Hybrid Bounds on Twisted L-Functions Associated to Modular Forms

For $f$ a primitive holomorphic cusp form of even weight $k \geq 4$, level $N$, and $\chi$ a Dirichlet character mod $Q$ with $(Q,N)=1$, we establish a new hybrid subconvexity bound for $L(1/2 + it, f_\chi)$, which improves upon all known hybrid bounds. This is done via amplification and taking advantage of a shifted convolution sum of two variables defined and analyzed in a recent paper of Hoffstein and Hulse.Comment: Updated version removes the restriction of level being square-fre

Subconvexity for twisted GL(3) L-functions

Using the circle method, we obtain subconvex bounds for GL(3) L-functions twisted by a character modulo a prime p, hybrid in the t and p-aspects.Comment: 18 page

Second moments in the generalized Gauss circle problem

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to P k ( n ) 2 , where P k ( n ) is the discrepancy between the volume of the k -dimensional sphere of radius √ n and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including ∑ P k ( n ) 2 e − n/X and the Laplace transform ∫ ∞ 0 P k ( t ) 2 e − t/X dt , in dimensions k ≥ 3. We also obtain main terms and power-saving error terms for the sharp sums ∑ n ≤ X P k ( n ) 2 , along with similar results for the sharp integral ∫ X 0 P 3 ( t ) 2 dt. This includes producing the first power-saving error term in mean square for the dimension-three Gauss circle problem

Sums of Cusp Form Coefficients Along Quadratic Sequences

Let $f(z) = \sum A(n) n^{(k-1)/2} e(nz)$ be a cusp form of weight $k \geq 2$ on $\Gamma_0(N)$ with character $\chi$. By studying a certain shifted convolution sum, we prove that $\sum_{n \leq X} A(n^2+h) = c_{f,h} X + O_{f,h,\epsilon}(X^{\frac{6k-8}{8k-11}+\epsilon})$ for $\epsilon>0$, which improves a result of Blomer from 2008 with error $X^{6/7+\epsilon}$. This includes an appendix due to Raphael S. Steiner, proving stronger bounds for certain spectral averages.Comment: 22 pages, with a 14 page appendix from Raphael S. Steiner. This version corrects a mistake in the previous, where lifts of holomorphic modular forms to Maass forms were omitte