Speaker
Aleksander Simonič
(UP FAMNIT)
Description
Ingham (1940) proved that $N(\sigma,T)\ll T^{3(1-\sigma)/(2-\sigma)}\log^{5}{T}$, where $N(\sigma,T)$ counts the number of the non-trivial zeros $\rho$ of the Riemann zeta-function with $\Re\{\rho\}\geq\sigma\geq 1/2$ and $0<\Im\{\rho\}\leq T$. Such estimates are often valuable in the distribution theory of prime numbers. In this talk I will present an explicit version of this result with the exponent $(7-5\sigma)/(2-\sigma)$ of the logarithmic factor. The crucial ingredient in the proof is an explicit estimate with asymptotically correct main term for the fourth power moment of the Riemann zeta-function on the critical line, a result which is of independent interest.
This is joint work with Shashi Chourasiya (UNSW Canberra).
Author
Aleksander Simonič
(UP FAMNIT)
Co-author
Shashi Chourasiya
(UNSW Canberra)
