Related papers: Average sizes of mixed character sums

In this paper we consider a variety of mixed character sums. In particular we extend a bound of Heath-Brown and Pierce to the case of squarefree modulus, improve on a result of Chang for mixed sums in finite fields, we show in certain…

We provide estimates for sums of the form \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|\] and \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|\] when $A,B,C,D\subset \mathbb F_p$, the field…

Let $\mathcal{A}$ be a union-closed family of sets with base set $b(\mathcal{A})=\bigcup_{A \in \mathcal{A}}A$ denoted by $[n]=\{1, \cdots, n\}$, and for any real $x>0$, let $\mathcal{A}_{<x} = \{A \in \mathcal{A} \ | \ |A| < x\}$. Also,…

For integer partitions $\lambda :n=a_1+...+a_k$, where $a_1\ge a_2\ge >...\ge a_k\ge 1$, we study the sum $a_1+a_3+...$ of the parts of odd index. We show that the average of this sum, over all partitions $\lambda$ of $n$, is of the form…

In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} \lambda_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\Big ( \frac{8d}{\cdot}\Big )$ is the…

Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,\chi)$ and $L(4,\chi)$ using certain…

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6,…

For $\zeta$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$. They measure how small $| P(\zeta)|$ can be for an integer polynomial $P$ of degree at most $n$ and naive…

We use free probability to compute the limiting spectral properties of the harmonic mean of $n$ i.i.d. Wishart random matrices $\mathbf{W}_i$ whose limiting aspect ratio is $\gamma \in (0,1)$ when $\mathbb{E}[\mathbf{W}_i] = \mathbf{I}$. We…

Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We show that number-theoretical properties of $\theta$ have a strong effect on the convergence when…

We consider the size of large character sums, proving new lower bounds for the quantity $\Delta(N,q) = \sup_{\chi\neq \chi_0 mod q} |\sum_{n < N} \chi(n)|$ for almost all ranges of $N$. The results are proven using the resonance method and…

We obtain an asymptotic formula for the L^1 norm of the exponential sum $M(\alpha) = \sum_{n\le X}\tau(n)e(n\alpha)$ where $\tau(n) = \sum_{d|n} 1$ is the divisor function. In particular, we show that it is $\sim C\sqrt{X}\log X$ with $C =…

Let $p$ be a prime number, $\mathbb{F}_{p^n}$ be the finite field of order $p^n$, and $\{\omega_1,\ldots\omega_n\}$ be a basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$. Let, further, $N_i,H_i$ be integers such that $1\leq H_i\leq p$,…

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^X\Delta(x)\Delta_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory…

We investigate existence and properties of discrete mixture representations $P_\theta =\sum_{i\in E} w_\theta(i) \, Q_i$ for a given family $P_\theta$, $\theta\in\Theta$, of probability measures. The noncentral chi-squared distributions…

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-\delta}$, for some $\delta>0$, and if the Dirichlet series…

Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with $d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with $i+j=1$. We prove that the set of $x \in \RR$ for which there exists some constant $c(x) >…

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the P\'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t}…

Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) \chi(k)= \varphi(n)\tau(n/d)$, which reduces to Menon's identity if $\chi$ is the principal…

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke…