Power Moments Of Hecke Eigenvalues For Congruence Group

Let N?1 be positive integer and H_l?N?be the set of normalized primitive holomorphic cusp forms of even integral weight l for the congruence group ?₀?N?.For any f?HlN),we have the Fourier expansion ??? where ?_f?n?are Hecke eigenvalues with ?_f?1?=1.It is well-known that the arith-metical function ?_f?n?is real multiplicative and verifies Deligne's inequality ??? for all n>1,where r?n?is the divisor function.In order to detect sign changes or cancellations among ?_f?n?,it is natural to study the summatory function In 1927,Hecke[7]proved that ??? for all f?H_l?N? and x>1.Many mathematicians studied the upper bound of S_f?x'N? later.For example,see[3,4.6,7,10,14-18].The best result to date is ??? proved by Wu[19].In the opposite direction,Ivi??? and Hafner[6] also showed that there is a positive constant D such that??? In fact,it is conjectured that S_f?x;N?<<x^1/4+?.In the absence a proof of conjecture,it is natural to consider the mean value of S_f?x;N?.For N=1,A.Walfisz[16] showed the mean square estimate ??? ,where ???.Cai[2]studied the third and fourth power mo-ments of S_f?x;1?.He proved that ???,where B₃,B₄ are computable constants.Later Ivic[9],Zhai[21]improved the two results by large value arguments.In this paper,we investigate the higher power moments of Hecke eigenvalue for the congruence group ?₀?N? and determine the explicit dependence on the level.We first study the mean square estimates of S_f?x;N? and obtain the following theorem.Theorem 1.For any f?H_l?N?,?N T^1-?,we have ??? where ?_f=±1.For higher moments,we start with the large value of S_f?x;N??see Theorem 4? by Halasz-Montgomery inequality and the estimates of exponential sums.Then we divide the interval[T/2,T]into subintervals [T/2+j-1,T/2+j],?j=1,2,...?,and pick the maximal |S_f??j;N?| in j intervals of length V.Finally,we can get the higher power moments of S_f?x;N?by Theorem 4.More precisely,we have the following theorem.Theorem 2.Suppose A>2 is a fixed constant and 1?N<<T^1-?.Then for any f?H_l?N?,we have???.Before stating the asymptotic results,we introduce some notations.Define ??? Suppose A₀>3 is a real number.Define ???.Theorem 3.Let A₀?8 be a real number such that ??? holds.Then for any integer 3<k<A₀,and 1?N<<T^1-?,we have the asymptotic formula ???Remark 1.1.For f?H_l?N?,the mean value of ?_f?n?in arithmetic progres-sions also attracts the attention of mathematicians,see[1,5,12,13,20,25].Let a and r be two positive integers.It is well-known that {?_f?n?|n?a mod r} determines a cusp form of higher level.More precisely,?e.g.,see[24,Lemma 3.1]???? is a cusp form on ?₀?Nr²?.If g is also a Hecke eigenform on ?₀?Nr²?,then we could get the higher power moments of Hecke eigenvalues in arithmetic progressions by Theorem 1,Theorem 2 and Theorem 3.