Large character sums: Burgess's theorem and zeros of LL-functions

We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds under the weaker assumption that `$100\%$' of the zeros of $L(s,\chi)$ up to height $\tfrac 14$ lie on the critical line; and establish various other consequences of having large character sums

On Bilinear Exponential and Character Sums with Reciprocals of Polynomials

We give nontrivial bounds for the bilinear sums $$\sum_{u = 1}^{U} \sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v))$$ where $\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial $f\in {\mathbb F}_p[X]$ and some complex weights $\{\alpha_u\}$, $\{\beta_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear sums with Kloosterman fractions. We also obtain new bounds for similar sums with multiplicative characters of ${\mathbb F}_p$

Asymptotics of q-Plancherel measures

In this paper, we are interested in the asymptotic size of rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order $n$, so it does not fit in the context studied by P. Biane and P. \'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method works also for other measures, for instance those coming from Schur-Weyl representations.Comment: 27 pages, 5 figures. Version 2: a lot of corrections suggested by anonymous referees have been made. To appear in PTR

The value distribution of incomplete Gauss sums

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author

RLZAP: Relative Lempel-Ziv with Adaptive Pointers

Relative Lempel-Ziv (RLZ) is a popular algorithm for compressing databases of genomes from individuals of the same species when fast random access is desired. With Kuruppu et al.'s (SPIRE 2010) original implementation, a reference genome is selected and then the other genomes are greedily parsed into phrases exactly matching substrings of the reference. Deorowicz and Grabowski (Bioinformatics, 2011) pointed out that letting each phrase end with a mismatch character usually gives better compression because many of the differences between individuals' genomes are single-nucleotide substitutions. Ferrada et al. (SPIRE 2014) then pointed out that also using relative pointers and run-length compressing them usually gives even better compression. In this paper we generalize Ferrada et al.'s idea to handle well also short insertions, deletions and multi-character substitutions. We show experimentally that our generalization achieves better compression than Ferrada et al.'s implementation with comparable random-access times

Multispecies Weighted Hurwitz Numbers

The construction of hypergeometric $2D$ Toda $\tau$-functions as generating functions for weighted Hurwitz numbers is extended to multispecies families. Both the enumerative geometrical significance of multispecies weighted Hurwitz numbers, as weighted enumerations of branched coverings of the Riemann sphere, and their combinatorial significance in terms of weighted paths in the Cayley graph of $S_n$ are derived. The particular case of multispecies quantum weighted Hurwitz numbers is studied in detail.Comment: this is substantially enhanced version of arXiv:1410.881