ﻻ يوجد ملخص باللغة العربية
In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three $L$-functions which can be factorized as a product of a degree one and a degree two $L$-functions. We emphasize that we can break the $1/2$-barrier in the error term, and we get an explicit exponent.
We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an $L$-function. As immediate applicat
Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlets divisor problem, it is conjectured that $S_f(X) ll X^{frac{k-1}{2} + frac{1}{4} + eps
Let $gcd(k,j)$ denote the greatest common divisor of the integers $k$ and $j$, and let $r$ be any fixed positive integer. Define $$ M_r(x; f) := sum_{kleq x}frac{1}{k^{r+1}}sum_{j=1}^{k}j^{r}f(gcd(j,k)) $$ for any large real number $xgeq 5$, where $f
Let $ mathfrak{f} $ run over the space $ H_{4k} $ of primitive cusp forms of level one and weight $ 4k $, $ k in N $. We prove an explicit formula for the mixed moment of the Hecke $ L $-function $ L(mathfrak{f}, 1/2) $ and the symmetric square $L$-f
The Zagier $L$-series encode data of real quadratic fields. We study the average size of these $L$-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.