Related papers: On the second integral moment of $L$-functions

The suggested approach is based on a known representation of Dirichlet $L$-functions via the incomplete gamma functions. Some properties of the Taylor coefficients of the lower incomplete gamma function at infinity seem to be new.…

In this paper, we consider the $L$-functions $L(s, f)$ where $f$ is an eigenform for the congruence subgroup $\Gamma_1(q)$. We prove an asymptotic formula for the sixth moment of this family of automorphic $L$-functions.

Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized…

In this article we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the…

We establish a second main theorem for algebraic tori with slow growth moving targets with truncation to level 1. As the first application of this result, we prove the Green-Griffith-Lang conjecture for projective spaces with $n+1$…

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |\zeta(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |\zeta(1/2+it)| may be framed in…

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$…

In his third notebook, Ramanujan claims that $$ \int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0. $$ In a following cryptic line, which only became visible in a…

Let $L(s,f)$ be an $L$-function associated to a primitive (holomorphic or Maass) cusp form $f$ on GL(2) over $\mathbb{Q}$. Combining mean-value estimates of Montgomery and Vaughan with a method of Ramachandra, we prove a formula for the…

In this paper, we introduce the little $\mu$-function, which is obtained as a degenerate limit of the generalized $\mu$-function. We derive the little $\mu$-function as the image of the $q$-Borel summation of a divergent solution to the…

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the…

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, \pi/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed…

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the…

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet $L$--functions over function fields. More specifically, we study the average of $L(1/2+\alpha,\chi_D)/L(1/2+\beta,\chi_D)$, when $D$ varies over monic,…

We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at…

In this paper, over an arbitrary number field, we prove subconvexity bounds for self-dual $\mathrm{GL}_3$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions in the $\mathrm{GL}_2$…

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an…

We evaluate asymptotically the negative first moment at points larger than $1/2$ of the family of quadratic twists of automorphic $L$-functions using multiple Dirichlet series under the generalized Riemann hypothesis and the…

We give an alternative computation of the twisted second moment of critival values of class group $L$-functions attached to an imaginary quadratic field. The method avoids long calculations and yields the expected polynomial growth in the…

We continue generalizing Altu\u{g}'s work on $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at $S=\{\infty,q_1,\dots,q_r\}$ with $2\in S$, after…