Related papers: Higher moments for symmetric powers of modular for…

In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form $\sigma_{2^{t+1}l-1, 2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and…

We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let $\{X_{i}\}$ be a sequence of independent, identically distributed mean zero random variables with finite variance $\sigma$ and…

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…

Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where…

We introduce a new method to bound bilinear (Type II) sums of Kloosterman sums with composite moduli $c$, using Fourier analysis on $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ and an amplification argument with non-abelian characters. For sums…

We study the $2k$-th moment at the central point of the family of symmetric square $L$-functions attached to holomorphic Hecke cusp forms of level one, weight $\kappa$. We establish sharp lower bounds for all real $k \geq 1/2$…

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…

For an odd integer $d > 1$ and a finite Galois extension $K/\mathbb{Q}$ of degree $d$, G. L\"{u} and Z. Yang \cite{lu3} obtained an asymptotic formula for the mean values of the divisor function for $K$ over square integers. In this…

We compute the critical $L$-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral $K$. In doing so, we prove closed form formulas for some moments of $K'^3$. Many of our…

Let $\pi$ be a cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish a conditional upper bound on the number of cuspidal isobaric summands in the symmetric $k$-th power lift of $\pi$, assuming that the…

Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq…

We derive asymptotic estimates for the coefficient of $z^{k}$ in $\left( f\left( z\right) \right) ^{n}$ when $n\rightarrow \infty $ and $k$ is of order $n^{\delta }$, where $0<\delta <1,$ and $f\left( z\right) $ is a power series satisfying…

Let $S(t,f)=\pi^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect…

The asymptotic formula of the fourth moment of Dirichlet $L$-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved…

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the…

Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-free monic polynomials $D(x) \in F_q[x]$ of degree $d$. Katz and Sarnak showed that the moments, over $H_{d,q}$, of the zeta functions associated to the curves…

In this paper we address the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet $L$-functions $L(1/2,\chi_{8d}\otimes\psi)$ when $\psi$ is a fixed even primitive…

The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$…

Let f be an arithmetic function satisfying certain conditions. In this paper, we give an asymptotic formula for the sum \[\sum_{n_1 n_2 \cdots n_r \leq x} f\left(\left\lfloor \frac{x}{n_1 n_2 \cdots n_r} \right\rfloor\right), \quad r \geq…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…