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Register a new userNon-vanishing of Maass form L-functions at the critical point
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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.
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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.
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime-power conductor.
We show that for a positive proportion of Laplace eigenvalues $lambda_j$ the associated Hecke-Maass $L$-functions $L(s,u_j)$ approximate with arbitrary precision any target function $f(s)$ on a closed disc with center in $3/4$ and radius $r<1/4$. The main ingredients in the proof are the spectral large sieve of Deshouillers-Iwaniec and Sarnaks equidistribution theorem for Hecke eigenvalues.
Let $pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${rm U}_{mathfrak n}$ defined over a number field $F$. Under the assumption that $pi$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(frac{1}{2},pitimeschi)$, with a certain automorphic character $chi$ of ${rm U}_1$, for the case of ${mathfrak n}=2,3,4$, and for the general ${mathfrak n}geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results for the central $L$-values by means of the theory of endoscopy.
We complete the proof of Proposition 5.3 of [GJR04].
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