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Register a new userThe first moment of Maass form symmetric square L-functions
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Authors
Olga Balkanova
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We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.
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We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog-Biro-Cherubini-Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.
In this paper, we consider the family ${L_j(s)}_{j=1}^{infty}$ of $L$-functions associated to an orthonormal basis ${u_j}_{j=1}^{infty}$ of even Hecke-Maass forms for the modular group $SL(2, Z)$ with eigenvalues ${lambda_j=kappa_{j}^{2}+1/4}_{j=1}^{infty}$. We prove the following effective non-vanishing result: At least $50 %$ of the central values $L_j(1/2)$ with $kappa_j leq T$ do not vanish as $Trightarrow infty$. Furthermore, we establish effective non-vanishing results in short intervals.
We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $Delta$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $Delta<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.
We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet L-functions at the central point, with an error term that is square-root of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term.
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 $s=1/2$.
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