Averages Of Shifted Convolution Sums For Some Arithmetic Functions

In this paper,we study shifted convolution sums for higher rank groups and triple correlations of ternary divisor functions.Many important problems in analyt-ic number theory concern the shifted convolution sum(?)where h?Z\{0} and ? and ? are some arithmetic functions.We would like to recall the following special cases that have been studied in the literature:·(?,? =(?,?):this is related to the twin prime conjecture(h = 2).·(?,?)=(?,?):this is related to the Chowla conjecture.·(?,?)=(?,?):this is known as binary additive divisor problem,which is related to the fourth moments of Riemann zeta function.One can also consider general additive divisor problem with ? replaced by rk(k ? 3),the ?-th divisor function,which is related to the higher moments of the Riemann zeta function.·(?,?)?(?,??):this is the classical Titchmarsh divisor problem.·(?,?)=(?f,?g):this is the shifted convolution problem for GL(2)x GL(2)as a cuspidal analogue of the additive divisor problem.Here A,?,?,? denote the von Mangoldt,Mobius and divisor functions,respec-tively,and ?f,?g denote the Fourier coefficients of the GL(2)cusp forms f,g.More details and relevant developments can be found in[1,3,7,9,11,12,14,15,17,21-24,28,31,33,34,36-38,40,44-48,51,53-57,60].Let AF(1,…,1,n)be normalized Fourier coefficients of a Hecke-Maass cusp form F for S Lm(Z),and let f be either ? or ??.This paper will be concerned with the shifted convolution sum(?)for almost all h?[-H,H].We have the following result.Theorem 0.1 Let AF(1,…,,n)be normalized Fourer coefficients of a Hecke-aass cusp form F for Lm(Z).Let ? be either the Von Mangoldt functionn ? or the?-th divisor function ?? Let>0,?>and suppose that N8/33+??N?1-?.hen we have(?)for all but O(Hlog-B N)values of h with |h|?H.On the other hand,we still have little information about triple correlations of arithmetic functions.Let ?,?,? be certain arathmetic functions.In this paper,we shall be concerned with the triple correlation(?)If one of the ?,? and y is identically 1,the triple correlation Th(a,?,?;N)reduces to the shifted convolution problem.Then the triple correlation Th(?,?,?;N)can be regarded as a natural extension of the shifted convolution problem.For instance,n extraordinarily interesting and challenging problem in analytic number theory is to find an asymptotic formula for Th(?,?,?;N).It has not been completed yet,ven for a fixed positive integer h.In fact,most of these problems are out of reach even if ?,?,? are taken as very simple arithmetic functions.It is expected that extra cancellation can be obtained by averaging over h.See e.g.[4,8,32,35,36,39,50,51].y incorporating the arguments in Blomer[4]with those in Topacogullari[55],we rove the following results.Theorem 0.2 Let 1?H?N/3,and W be a smooth function with compact upport in[1,2]and Mellin transform W.Let a(n)be arbitrary arithmetic function and rd(n)be the Ramanujan sum.Then we have(?)where P2,d is a polynomial of degree 2,y is the Euler constant,and the O-constant depends on W and ?.Here ???2=(?|?(n)|2)1/2 is the l2-norm.By noting the similarities between divisor functions and Fourier coefficients of cusp forms,one naturally would ask what will be the case if the divisor function? is replaced by Fourier coefficient ?f of cusp form f.For instance,we have the following result which follows from the same line of proof of Theorem 0.2.Theorem 0.3 Let 1 ? H?N/3,and W be a smooth function with compact support in[1,2].Let a(n)be arbitrary arithmetic function.Let ?f(n)be normalized Fourier coefficient of a holomorphic cusp form f for the full modular group S L2(Z).Then we have(?)where the implied constant depends on W,f and ?.By we denote the bound in the Ramanujan-Petersson conjecture.In any case,v =7/(64)is admissible.Our result is non-trivial as long as N5/9+?? N1-?.