Behavior of Zeta and L-functions and their arithmetic meaning

There is a strong analogy between the divisor problem(the evaluation ofΔy D2a i D2(X))and the evaluation of the remainder term E ii D2σii(T)in the mean square formula for the Riemann zeta-functionζ(S)。The basic tools for the study of them are Voronoi‘s formula and Atkinson’s formula，respectively。The results we have obtained on this topic are：1.By using Voronoi‘s and Atkinson’s formulas，we obtained the mean square formulas of the differences ofΔI D2a西耶D2(X)，or E怡D2σii(T)，in short intervals.Also we proved similar results for the remainder term in the approximate functional@equation forζI D12个D 1(S).2。We have studied the generalization to the cases with characters.3。We showed the Voronoi-type formula for the Riesz sum of the coefficient of Rankin-Selberg L-functions，and proved their mean square formulas.4。We developed the method of using Mellin-Barnes type of integrals，and established the usefulness of this method for the study of analytic continuation and asymptotic expansions.Also，we could prove the joint universality for Lerch zeta-functions，and the universality of L-functions attached to modular forms。

There is a strong analogy between the divisor problem (the evaluation of ΔィイD2aィエD2 (X)) and the evaluation of the remainder term EィイD2σィエD2 (T) in the mean square formula for the Riemann zeta-function ζ (S). The basic tools for the study of them are Voronoi's formula and Atkinson's formula, respectively. The results we have obtained on this topic are :1. By using Voronoi's and Atkinson's formulas, we obtained the mean square formulas of the differences of ΔィイD2aィエD2 (X), or EィイD2σィエD2 (T), in short intervals. Also we proved similar results for the remainder term in the approximate functional equation for ζ ィイD12ィエD1(S).2. We have studied the generalization to the cases with characters.3. We showed the Voronoi-type formula for the Riesz sum of the coefficient of Rankin-Selberg L-functions, and proved their mean square formulas.4. We developed the method of using Mellin-Barnes type of integrals, and established the usefulness of this method for the study of analytic continuation and asymptotic expansions.Also, we could prove the joint universality for Lerch zeta-functions, and the universality of L-functions attached to modular forms.