Related papers: Sums of Hecke eigenvalues over quadratic polynomia…

Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $\rho$ with $\rho^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,\rho}$ on…

Let S_{w+2}(\Gamma_0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup \Gamma_0(N). We first determine explicit formulas for period polynomials of elements in S_{w+2}(\Gamma_0(N)) by means of Bernoulli…

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers,…

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall's identity for the generalized hypergeometric function…

We use one-dimensional double affine Hecke algebras to introduce q-counterparts of the Gauss integrals and new types of Gauss-Selberg sums at roots of unity.

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for…

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of…

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…

An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its…

Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known…

For each odd prime power q, and each integer k, we determine the sum of the k-th powers of all elements x in F_q for which both x and x+1 are squares in F_q^*. We also solve the analogous problem when one or both of x and x+1 is a…

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_{n,m,i} for a set of small 0 <= n-m <= 6…

This is the second in a series of three papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We give sharp conditions on the entries of a positive semidefinite NxN matrix function F on n-dimensional…

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s\textgreater{}\tfrac{1}{2}$.…

In this paper, we prove that if $f(x)=\sum_{k=0}^n{n\choose k}a_kx^k$ is a polynomial with real zeros only, then the sequence $\{a_k\}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k})^2/a_k^2…

In this paper, we will prove the non-trivial bound for the weighted average version of shifted convolution sum for $GL(3)\times GL(2)$, i.e. for any $\epsilon >0$ and $X^{1/4+\delta} \leq H \leq X$ with $\delta >0$, \[…

We study the behavior of the shifted convolution sum involving fourth power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the $k$-full kernel function for any fixed integer $k\geq2$.

We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

Given two distinct newforms with real Fourier coefficients, we show that the set of primes where the Hecke eigenvalues of one of them dominate the Hecke eigenvalues of the other has density at least 1/16. Furthermore, if the two newforms do…