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.

Bibliographie

[1] Hardy G. H., Littlewood J. E., “The zeros of Riemann zeta–function on the critical line”, Math. Zs., 10 (1921), 283–317 | DOI | MR | Zbl

[2] Selberg A., “On the zeroes of Riemann's zeta–function”, Skr. Norske Vid. Akad. Oslo, 10 (1942) | MR

[3] Karatsuba A. A., “Fractional Moments and Zeros of $\zeta(s)$ on the Critical Line”, Math. Notes, 72:4 (2002), 466–472 | DOI | DOI | MR | Zbl

[4] Gritsenko S. A. 2017, “On the zeros of the Davenport–Heilbronn function”, Proc. Steklov Inst. Math., 296, 65–87 | DOI | MR | Zbl

[5] Karatsuba A. A., “On the zeros of the Davenport–Heilbronn function lying on the critical line”, Math. USSR-Izv., 36:2 (1991), 311–324 | DOI | MR | Zbl

[6] Karatsuba A. A., “Zeros of certain Dirichlet series”, Russian Math. Surveys, 45:1 (1990), 207–208 | DOI | MR | Zbl

[7] Karatsuba A. A., “On the zeros of a special type of function connected with Dirichlet series”, Math. USSR-Izv., 38:3 (1992), 471–502 | DOI | MR

[8] Karatsuba A. A., “A new approach to the problem of the zeros of some Dirichlet series”, Proc. Steklov Inst. Math., 207 (1995), 163–177 | MR | Zbl

[9] Karatsuba A. A., “On the zeros of arithmetic Dirichlet series without Euler product”, Russian Acad. Sci. Izv. Math., 43:2 (1994), 193–203 | MR

[10] Karatsuba A. A., “On the zeros of a class of functions generated by the Hurwitz function”, Russian Math. Surveys, 48:5 (1993), 175–176 | DOI | MR | Zbl

[11] Karatsuba A. A., “The Riemann zeta function and its zeros”, Russian Math. Surveys, 40:5 (1985), 23–82 | DOI | MR | Zbl | Zbl

[12] Karatsuba A. A., Basic Analytic Number Theory, Springer-Verlag, Berlin–Heidelberg, 1983, 222 pp. | MR

[13] Titchmarsh E. C., The Theory of the Riemann Zeta–function, 2-nd edition, Oxford University Press, New York, 1986 | MR | Zbl

[14] Heath–Brown D. R., “Fractional moments of Riemann zeta–function”, J. Lond. Math. Soc., 24:2 (1981), 65–78 | DOI | MR | Zbl

[15] Iviĉ A., The Riemann Zeta–function, John Wiley and Sons, N. Y., 1985 | MR | Zbl