Showing papers by "Henryk Iwaniec published in 2006"

Prime numbers and L-functions

Abstract: The classical memoir by Riemann on the zeta function was motivated by questions about the distribution of prime numbers. But there are important problems concerning prime numbers which cannot be addressed along these lines, for example the representation of primes by polynomials. In this talk Iwill showa panorama of techniques, which modern analytic number theorists use in the study of prime numbers. Among these are sieve methods. I will explain how the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent progress in traditional questions about primes, such as small gaps, and fundamental ones such as equidistribution in arithmetic progressions. However, my primary objective is to indicate the current directions in Prime Number Theory.

Conversations on the Exceptional Character

Abstract: Everything has its exceptional character, and the analytic number theory is no exception, it has one which is real and most perplexed. In this article I will tell the story how the existence or the non-existence of such a character shaped developments in arithmetic, especially for studies in the distribution of prime numbers. Many researchers are affected by this dangerous yet beautiful beast, and this author is no exception. I shall address questions and present results which I witnessed during my own studies. Of course, the Grand Riemann Hypothesis for the Dirichlet L-functions rules out any exception! Nevertheless, after powerful researchers made serious attacks on the beast and got painfully defeated, it is now understandable that these people consider the problem to be as hard as the GRH itself. Some experts go further with prediction that the GRH will be established first for complex zeros, while the real zeros may wait long for a different treatment. In the meantime we have many ways of living with or without the exceptional character. In this article I try to show that this little dose of uncertainty is enjoyable and stimulating for many new ideas.

A polynomial divisor problem

Abstract: Abstract We prove the asymptotic formula for the sum and we announce some “illusory” consequences for prime numbers and elliptic curves.

Divisor weighted sums

Abstract: Estimates for sums of divisor functions are required in many arguments of analytic number theory. Linnik was among the first to show how to handle such questions. We provide bounds that are particularly suitable for sparse sequences. Bibliography: 5 titles.