Estimate of the zeta-functions of quadratic forms negative discriminant on the unit line

I. F. Avdeev

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 23-36.

Résumé

Research on the theory of the Riemann zeta function are carried out with great intensity that's been going on for one and a half centuries, and some parts of the theory became independent scientific directions of modern analytic number theory. An important role among these areas play a theorem about the zeros of the density distribution of the Riemann zeta function in the critical strip. During the last decades, the topic in a large number of scientific articles. She repeatedly touched in scientific monographs and special books on various issues of analytic number theory. Studies of the behavior of the Riemann zeta function $\zeta(s)$ in the critical strip essentially based on its proximity segment of the Dirichlet series. The main result of this work is using of the Vinogradov's method for estimation of $\zeta(s,k)$-Zeta function of the quadratic form $K$ and the growing negative discriminant $(-d)$. In the article is given the use of Vinogradov's method for estimating $\zeta(s,k)$-Zeta function of the quadratic form $K$ and the growing negative discriminant $(-d)$. Application Vinogradov method for estimating $\zeta(s,k)$-Zeta function of the quadratic form $K$ and the growing negative discriminant $(-d)$. is difficult due to lack of suitable for the purpose of the approximate functional equation. Typically, members of this equation include the factor, which is the value of the character group of divisor classes of the field $Q(\sqrt{-d})$ for positive definite quadratic forms of discriminant $(-d)$. This fact is the main obstacle to the effective application of the method of trigonometric sums. C. M. Voronin in his work [1] an approximate functional equation for $\zeta(s,k)$ principal term of which represent the initial segment of the Dirichlet series of functions, which are not members of the «twisted» with any character. This allows reducing the question about his assessment to the assessment of the amount of double dzetovoy. The proof is carried out by bringing the zeta-functions of quadratic forms a segment of a Dirichlet series. Also in the article describes the history of the problem behavior of the Riemann zeta function $\zeta(s)$ in the critical strip. The basic results of relevance today, shows the application results found. Bibliography: 15 titles.

Détail

Keywords: Riemann zeta-function, approximate functional equation of Hardy–Littlewood, quadratic form, the Dirichlet's series.

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