Mean Value Of A Kind Of Zeta Functions And Exponential Sums Involving Fourier Coefficients Of Cusp Forms

In Chapter1of this paper, we consider the following kind of Zeta function where α, β are fixed rational numbers satisfying0<α<β and dα,β(n) is defined by We show that (α,β(s) has an analytic continuation to (?)s>1/3and prove the following asymptotic formula for the mean square of (?)α,β(s).Theoreml.2. Suppose T≥2and s=σ+it, then for any1/2<σ<1, there exists a constant ε (σ)>0such that As an application of Theorem1.2, we consider the distribution of primitive Pythagore-an triangles which are triples (a,b,c) of positive integers with a2+b2=c2, a<b and gcd(a, b, c)=1. Let P(x) denote the number of primitive Pythagorean triangles with perimeter less than x. In this paper, we prove Theoreml.3. Assuming Riemann Hypothesis, for x≥2, we have which improved the power of the error term5805/15408in [18].In Chapter2. we consider quadratic exponential sums involving Fourier coeffi-cients of cusp forms. Let f(z) be a holomorphie cusp form of weight k for SL2(Z) Then f(z) has a Fourier expansion where e.(z)=e2πiz. Analogously, let u(z) be a Maass cusp form with Laplacian eigenvalue1/4+r2for SL2(Z), then u(z) has the Fourier expansion where K denotes the K-Besscl function. In [26], Pitt estimated the following quadrat-ic exponential sums involving a(n) and proved that uniformly in α, β∈R, where ε>0is arbitrary and the implied constant depends only on ε and the cusp form. We prove the following improvement on Pitt's result.Theorem2.1Let X≥2and a(n) be defined as in (2.1) or (2.2). Then uniformly in α, β∈R we have where ε>0is arbitrary and the implied constant depends only on ε and the cusp form in (2.1) or (2.2).