A Polynomial With Prime Variables Attached To Cusp Forms

Let f be a holomorphic cusp form for the group SL2(Z)of even integral weight k,with Fourier coefficients a(n):We normalize f with the first coefficient being 1,and set ?(n)= a(n)/n(k-1)/2.From the Ramanujan Conjecture it is easy to know that|?(n)| ?d(n),(1.1)where d(n)is the Dirichlet divisor function(this result is due to Deligne).Many scholars are interested in researching the properties of quadratic forms.In 1963,Vinogradov[17]and Chen[4]independently studied the number of lattice Subsequently,the exponent 2/3 in the above error term was improved to 29/44 by Chamizo and Iwaniec[2],and to 21/32 by Heath-Brown[8].Friedlander and Iwaniec[5]studied the number of prime vectors among integer lattice points in 3-dimensional ball.Let ?3(x)denote the number of integer points(m1,m2,m3)? Z3 with m12 + m22 + m32=p ? x.They proved that?3(x)?4?/3 x3/2/logx,which can be viewed as a generalization of the prime number theorem.Let A(n)stands for von Mangoldt function,i.e.Guo and Zhai[6]studied the asymptotic behavior of sum and obtained S(x)= 8C3I3x3/2 + 0(x3/2 log-4 x),where A>0 is a fixed constant,andIn 2015,Hu[9]studied the sum and obtained its upper bound x3/2logc1x,where c1 is a suitable constant.Later,G.Zaghloul[19]improved this result to x3/2 exp(-c2 logx),where c2 is arbitrary.In 2017,Zhang and Wang[20]studied the sum and proved S?{x)?x3/2-267+?.In this paper,motivated by above results we firstly study hybrid problems of the Fourier coefficients ?(n)and a polynomial with prime variables p1k+p22+p32 and our first result is as follows.Theorem 1.Let Then we haveWe also consider a similar problem of the Fourier coefficients twisted by the von Mangoldt function,and we have the following theorem.Theorem 2.Let Then we have where c>0 is arbitrary.remark.Taking k = 2,Theorem 1 and Theorem 2 matches the bound obtained by zhang and wang[20]and the result of Zaghloul[19],respectively.To prove our Theorems,we follow the classical line of the circle method,the main trouble here is that we need to establish the Voronoi's summation formula of?(n)over the arithmetic progression,which is needed on the major arcs.