Description
The most important approximations to the celebrated Riemann hypothesis are probably the so-called density theorems. They assert that if we consider any a>1/2 then most of the zeros of the zeta-function are in the half plane Res < a up to any large height T. They have a very important application in the theory of prime numbers since the strongest form of the hypothesis, the so called density hypothesis gives an almost as sharp upper estimate for the gaps between consecutive primes as the Riemann Hypothesis itself (which is still not sufficient to show the famous conjecture of Landau that there are always primes between neighboring squares). 110 years after Riemann's pioneering paper introducing the zeta function as a complex function and many important properties of it, Gábor Halász and Pál Turán succeeded in showing the density hypothesis in a narrow halfplane 1-c < Re s < 1 with a small, unspecified (but theoretically effective) positive constant. In the proof, an important role was played by Korobov-Vinogradov's estimate, Turán's power sum method, and a simple but ingenious idea of Halász.
During the last more than half century, their result was extended by many authors (mostly) based on the large sieve. The record is due to a work of Bourgain from the year 2000, that the density hypothesis is true for any c<0.22, i.e., for Re s>0.78. The goal of our lecture is to give a proof that uses no special property of the zeta function (just classical general complex function theoretic arguments) apart from the functional equation proved by Riemann in his famous work in 1859. We do not need a "non-trivial " (that is, subconvex) estimate for the growth of the zeta-function on the critical line, neither Turán's method nor the large sieve, just the idea of Halász, which was an indispensable tool in all earlier proofs as well.
Zoom:
https://us06web.zoom.us/j/81210137074?pwd=Ye3iWTl6qor3rvZv43rsYBZG9thy7w.1
Meeting ID: 812 1013 7074
Passcode: 582986