The Spectral Mean Value For Linear Forms In Twisted Coefficients Of Cusp Forms

The nature of the Fourier coefficient of the Maass cusp forms is an important research content of the Automorphic form theory,which has important theoretical significance for the study of the Langlands program.Let {uj(z)}j?1 be an orthonor-mal basis of Maass cusp forms for the Hecke congruence group ?O(q),Thus {uj{z}}j?1 is an eigenfunction of the Laplace operator with eigenvalue ?j =sj(1-sj),where sj= 1/2+itj with tj>0,and it has the Fourier expansionaccording to whether {uj(z)} is even or odd,where K,is the K-Bessel function.The Weyl law(proved by A.Selberg[8]in 1989)#{j:tj?T}?T2/12,shows that there are infinitely many linearly independent cusp forms but none of them have ever been constructed.The nature of Fourier coefficients ?j(n)has always been a hot topic for number theorists.There are still basic questions to be answered,such as the order of magnitude of |?i(n)|.From the asymptotic formula(proved by Rankin[9]and Selberg[10]in 1965)and the formula(proved by Kuznetsov[1]in 1980)it follows that(cosh?tj)-1/2 |?i(n)| is bounded on average in n and tj.The oscillatory behavior of ?j(n)is revealed in the large sieve type iuequality of H.Iwaniec[11]where an is a complex sequence.Another large sieve type inequality for the twisted coefficients ?j(n)nitj was established by J.-M.Deshouillers and H.Iwaniec in 1982.They proved,thatfor arbitrary complex numbers an.In 1995,W.Luo[3]provedfor arbitrary complex numbers an.In this paper,we will use Luo's method to study the spectral mean of the linear forms in twisted coefficients of cusp forms on the congruent group ?O(q),and our main results are as follows.?? 1.Let an be a complex sequence,a be a cusp and ?(t)=2sinh(?-2?)t)/sinh 2?t,2? = T-1.let ?t(n)=?t(1)?t(n)as s = 1/2 +it,where ?t(1)?2?s?(2s)-1?(s)-1,?t(n)= ?d1d2=n(di/d2)it.Let the corresponding contribution from the continuous spectrum beIf N>>T,we havewhere(?)(n,m,c1,C2)= K(n,m,c1,C2)-q-1+2itK(n,m,c1,c2q)-q-1-2itK(n,m,c1q,c2)+q-2K(n,m,c1q,c2q),K(n,m,a,b)= S(0,n;a)S(0,m;b),S(0,n;a)is the Ramanujan sum and A =(an)is a finite sequence,real number n,satisfies N<n? 2N,||A||2 = ?an2.?? 2.Let an be a complex sequence,?(t)=2sinh((?-2?)t)/sin h 2?t,2?=T-1.Let a smooth sumIf N>>T,we have S(A)= H(n,m,r,q)+ L(A,X)+ O((T2 + N3/2T-1q-1/2 + Nq-1 + NTX-1q-1)N?||A||2),where L(A,X)? LO(A,X)+ L1(A,X)+ L2(A,X),hereand A =(an)is a,finite sequence,real number n satisfies N<n ? 2N,?A?2 =?an2.