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Register a new userOn moments of twisted $L$-functions
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We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ and $g$ is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted $L$-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M.~Youngs asymptotic evaluation of the fourth moment of Dirichlet $L$-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.
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We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.
We prove a spectral decomposition formula for averages of Zagier L-series in terms of moments of symmetric square L-functions associated to Maass and holomorphic cusp forms of levels 4, 16, 64.
We establish sharp bounds for the second moment of symmetric-square $L$-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$, where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the $L$-function, our interval is smaller than previous known results. More specifically, for $|t_j|$ of size $T$, our interval is of size $T^{1/5}$, while the previous best was $T^{1/3}$ from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square $L$-function. More specifically, we get subconvexity at $s=1/2+it$ provided $|t_j|^{6/7+delta}le |t| le (2-delta)|t_j|$ for any fixed $delta>0$. Since $|t|$ can be taken significantly smaller than $|t_j|$, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square $L$-function in the spectral aspect at $s=1/2$.
We calculate certain wide moments of central values of Rankin--Selberg $L$-functions $L(piotimes Omega, 1/2)$ where $pi$ is a cuspidal automorphic representation of $mathrm{GL}_2$ over $mathbb{Q}$ and $Omega$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain weak simultaneous non-vanishing results, which are non-vanishing results for different Rankin--Selberg $L$-functions where the product of the twists is trivial. The proof relies on relating the wide moments to the usual moments of automorphic forms evaluated at Heegner points using Waldspurgers formula. To achieve this, a classical version of Waldspurgers formula for general weight automorphic forms is proven, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error-terms) together with non-vanishing results for certain period integrals. In particular, we develop a soft technique for obtaining non-vanishing of triple convolution $L$-functions.
Let $qge3$ be an integer, $chi$ be a Dirichlet character modulo $q$, and $L(s,chi)$ denote the Dirichlet $L$-functions corresponding to $chi$. In this paper, we show some special function series, and give some new identities for the Dirichlet $L$-functions involving Gauss sums. Specially, we give specific identities for $L(2,chi)$.
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