On fractional moments of the mollified Dirichlet $L$-functions

S. A. Gritsenko

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 168-187

Résumé

Let $\chi_1(n)$ be the character of Dirichlet mod 5 such that $\chi_1(2)=i$, $$ \varkappa=\frac{\sqrt{10-2 \sqrt{5}}-2}{\sqrt{5}-1}. $$ Davenport–Heilbronn function is defined below $$ f(s)=\frac{1-i\varkappa}{2}L(s,\chi_1)+\frac{1+i\varkappa}{2}L(s,\overline{\chi}_1). $$ The function $f(s)$ was introduced and investigated by Davenport and Heilbronn, in 1936. It satisfies the functional equation of Riemann's type $$ g(s)=g(1-s), $$ where $g(s)=(\frac{\pi}{5})^{-s/2}\Gamma(\frac{1+s}{2})f(s)$. It is well-known however, that not all non-trivial zeros of $f(s)$ lie on the line $\Re s=\frac{1}{2}$. In the region $\Re s>1$, $0\Im s\le T$ the number of zeros of $f(s)$ exceeds $cT$, where $c>0$ is an absolute constant (Davenport and Heilbronn, 1936). Moreover, the number of zeros of $f(s)$ in the region $\frac{1}{2}\sigma_1\Re s\sigma_2$, $0\Im s\le T$ exceeds $c_1T$, where $c>0$ is an absolute constant(S. M. Voronin, 1976). In 1980, S. M. Voronin proved that «abnormally many» zeros of $f(s)$ lied on the critical line $\Re s=\frac{1}{2}$. Let $N_{0,f}(T)$ be the number of zeros of $f(s)$ on the segment $\Re s=\frac{1}{2}$, $0\Im s\le T$. S. M. Voronin got the estimate $$ N_{0,f}(T)>c_2T\exp\{\frac{1}{20}\sqrt{\log\log\log\log T}\}, $$ where $c_2>0$ is an absolute constant. In 1990, A. A. Karatsuba significantly improved Voronin's estimate and got the inequality $$ N_{0,f}(T)>T(\log T)^{1/2-\varepsilon}, $$ where $\varepsilon>0$ is an arbitrary small constant, $T>T_0(\varepsilon)>0$. In 1994, A. A. Karatsuba got somewhat more accurate estimate $$ N_{0,f}(T)>T(\log T)^{1/2}\exp\{-c_3\sqrt{\log\log T}\}, $$ where $c_3>0$ is an absolute constant. In 2017, the author got the following estimate $$ N_{0,f}(T)> T (\log T)^{1/2+1/16-\varepsilon}\quad (\varepsilon>0). $$ In this paper we obtain new upper and lower estimates of the fractional moments of mollified Dirichlet series, from which it follows that $$ N_{0,f}(T)> T (\log T)^{1/2+1/12-\varepsilon}\quad (\varepsilon>0). $$

Détail

Keywords: Davenport–Heilbronn function, zeroes on the critical line, fractional moments of mollified moments of Dirichlet series.