Henryk Iwaniec

Low lying zeros of families of L-functions

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.

Primes in arithmetic progressions to large moduli

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Perspectives on the Analytic Theory of L-Functions

To the general mathematician L-functions might appear to be an esoteric and special topic in number theory. We hope that the discussion below will convince the reader otherwise. Time and again it has turned out that the crux of a problem lies in the theory of these functions. At some level it is not entirely clear to us why L-functions should enter decisively, though in hindsight one can give reasons. Our plan is to introduce L-functions and describe the central problems connected with them. We give a sample (this is certainly not meant to be a survey) of results towards these conjectures as well as some problems that can be resolved by finessing these conjectures. We also mention briefly some of the successful present-day tools and the role they might play in the big picture.

The cubic moment of central values of automorphic

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:

L

This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} \Lambda(a^2 + b^4) = 4\pi^{-1}\kappa x^{3/4} (1 + O(\log\log x / \log x)) where a, b run over positive integers and \kappa = \int^1_0 (1 - t^4)^{1/2} dt = \Gamma(1/4)^2 /6\sqrt{2\pi}. Here of course, \Lambda denotes the von Mangoldt function and \Gamma the Euler gamma function.

-functions

One of the oldest problems in analytic number theory consists of counting points with integer coordinates in the d-dimensional ball. It is very easy to find a main term for the counting function, but the size of the error term is difficult to estimate (...).

The polynomial X2+ Y4 captures its primes

We give a number of estimates for character sums Z EX(a+b) aE.V bER for rather general sets X, 7 . These give, in particular, a modified proof of the inequalities of P6lya-Vinogradov and of Burgess, which displays the latter as a generalization of the former.

On the Sphere Problem

Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the L-functions L (s, ψ), whereψ is a character of the ideal class group of the imaginary quadratic fieldQ( √ −D) (D squarefree, D> 3, D ≡ 3 (mod 4)). We prove that, in the vicinity of the central point s = 1/2, the average distribution of these zeros (for D−→ ∞) is governed by the symplectic distribution. By averaging over D, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes p of the form4p = m2 + Dn2, and sieve techniques are applied.

Estimates for character sums

Let π(x) stand for the number of primes not exceedingx. In the present work it is shown that if 23/42≤Θ≤1,y≤xθ andx>x(Θ) then π(x)−π(x−y)>y/(100 logx). This implies for the difference between consecutive primes the inequalitypn+1−p n ≪p n 23/42 .

Low-lying zeros of dihedral L-functions

for any e > 0 and A > 0, the implied constant in the symbol <g depending at most on E and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applications of sieve methods. Having this in mind distinct generalizations have been investigated (see for example [15] and [2]). Y. Motohashi established a general theorem which, roughly speaking, says that if (1) holds for two arithmetic functions then it also holds for their Dirichlet convolution; for precise assumptions and statement see [11]. So far, all methods depend on the large sieve inequality (see [10])