Exponential Sums Over Primes Formed With Fourier Coefficients Of Maass Cusp Forms

The Fourier coefficients of Maass cusp form has attracted the attention of manyscholars and has been widely studied[9,21]. In this paper, according to the zero density estimation of automorphic L-function, Abel partial summation formula, Vaughan identities, exponential sums, we study the exponential sums over primes formed with Fourier coefficients of Maass cusp forms when the Ramanujan-Petersson conjecture is valid and the conjecture is not valid. This enriches the results on the properties of the Fourier coefficients.let f(z) be a Maass cusp form for SL(2. Z) with Laplace eigenvalue 1/4 +r2, then its Fourier expansion at infinity is where Kir denotes the K-Bessel function and af(n) is the normalized Fourier coef-ficient.For ? ?(?)s >1?let L(f,s) be the corresponding Hecke L-function associated to f(z), then where ?f(p) and ?f(p) are local roots at p, and?f(p) + ?f(p) ??f(p), ?f(p)?f(p) = 1.The Ramanujan-Petersson conjecture for Maass cusp form, which states that of(n)? n? for any fixed ?> 0, has not been proved yet. The best record till now is ?f(n)?n7/64+?,which is due to Kim and Sarnak[1].According to ?f(p) and ?(p), we can get For convenient calculation, we rewrite the formula asIn order to associate L(f,s) with prime numbers, we take the logarithm of L(f,s) expression, we can getFor the Dirichlet series of L(f,s)-1, its nth coefficient ?(n, f) is related to the normalized Fourier coefficients of f(z) by 1985, Vinogradov[2] is the first person to study the following sum S(x) =(?)?(n)e(?(?))where ?(n) refers to the Mangoldt function. And it was shown that S(x)<<x7/8+?On tlhis basis, many scholars have improvecd the results[5,6]. In this paper, we generalize the Vinogradov exponential summation problem. We want to study the mean value estimate for the coefficients of Maass cusp form in exponential sums over primes. With an additive character e(?(?)), ?> 0, we have Sf(x)=(?)?(n,f)e(?(?)),where: x?2. Note that Sf(x)=(?)?f(p)log pe(?(?))+O(x1/2logx)The Ramanujan-Petersson conjecture for Maass cusp form, has not been proved yet. So in this paper, we study the results of Maass cusp form when the Ramanujan-Petersson conjecture is valid and the conjecture is not valid. In the case of the conjecture is not valid, the results are slightly worse. This is because in the case of conjecture, we can use (3.7), in the case of the conjecture is not valid, we can not use (3.7), which makes it more difficult to estimate the Fourier coefficients of Maass cusp form in the exponent of the prime variables.When, the Ramanujan-Petersson conjecture is valid:Theorem 1 When( ? > 0, x ? 2 and any sufficiently small ? > 0, we have Sf(x)=(?)?(n,f)e(?(?))?x5/6+?where the implied constant depends on ? and the cusp form f.When the Ramanujan-Petersson conjecture is not valid:Theorem 2 When 0<?<x-17/36,x?2, any sufficiently small ? > 0, we have where the implied constant depends on a and the cusp form f.