider the functions ez, p obtained by setting the complex variable s in the eisenstein series ez, s equal to a zero of the riemann zeta function and will show that these functions satisfy a number. In chapter three, we state and prove the deligneserre theorem. Archimedean rankinselberg integrals 3 let v0 be the space of kn.

It is known that s converges absolutely for the real part of sgreater than. Pdf we obtain the approximate functional equation for the rankin selberg zeta function in the critical strip and, in particular, on the critical line. Some of these results can collectively be seen in 4 and 14. It turns out that the theory of the rankinselberg convolution delivers more. Introduction e generating function attached to pairs of automorphic representations has its origins in the papers of rankin and selberg. Pdf on the rankinselberg problem in short intervals. The paper gives complete proofs of the properties of the rankin. Jan 10, 2011 hida, on the other hand, constructs in great generality 7, 9 a padic lfunction that interpolates the central critical value of the rankinselberg convolution of two independent padic families of modular forms and our padic lfunction is essentially the square root of the restriction of his general padic lfunction to one of the. Apr 27, 2014 if denotes the cardinality of the residue field, then we define the rankin selberg function we again obtain absolute convergence for. The mean values and the universality of rankinselberg lfunctions. In this paper, we prove a similar result for any cusp form over the full modular group.

The mean values and the universality of rankinselberg l. It is analogous to the famous riemann zeta function where is the set of prime numbers. A short version will appear in the ramanujan journal. Aug 24, 2009 the proof of the theorem is based on a cohomological interpretation of the rankinselberg zeta integral. Theta and selberg zeta function ulrich bunke june 22, 2015 abstract contents 1 the theta function 1 2 hyperbolic space 5 3 the selberg trace formula 8 4 the selberg zeta function 10 5 a panorama of results on z s 11. Expression of the selberg ruelle zeta function as an euler product of characteristic polynomials is due to. The exposition is selfcontained, and required a preliminary knowledge of only the ele. We give a simple proof of a standard zerofree region in the taspect for the rankin selberg l function ls e. We improve the existing upper bound for the meansquare of the absolute value of the rankinselberg zetafunction attached to a holomorphic cusp form defined for the full modular group in the. Modulartype relations associated to the rankinselberg l. He was the first to prove that the riemann zeta function has a positive proportion of its zeros on the critical line and laid the groundwork for the proof that the logarithm of the riemann zeta function is normally distributed on the critical line. Standard zerofree regions for rankinselberg lfunctions via. Applications to the koechermaass series associated with nonholomorphic siegeleisenstein series are given. As a consequence of the knowledge of the location and multiplicities of the poles of these functions.

The universality of symmetric power l functions and their rankin selberg l functions li, hongze and wu, jie, journal of the mathematical society of japan, 2007. Standard zerofree regions for rankin selberg l functions via sieve theory peter humphries, with an appendix by farrell brumley abstract. On the generalized eulerstieltjes constants for the rankin. We obtain the approximate functional equation for the rankinselberg zetafunction on the 12line. We follow the arguments of ramachandra and sankaranarayanan see 6. I think this is easier to understand from a genuinely adelic viewpoint, since that makes the proofs be the same as for the classic rankin selberg story from 1939, in the same way that iwasawatates viewpoint on hecke l functions makes heckes general case nearly identical to riemanns original argument.

Nonvanishing of the central value of the rankinselberg l. Generalized frobenius schur numbers with david ginzburg. Riemann zetafunction, automorphic lfunctions, beurlingselberg extremal problem. Functional equation of rankin selberg zeta function. Euler systems for rankinselberg convolutions of modular. Nonvanishing of the central value of the rankinselberg l functions dihua jiang school of mathematics, university of minnesota minneapolis, mn 55455, usa april 12, 2005.

Standard zerofree regions for rankinselberg l functions. Furthermore, we have the following conditional automorphy result for certain rankin selberg l functions. We give a simple proof of a standard zerofree region in the taspect for the rankinselberg lfunction ls e. The mellin transform of the constant term in the fourier expansion of an automorphic function can be written as the convolution of the automorphic function with an eisenstein series. Test vectors for rankinselberg lfunctions sciencedirect.

Zeta regularization, spectral zeta functions, eisenstein series, and casimir energy 57 8. Research article on a rankinselberg function over different. Residue of rankinselberg lfunction for nontrivial nebentypus. The selberg zetafunction uses the lengths of simple closed geodesics instead of the primes numbers. Journal of number theory vol 209, pages 1530 april. Basic rankinselberg july 9, 2010 be the usual zeta function. Through a process of unipotent averaging we produce a pair of corresponding whittaker functions whose zeta integral is nonzero, and we express this integral in terms of the langlands parameters of. Let f be a number field of a finite degree and let ls. We study certain lseries of rankin selberg type associated to siegel modular forms of degree 2. Basic rankinselberg july 9, 2010 be the usual zeta function s x n 1 1 ns along with its gamma factor. Johan andersson summation formulae and zeta functions diva. The rankinselberg convolution 163 define the function 4 by tv. The author gives the analytic properties of the rankinselberg convolutions of two halfintegral weight maass forms in the plus space. This has appeared in the journal of the ramanujan math.

