Irrationality of values of zeta-function

We present several results on the number of irrational and linear independent values among $\zeta(s),\zeta(s+2),...,\zeta(s+2n)$, where $s>2$ is an odd integer and $n>0$ is an integer. The main tool in our proofs is a certain generalization of Rivoal's construction (math.NT/0008051, math.NT/0104221).Comment: 8+8 pages (English+Russian); to appear in the Proceedings of the Conference of Young Scientists (Moscow University, April 9-14, 2001

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

This paper is third in a series of three, following "Summation Formulas, from Poisson and Voronoi to the Present" (math.NT/0304187) and "Distributions and Analytic Continuation of Dirichlet Series" (math.FA/0403030). The first is primarily an expository paper explaining the present one, whereas the second contains some distributional machinery used here as well. These papers concern the boundary distributions of automorphic forms, and how they can be applied to study questions about cusp forms on semisimple Lie groups. The main result of this paper is a Voronoi-style summation formula for the Fourier coefficients of a cusp form on GL(3,Z)\GL(3,R). We also give a treatment of the standard L-function on GL(3), focusing on the archimedean analysis as performed using distributions. Finally a new proof is given of the GL(3)xGL(1) converse theorem of Jacquet, Piatetski-Shapiro, and Shalika. This paper is also related to the later papers math.NT/0402382 and math.NT/0404521.Comment: 66 pages, published versio

Transcendental equations satisfied by the individual zeros of Riemann ζ\zeta, Dirichlet and modular LL-functions

We consider the non-trivial zeros of the Riemann $\zeta$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $\zeta$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $\tau$-function, which is closely related to the bosonic string partition function.Comment: Matches the version to appear in Communications in Number Theory and Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and arXiv:1307.8395 [math.NT

Binomial sums related to rational approximations to ζ(4)\zeta(4)

For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $\zeta(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.Comment: 5 pages, AmSTe