Extreme values of Epstein zeta-functions

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 31, Tome 445 (2016), pp. 250-267

Résumé

Let $Q(u_1,u_2,\dots,u_l)$ be a positive definite quadratic form in $l(\geq2)$ variables and with integer coefficients. Put $$ \zeta_Q(s)=\sum'(Q(u_1,u_2,\dots,u_l))^{-s} $$ where the accent indicates that the summation is over all integer $l$-tuples $(u_1,u_2,\dots,u_l)$ with the exception of $(0,0,\dots,0)$. It is known that $\zeta_Q(s)\big(s-\frac l2\big)$ is an entire function. We treat $\Omega$-theorems for $\zeta_Q(s)l\leq3)$ and for some $\zeta_Q(s)(l=2)$. Let $l\leq4$ and $F_Q(s)=\zeta_Q\big(s+\frac l2-1\big)$. As $t$ tends to infinity, we have $$ \log\bigg|F_Q\biggl(\frac12+it\biggr)\bigg|=\Omega_+\bigg(\bigg(\frac{\log t}{\log\log t}\bigg)^{1/2}\bigg), $$ and $$ \log |F_Q(\sigma_0+it)|=\Omega_+\bigg(\frac{(\log t)^{1-\sigma_0}}{\log\log t}\bigg) $$ for fixed $\sigma_0\in\big(\frac12,1\big)$.