The rankin selberg convolution is usually normalized by the multiplication of a zeta factor. This is not the same as the rs zeta function in ivics paper which is what i think of as the rs zeta function, though it is not wildly different unfortunately, im having trouble finding online descriptions of it, and even worse, shimura has two rankin selberg zeta functions attached to his name, one for evenintegral weight and one for. Thanks again to ghfrommo for explaining the convention in the relevant paper. Contemporary mathematics archimedean rankinselberg integrals herve jacquet this paper is dedicated to stephen gelbart. We prove the absolute convergence of these functions for. Bandlimited approximations and estimates for the riemann zetafunction carneiro, emanuel, chirre, andres, and milinovich, micah b. In mathematics, the rankin selberg method, introduced by rankin 1939 and selberg, also known as the theory of integral representations of lfunctions, is a technique for directly constructing and analytically continuing several important examples of automorphic lfunctions. Citeseerx document details isaac councill, lee giles, pradeep teregowda. We establish the universality theorem for the first four symmetric power l functions of automorphic forms and their associated rankin selberg l functions. The prime number theorem, hardys theorem on the zeros of. On the special values of certain rankinselberg lfunctions. Selbergs orthonormality conjecture and joint universality of lfunctions. Eisenstein series and the riemann zetafunction by d. Real zeros and size of rankin selberg l functions in the level aspect ricotta, g.

Let f be a totally real number eld of degree n over q with ring of integers o. Dedekind zeta functions of algebraic number fields, lseries of hecke grossencharacters, from representation theory and the theory of automorphic forms hecke lseries and rankin selberg l functions of classical modular forms and their generalizations to hilbert and siegel modular forms and to automorphic forms on more complicated groups. Nonvanishing of rankin selberg lfunctions for hilbert modular forms shengchi liu and riad masri abstract. We give a simple proof of a standard zerofree region in the taspect for the rankin selberg l function ls. Let and be finite galois extensions of of degrees and, and let and be unitary automorphic cuspidal representations of and, respectively. Introduction of rankin selberg l functions in the level aspect by kowalski, michel, vanderkam. An explicit formula for the zeros of the rankinselberg lfunction via the projection of c. Z, treated independently in 1939 by rankin and 1940 by selberg. Zeros of the selberg zeta function for noncompact surfaces. And the rankin selberg convolution, a method for creating new l functions out of old, opened the. The main difficulty in our case is that the two extension fields may be completely unrelated, so we are. Poles of the rankinselberg zeta function associated to. This function is called nonholomorphic eisenstein series or epstein zeta function of. Selberg and ruelle zeta functions for compact hyperbolic manifolds.

Zeros of the selberg zeta function for noncompact surfaces abstract in the present paper we provide a rigorous mathematical foundation for describing the zeros of the selberg zeta functions z x for certain compact surfaces x with boundary, corresponding to convex cocompact fuchsian groups g without parabolic points. This generalizes some results of laurincikas and matsumoto and of matsumoto, respectively. On the rankinselberg method for functions not of rapid decay. The rankin selberg convolution is usually normalized by the mul tiplication of a zeta factor. Transactions of the american mathematical society volume 348, number 3, march 1996 on the poles of rankin selberg convolutions of modular forms xianjin li abstract. There is an analogous relationship between the gu2. Pdf for any congruence subgroup of the modular group, we extend the region of convergence of the euler product of the selberg zeta. Ams transactions of the american mathematical society. Given two unitary automorphic cuspidal representations and defined on and, respectively, with and being galois extensions of, we consider two generalized rankin selberg functions obtained by forcefully factoring and. Anticyclotomic padic lfunction of central critical rankin.

Introduction integral representations of rankin selberg type have been used over the years to relate the. Ramesh gangolli, zeta functions of selberg s type for compact space forms of symmetric spaces of rank one, illinois j. Lectures on zeta functions, lfunctions and modular forms. Inspired by a construction by bump, friedberg, and ginzburg of a twovariable integral representation on gsp 4 for the product of the standard and spin l functions, we give two similar multivariate integral representations. The universality of symmetric power lfunctions and their rankinselberg lfunctions li, hongze and wu, jie, journal of the mathematical society of japan, 2007. We improve the existing upper bound for the meansquare of the absolute value of the rankin selberg zeta function attached to a holomorphic cusp form defined for the full modular group in the. Moreover, applying the rankin selberg method to the function fg i gw one finds that the integral of this function against eg,s equals the rankin zeta function rfs roughly speaking, the dirichlet series l i.

The involution will play a key role in the functional equation for the. Standard zerofree regions for rankinselberg lfunctions via sieve theory peter humphries, with an appendix by farrell brumley abstract. Citeseerx on the poles of rankinselberg convolutions of. Period relations and special values of rankin selberg lfunctions3 3by applying the descent described in 2, the rankin selberg l function of glpnq glp1qcan be identi ed with the standard l function of the unitary group u, whose integral representation by means of the doubling method identi es its critical values with peri. During world war ii, selberg worked in isolation due to the german occupation of norway. Aug 11, 2016 hafner and stopple proved a conjecture of zagier relating to the asymptotic behaviour of the inverse mellin transform of the symmetric square l function associated with the ramanujan tau function. Moscow journal of combinatorics and number theory, vol. July 9, 2010 basic rankinselberg university of minnesota. The rankinselberg zeta function associated with the eisenstein series is easiest to treat. The aim of these lectures is to provide an intorduc tion to the theory of the riemann zetafunction for stu dents who might later want to do research on the subject. Atle selberg 14 june 1917 6 august 2007 was a norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. We present the simplest possible example of the rankin selberg method, namely for a pair of holomorphic modular forms for sl2. Let fbe a classical holomorphic newform of level qand even weight k.

Aug 17, 2018 for s in this region in particular, we improve the zerofree region and lower bound of goldfeld and li to and respectively while removing goldfeld and lis restriction that \f \mathbb q\ and that \\pi \ is unramified at every place. One naturally expects that the nonnormalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. An l function of degree 27 for spin9, joint with ginzburg. In mathematics, the rankinselberg method, introduced by rankin 1939 and selberg, also known as the theory of integral representations of lfunctions, is a technique for directly constructing and analytically continuing several important examples of automorphic lfunctions. The rankinselberg method for automorphic functions which are not of. The rankin selberg method is one of the most important ideas in the theory of automorphic forms.