| The Fourier coefficients of automorphic forms are very important arithmetic functions. The L-functions come from these automorphic forms have a lot pro-found properties. The Taniyama-Shimura conjecture built a connection between modular forms and elliptic curves according to their L-functions. In 1995, Andrew Wiles proved this conjecture as the key step in the proof of the Fermat’s last theo-rem. Therefore the problems about the Fourier coefficients of automorphic forms have always been the major topic in number theory. Also, problems concerning the asymptotic distribution of arithmetic functions over the arithmetic progression are very classical in analytic number theory, and have been considered from many different points of view. For example, establishing the distribution of primes in the arithmetic progression is very important to the Goldbach conjecture and twin primes problem. Recently, Yitang Zhang improved some result about the distribu-tion of primes in the arithmetic progression, and achieved a breakthrough in the twin prime problem.Throughout this paper, we study the nonlinear exponential sum twisting the Fourier coefficients of Maass forms over the arithmetic progression. The Maass cusp forms are very different from the holomorphic modular forms. Ramanujan-Petersson conjecture predicts that the n-th normalized Fourier coefficients of a cusp form is bounded by nε for any ε>0 sufficiently small. In the case of holomorphic cusp forms of SL(2, Z), this was proved by Deligne [7] in 1974. However, the case of Maass forms is still open. The best result is the bound n7/64+ε which was proved by Kim-Sarnak [22]. Although the Ramanujan-Petersson conjecture is still open in this case, one can prove it for mean values of some kind. More precisely, let Ag(n) be the n-th Fourier coefficient of a automorphic form g for SL(2, Z). We have the estimate [12] We can even get a better upper bound in the mean values which means that the Fourier coefficients have the oscillation behavior.Sometimes we twist the Fourier coefficients by exponential functions, and seek the resonance phenomenon. We consider the exponential sums below. where 0≠α∈R,0≤β≤1, e(x)=e2πix, and n～X means X< n≤2X. We have a lot of results for different values of a and β[33] [34] [36].In this paper, we generalize this problem to the arithmetic progressions. More precisely, we will study the exponential sum: where0≠α∈R,0≤β≤1, e(x)=e2πix.Firstly, we can get a upper bound estimate in chapter 2 when the value of |α|βXβ is small.定理0.1 Let Ag(n) be the normalized n-th Fourier coefficient of a Hecke-Maass cusp form for S L(2, Z) with Laplace eigenvalue 1/4+r2. Let X> 1,0<β<1 and 0≠α∈R. Let l, q∈N and l≤q≤X1/2.If |α|βXβ<(?)/2q, then we haveWhen q= 1, we shall use the result of Lau & Lii [25] about the eighth mean-value for λg(n) instead of the individual bound λg(n) <<n7/64+ε. In this way, we can improved the result in [36]. When q=1, one can see that we have the error term of size X1027/2827 instead of X71/192 in [36].There will be different results for |α|βXβ is big. We will prove theorem 1.2-1.5 in chapter 5. When β≠1/2, we can still get an upper bound:定理0.2If |α|βXβ≥(?)/2q and β≠1/2 then we haveWhen β=1/2, we discuss two conditions according to the value of a. One can see that we will get an upper bound for S (X) when|α| is too big or small, and we can get a asymptotic formula when the value of |α| is appropriate.定理0.3where and δc= 1 or 0 according to if there exists a positive integer nc for c|q satisfying or not.We point out that we can get a asymptotic formula similar to [38], and one can see the resonance phenomenon occurs, just like the result in [33].定理0.4In particular, if |α|=2(?)/q with k∈N and k< X/4, then (1.4) becomesTo prove Theorem 1.1-1.4, we shall use the method in [33] and [36]. The techniques we will use include estimation of exponential sum, Voronoi summation formulas, and the weighted stationary phase. Because of the orthogonality of the additive character, we will get the Kloosterman sum after applying the Voronoi summation formula. Thus we will use the Weil’s bound to get the saving in the q-aspect, and obtain a similar main term of (1.4). In addition, we can see the resonance phenomenon also occurs only when β=1/2 and |α| is close to 2(?), k∈ Z+, just like the result in [33]Finally, we consider the case of holomorphic cusps forms. In other word, we assume that the Ramanujan-Petersson conjecture is ture. We can get the results below:定理0.5Assume Ramanujan-Petersson conjecture2.1 is ture, or suppose thatg is a holomorphic modular cusp form. Let Ag(n) be the n-th Fourier coefficient of g, then we have(i)if |α|βXβ<(?)/2q, then (1.5) is(ii)If |α|βXβ<(?)/2q and β= 1/2, when |α|<1/qor |α|>(?)/q, we have when 1/q≤|α|<(?)/q, we have(iii)If |α|=2(?)/q with k∈N and k<X/4, then (1.9) will be... |
PDF Full Text Request