Research Projects

Let F be a Hecke-Maass cusp form for SL(3,Z) and let A(m,n) be the normalized (A(1,1)=1) Fourier coefficients of F. L(F,s), the L-function associated to the automorphic form F is a degree three L-function which can be expressed as a Dirichlet series of A(1,n). To understand the L-function L(F,s), it is essential to estimate the first moment of the shifted convolution sums of A(1,n) of the form $$\sum_{h\sim H}\sum_{n\sim N}A(1,n)A(1,n+h)$$ which can be trivially bounded by $(NH)^{1+\varepsilon}$, for any $\varepsilon\geq 0$. The notation $\sum_{n\sim N}$ signifies an underlying smooth function V(n) supported in the range [N,2N]. There have been some results regarding this problem in recent days. PROBLEM 1. But the estimation of the second moment of the shifted convolution sum with respect to the shift h: $$\sum_{h\sim H}|\sum_{n\sim N}A(1,n)A(1,n+h)|^2$$ and improvement of its trivial upper bound of $(N^2H)^{1+\varepsilon}$ is still an open problem. This is one of the problems I want to deal with in this project. If we consider the non-cuspidal scenario, i.e., replacing A(1,n) with $d_3(n)$, Baier et al. have obtained power saving error term over the trivial bound $(N^2H)^{1+\varepsilon}$ for $N^{1/3+\varepsilon}\leq H\leq N^{1-\varepsilon}$. However, their method goes through existing results of the Riemann zeta function, whose analogue is unavailable for our cuspidal case. I want to approach this problem through the delta method along with the application of the Poisson summation formula, Voronoi summation formula, duality principle and large sieve inequality. PROBLEM 2. Another interesting variation of the shifted convolution sum problem, which is open, is the estimation of the first moment of the shifted convolution sums of A(1,n), where the sum over shift h is taken over the set of squares of integers: $$\sum_{h\sim H_1}\sum_{n\sim N}A(1,n)A(1,n+h^2).$$ It can be trivially bounded by $(NH_1)^{\varepsilon}$. With some manipulations involving Cauchy's inequality, one can derive a non-trivial bound of this first moment for $H_1\geq N^{A}$ if one has a non-trivial bound of the second moment of the shifted convolution sum considered above for $H\geq N^{2A}$. But, in this project, I intend to directly approach this problem through the delta method instead of going through the second moment of the shifted convolution sums. PROBLEM 3. Finally, in this project, I want to improve upon the existing estimate of the averaged shifted convolution sum $$\sum_{h\sim H}\sum_{n\sim N}A(1,n)A(1,n+h)$$ for $H\sim N^{1/3}$, which will result in an improvement of the current best known upper bound for the second moment of degree three L-functions: $\int_T^{2T}|L(F,1/2+it)|^2dt\ll T^{4/3+\varepsilon}$, by Dasgupta-Leung-Young. This result was an improvement of my result, the first non-trivial upper bound for the second moment of degree three L-functions $\int_T^{2T}|L(F,1/2+it)|^2 dt\ll T^{3/2-3/32+\varepsilon}$.