An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ₀(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once.     

The Riemann hypothesis for certain integrals of Eisenstein series

This paper studies the nonholomorphic Eisenstein series E(z,s) for the modular surface PSL(2,Z)\H, and shows that integration with respect to certain nonnegative measures μ(z) gives meromorphic functions Fμ(s) that have all their zeros on the line . For the constant term a₀(y,s) of the Eisenstein series the Riemann hypothesis holds for all values y?1, with at most two exceptional real zeros, which occur exactly for those y>4πe−γ=7.0555+. The Riemann hypothesis holds for all truncation integrals with truncation parameter T?1. At the value T=1 this proves the Riemann hypothesis for a zeta function Z2,Q(s) recently introduced by Lin Weng, associated to rank 2 semistable lattices over Q.     

On the zeta function associated with module classes of a number field

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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE.     

The joint value distribution of the Riemann zeta function and Hurwitz zeta functions II

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions. Received: 17 Apri1 2007     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Universality of the Riemann zeta-function

In 1975, S.M. Voronin proved the universality of the Riemann zeta-function ζ(s). This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts ζ(s+iτ). In the paper, we consider the functions F(ζ(s)) which are universal in the Voronin sense.     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Large gaps between the zeros of the Riemann zeta function

We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the zeta function whose spacing is three times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.     

Large spaces between the zeros of the Riemann zeta-function and random matrix theory

On the hypothesis that the 2k-th mixed moments of Hardy's Z-function and its derivative are correctly predicted by random matrix theory, it is established that large gaps (depending on, and apparently increasing with k) exist between the zeta zeros. The case k=3 has been worked out in an earlier paper (in this journal) and the cases k=4,5,6 are considered here. When k=6 the gaps obtained have >4 times the average gap length. This depends on calculations involving Jacobi-Schur functions and formulae for these functions due to Jacobi, Trudi and Aitken in the classical theory of equations.     

Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.     

Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL₂ is toroidal if all its right translates integrate to zero over all non-split tori in GL₂, and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g?1, and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an “automorphic” proof for the Riemann hypothesis for the zeta function of those curves.     

Determination of algebraic relations among special zeta values in positive characteristic

As an analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values ζC(n) at positive integers n. By constructing t-motives after Papanikolas, we prove that the only algebraic relations among these characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius pth power relations.     

Müntz formula and zero free regions for the Riemann zeta function

A formula first derived by Müntz which relates the Riemann zeta function ζ times the Mellin transform of a test function f and the Mellin transform of the theta transform of f is exploited, together with other analytic techniques, to construct zero free regions for ζ(s) with s in the critical strip. Among these are regions with a shape independent of Res.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

The joint universality theorem for a pair of Hurwitz zeta functions

In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α₁ and α₂ satisfy α₂∈Q(α₁). Especially we establish the joint universality theorem for these zeta functions.     

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In [P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982) 229-247], it was proved that the Selberg zeta function for SL₂(Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL₂(Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem and the asymptotic formula of the sum of the class number.     

The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.     

Solomon's zeta functions over algebraic function fields

L. Solomon recently introduced a wide-ranging but concrete generalization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solomon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi-simpleF _q (X) -algebra, whereF _q (X) is a field of rational functions over a finite fieldF _q . Use is made of the analogues for function fields of results on his zeta functions which were first conjectured by Solomon, and later established by C J Bushnell and l Reiner.     

Deux extensions de Théorèmes de Hamburger (portant sur l’équation fonctionnelle de la fonction zêta)

We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with a functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary between the moderate meromorphic functions with the functional equation and the tempered distributions with an extended S$S$-support condition.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.