Related papers: Character sums of composite moduli and hybrid subc…

Let $\chi=\chi_q$ be a primitive character mod $q$ and fix $\Delta>0$. In 1989 Graham and Ringrose gave strong bounds on character sums $\sum_{M<n\leq M+N} \chi(n)$ in intervals of length $N=q^\Delta$ whenever $q$ is squarefree and is…

We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $\chi$ is a primitive Dirichlet character modulo a squarefree, $q^\delta$-smooth integer $q$ we show that $$L(\frac12,\chi)\ll_\epsilon…

Let $f$ be a holomorphic or Maass cusp forms for $ \rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let…

In recent years the question of maximizing GCD sums regained interest due to its firm link with large values of $L$-functions. In the present paper we initiate the study of minimizing for positive weights~$w$ of normalized $L^1$- norm the…

We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In…

Given a Dirichlet character $\chi$ modulo $q$ and its associated $L$-function, $L(s,\chi)$, we provide an explicit version of Burgess' estimate for $|L(s, \chi)|$. We use partial summation to provide bounds along the vertical lines $\Re{s}…

We study a double Dirichlet series of the form $\sum_{d}L(s,\chi_{d}\chi)\chi'(d)d^{-w}$, where $\chi$ and $\chi'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to…

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,…

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 $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^\eta$ with $0 < \eta < 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimes\chi)$ when $f$ is a primitive…

We consider sums of the form $$F_\chi(\alpha,\beta;\theta) := \sum_{\alpha p<n\le\beta p}\chi(n)e(n\theta),$$ where $\chi$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \sqrt p \log \log p \ll \max_{0…

For a primitive Dirichlet character $\chi$ modulo $q$, we define $M(\chi)=\max_{t } |\sum_{n \leq t} \chi(n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\geq 3$. Our main result provides a further…

Let $q\geqslant2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)\leqslant q.$ In this paper the authors prove some estimates for character sums of the form…

For Dirichlet characters $\chi$ mod $k$ where $k\geq 3$, we here give a computable formula for evaluating the mean square sums $\sum\limits_{\substack{\chi \text{ mod }k\\\chi(-1)=(-1)^r}}|L(r,\chi)|^2$ for any positive integer $r\geq 3$.…

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This…

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}…

We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} \chi(n) e(n\theta) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine…

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

Let $g \geq 3$ be fixed and odd, and for large $q$ let $\chi$ be a primitive Dirichlet character modulo $q$ of order $g$. Conditionally on GRH we improve the existing upper bounds in the P\'{o}lya-Vinogradov inequality for $\chi$, showing…

We use the large sieve inequality for smooth numbers due to S. Drappeau, A. Granville and X. Shao (2017), together with some other arguments, to improve their bounds on the frequency of pairs $(q,\chi)$ of moduli $q$ and primitive…