Graduate Student Seminar

Discrepancy Bounds of the Logarithmic Derivative of the Riemann Zeta Function

Lungstrum, Clayton

Monday, March 25th, 2019
12:00 PM - 1:00 PM
Hylan 1104

The problem of the distribution of the logarithmic derivative of the zeta function dates as far back as Kershner and Wintner’s original paper from 1937, where they prove that \(\zeta'/\zeta(\sigma+it)\) has an asymptotic distribution for any fixed \(\sigma>\frac12\). Nearly 60 years later, C. R. Guo improved this result by providing an explicit error term for the distribution. Lester improved Guo’s result further and determined that the distribution is, in fact, a Gaussian distribution. In the literature, this error term has become known as the discrepancy as it quantifies how much is lost when approximating a function by its appropriate random model. In this paper, we investigate the discrepancy of the logarithmic derivative of Riemann’s zeta function on the vertical line \(\Re(s)=\sigma>\frac12\) when compared to its random model \(\zeta'/\zeta(\sigma,X)\). We then look at extending the result to a subset of \(L\)-functions in the Selberg class, improving upon Masahiro Mine’s recent results.

Event contact: junwen dot peng at rochester dot edu