Related papers: Central values of additive twists of cuspidal $L$-…

Let $\mathit{\Pi}$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $\omega_{\mathit{\Pi}}$ over a totally real number field ${\mathbb F}$. In this paper, we…

Let $f$ be a normalized holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k$ with $k\equiv0\bmod 4$. By the Kuznetsov trace formula for $GL_3(\mathbb R)$, we obtain the first moment of central values of $L(s,f\otimes \phi)$, where…

"Murmurations" are a recently-discovered type of fine structure in sums of Dirichlet coefficients averaged over families of $L$-functions. The root cause of this phenomenon remains mysterious. In the present paper, we demonstrate how…

While several instances of shifted convolution problems for GL(3) x GL(2) have been solved, the case where one factor is the classical divisor function and one factor is a GL(3) Fourier coefficient has remained open. We solve this case in…

Let $F$ be a cuspidal Hermitian eigenform of degree two over $\mathbb{Q}(i)$, with first Fourier-Jacobi coefficient not identically zero. Building on a paper by Das and Jha, we prove the meromorphic continuation to $\mathbb{C}$ and the…

In the spirit of Arthur's trace formula, we establish a general trace formula for symmetric spaces associated with the variety of involutions of a finite $D$-module where $D$ is a division algebra central over a number field $F$. Such a…

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations.…

The standard twist of $L$-functions plays a fundamental role in the Selberg class theory. It is defined as an absolutely convergent Dirichlet series and admits meromorphic continuation beyond the half-plane of absolute convergence.…

The Katz-Sarnak philosophy states that statistics of zeros of $L$-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity.…

In this paper, over imaginary quadratic fields, we consider the family of $L$-functions $L (s, f)$ for an orthonormal basis of spherical Hecke--Maass forms $f$ with Archimedean parameter $t_f$. We establish asymptotic formulae for the…

In this article, we study the mixed fourth moments of Hecke--Maass cusp forms and Eisenstein series with type $(2, 2)$. Under the assumptions of the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC), we…

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate…

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a…

Let $g$ denote a fixed holomorphic Hecke cusp form of weight $k \equiv 0 \pmod{4}$ on $\mathrm{SL}_2(\mathbb{Z})$, and let $\pi$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_3$. In this paper, we establish an asymptotic…

Based on the theory of $L$-series associated with weakly holomorphic modular forms in \cite{DLRR}, we derive explicit formulas for central values of derivatives of $L$-series as integrals with limits inside the upper half-plane. This has…

In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet…

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:

Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$,…