On The Weighted Mean Of L-functions And Exponential Sums

The L-functions and the exponential sums are two important research objects in analytic number theory,where the latter often appears in the functional equation of the former.Their mean value problems are widely applied in arithmetic geometry,coding theory,and cryptography.The L-function and the exponential sums are associated to the recurrence sequences since many special L-functions and exponential sums satisfy recursive relations.This paper mainly studies the mean value and estimation of some closely related L-functions and exponential sums such as Riemann zeta-function,Dirichlet L-function,generalized Gauss sums,and so on.In addition,the recursive properties of two recurrence sequences are studied.The main works of this paper are summarized as follows.1.Obtain the upper and lower bound estimations for the tails of the Riemann zeta-function and Mathieu series,and acquire the computational formulae for the floor function of their reciprocal sums.The estimations reflect the distribution of the remainders and the convergence rate of the Riemann zetafunction and Mathieu series.The computational formulae ensure the accuracy of the estimation formulae we obtained.In addition,this work provides a new elementary proof for Mathieu conjecture,and optimizes the corresponding results by Alzer,Brenner,Ruehr,Mortici et al.under certain constraints.2.Completely solve the computational problem of the mean square value of Dirichlet L-function at positive integers,which in hence unified and generalized Paley,Selberg,Ankeny,Chowla,Walum,Slavutskii,Louboutin,Alkan,and Zhang's work in this field.Compared with the existing results,this work extends the the variable n to any positive integer.Besides,The numerical results of this work can be calculated by mathematical software.As corollaries,we obtain some identities related to trigonometric functions.3.Study the mean value problem of two generalized forms of Gauss sums.Specifically,we obtain the computational formulae for the fourth power mean of generalized two-term exponential sums,which generalizes the moduli from odd prime to positive integer.Moreover,we study the hybrid mean value of Gauss sums and generalized Kloosterman sums,and get the corresponding computational and asymptotic formulae depending on the congruence of the moduli,which constructs the complementarity of Gauss sums and generalized Kloosterman sums.4.Investigate the recursive properties of the Narayana's cows sequence at negative indices and the convolved Narayana's cows sequence and Fubini polynomial.The studies about the Narayana's cows sequence solve an open problem proposed by Professor Tianxin Cai and illustrate the connection between the Narayana's cows sequence at the positive index and the negative index.The research of the Fubini polynomial proves an existing conjecture,whose corollaries give the congruence expression related to the Fubini number and Euler number.