Some Problems On The Distribution Laws For Coefficients Of L-functions

The theory of L-functions plays an important role in number theory.Many mathematicians are interested in shifted convolution sums,the subconvexity problem and moment of L-functions.The distribution laws for coefficients of L-functions are almost inextricably bound up with such important problems.In this paper,we study the estimation on exponential sums with the coefficients of some kinds of L-functions and the random ordering in modulus of consecutive coefficients of some kinds of L-functions,which both reveal to us some deep distribution laws for coefficients of L-functions.In Chapter 1,we apply the Hardy-Littlewood circle method to obtain upper bounds for exponential sums of some kinds of Rankin-Selberg L-functions associated with two automorphic representations.When the parameter belongs to minor arcs,we apply a new method to obtain upper bounds for exponential sums.When the parameter belongs to the major arcs,we use the analytic property of the Rankin-Selberg L-functions to obtain upper bounds for the exponential sums.Thus,uniformly in the parameter,we can obtain nontrivial bounds for the exponential sums with the coefficients of some kinds of Rankin-Selberg L-functions that have not been obtained previously.More precisely,π denotes an irreducible cuspidal automorphic representation of GLm over Q with unitary central character and π’ denotes an irreducible cuspidal automorphic representation of GLm’ over Q with unitary central character.Denote by λπ×π’(n)the coefficients of the Dirichlet expansion of L(π×π’,s),then we give a notably milder hypothesis on the size of the second power-moment of λπ×π’(n),i.e.where δ is a positive constant depending on π and π’.Then we have uniformly in α∈R.In Chapter 2,we can estimate the sum where Λπ(n)is defined as the nth coefficient in the Dirichlet series expansion of the logarithmic derivative of L(s,π)associated with an automorphic irreducible cuspidal representation.We can grasp the analytic meaning of the Grand Riemann Hypothesis by predicting the behavior of the above sum.Assume that 1/2<θ<1 and α is an irrational number of type 1,under the condition we have for ε>0.In Chapter 3,we expect to generalize the random ordering in modulus of consecutive Fourier coefficients on full module groups to higher-rank groups.Automorphic representations can be considered as the representations arising from the automorphic forms.So there is also some relationship between the Fourier coefficients of automorphic forms and the coefficients in the Dirichlet series expansion of automorphic L-functions associated with automorphic representations.We use the theory of automorphic representations,the Sato-Tate conjecture and the sieve method to study distribution laws for Fourier coefficients of holomorphic cusp forms on higher-rank groups.Using the Rankin-Selberg theory,we study the random ordering of absolute values of non-zero coefficients of the Rankin-Selberg L-function on some higher-rank groups.