ncku-talk2023-03-16

Colloquium

DATE 2023-03-16　16:10-17:00

PLACE 數學系館 1F3174教室

SPEAKER 翁鵬絜教授（國立中山大學應用數學系）

TITLE The Eighth Moment of the Riemann Zeta Function

ABSTRACT The study of the value distribution of the Riemann zeta function $\zeta(s)$ is central to analytic number theory, and the moment problem is one of the essential topics. The second and fourth moments of $\zeta(\frac{1}{2}+it)$ were established asymptotically by Hardy-Littlewood (1918) and Ingham (1926), respectively, and there has been folklore that sixth and higher moments are beyond current techniques. Nonetheless, a few years ago, Ng proved an asymptotic formula for the sixth moment under a conjecture for ternary additive divisor sums.
In this talk, I will review some background knowledge and then explain how the Riemann hypothesis and a conjecture for quaternary additive divisor sums imply the conjectural asymptotic for the eighth moment of the Riemann zeta function. This is joint work with Nathan Ng and Quanli Shen, and it builds on the above-mentioned work of Ng. A key new idea is to use sharp bounds for shifted moments of the zeta function on the critical line.

Colloquium

DATE	2023-03-16　16:10-17:00

PLACE	數學系館 1F3174教室

SPEAKER	翁鵬絜教授（國立中山大學應用數學系）

TITLE	The Eighth Moment of the Riemann Zeta Function

ABSTRACT	The study of the value distribution of the Riemann zeta function $\zeta(s)$ is central to analytic number theory, and the moment problem is one of the essential topics. The second and fourth moments of $\zeta(\frac{1}{2}+it)$ were established asymptotically by Hardy-Littlewood (1918) and Ingham (1926), respectively, and there has been folklore that sixth and higher moments are beyond current techniques. Nonetheless, a few years ago, Ng proved an asymptotic formula for the sixth moment under a conjecture for ternary additive divisor sums. In this talk, I will review some background knowledge and then explain how the Riemann hypothesis and a conjecture for quaternary additive divisor sums imply the conjectural asymptotic for the eighth moment of the Riemann zeta function. This is joint work with Nathan Ng and Quanli Shen, and it builds on the above-mentioned work of Ng. A key new idea is to use sharp bounds for shifted moments of the zeta function on the critical line.