We discover a surface related to the pair correlation of zeros of the Riemann zeta function. We make a conjecture on the shape of the surface and present partial results and numerical evidence towards the conjecture.
Tag - Riemann zeta-function
Smooth numbers are integers whose prime factors are smaller than a threshold y. In the 80s they became important outside of pure math, as Pomerance's quadratic sieve for factoring integers relied on their distribution. The density of smooth numbers up to x can be approximated, in some range, using a peculiar function ρ called Dickman's function, defined via a delay-differential equation. All of the above is also true for smooth polynomials over finite fields. We'll survey these topics and discuss recent results concerning the range of validity of the approximation of the density of smooth numbers by ρ, whose proofs rely on relating the counting function of smooth numbers to the Riemann zeta function and the counting function of primes. In particular, we uncover phase transitions in the behavior of the density at the points y=(log x)2 (as conjectured by Hildebrand) and y=(log x)3/2, when previously only a transition at y=log x was known and understood. These transitions also occur in the polynomial setting. We'll also show that a standard conjecture on the error in the Prime Number Theorem implies ρ is always a lower bound for the density, addressing a conjecture of Pomerance.
Many important consequences of the Riemann Hypothesis would remain true even if there were some zeros off the critical line, provided these exceptions to the Riemann Hypothesis are suitably rare. We can unconditionally prove some results on the rarity of possible exceptions, which give partial control over the distribution of primes, but the central estimates have resisted improvement for 50 years.
I'll describe a new approach to this problem based on a principle of finding DIophantine structure in such possible counterexamples, which overcomes some key obstructions to progress on applications to primes. As a consequence, if we assume that the zeros of the Zeta function lie on finitely many vertical lines then we obtain consequences for primes which are almost as strong as those implied by the Riemann Hypothesis itself.
I will talk about work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how one can obtain non-trivial upper bounds for smaller shifts. Joint work with H. Bui.
Values of the Riemann zeta function at odd positive integers have proved enigmatic over several centuries of study. In 1740, Euler asked whether ζ(3) could be expressed algebraically in terms of log 2 and π. In this talk, we shall show that the Grothendieck period conjecture applied to certain mixed motives answers Euler's question in the negative.
You must be logged in to post a comment.