Neea Palojarvi

On the Error Term of the Fourth Moment of the Riemann Zeta-function

The k-th moment of the Riemann zeta function on the half-line is defined as $\int_0^T |\zeta(1/2+it)|^k dt$. It describes the average size of the Riemann zeta function in the respective region, and has connections to matrix theory [2].
In this talk, I will discuss about the error term of the fourth moment of the Riemann zeta function. Using spectral-theoretic approach, Motohashi [3] was able to show that the error term, denoted as $E_2(T)$, is $\ll T^{2/3}(\log{T})^8$ and $\int_0^T E_2(T)^2 dt \ll T^2 (\log{T})^22$. Using Ivić bounds for certain sums [1] and our own refinements for Motohashi's method, I and Tim Trudgian were able to improve the exponents of the previous logarithms from 8 and 22 to 3.5 and 9, respectively.

[1] A. Ivić, On the moments of Hecke series at central points. Funct. Approx. Comment. Math., 30:49-82, 2002.
[2] J. P. Keating, & N. C. Snaith, Random Matrix Theory and . Commun. Math. Phys., 214: 57-89, 2000.
[3] Y. Motohashi, Spectral Theory of the Riemann Zeta-function. Cambridge Tracts in Mathematics, 127. Cambridge University Press, Cambridge. 1997.