P\'olya-Vinogradov and the least quadratic nonresidue

It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess' work, so it is desirable to find a new approach to nonresidues. The goal of this note is to demonstrate a new line of attack via long character sums, a currently active area of research. Among other results, we demonstrate that improving the constant in the P\'{o}lya-Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod $k$) is bounded by $(\log k)^{1.4}$.Comment: 9 pages; a few small corrections from the previous versio

The distribution of the maximum of character sums

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.Comment: 16 pages, 1 figure, new version with correction

Factorial ratios, hypergeometric series, and a family of step functions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135221/1/jlms0422.pd

On a discrete version of Tanaka's theorem for maximal functions

In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\widetilde{M}$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ of bounded variation, $$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$ where $\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C \|f\|_{\ell^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.Comment: V4 - Proof of Lemma 3 update