Four Constructions of Asymptotically Optimal Codebooks via Additive Characters and Multiplicative Characters

In this paper, we present four new constructions of complex codebooks with multiplicative characters, additive characters, and quadratic irreducible polynomials and determine the maximal cross-correlation amplitude of these codebooks. We prove that the codebooks we constructed are asymptotically optimal with respect to the Welch bound. Moreover, we generalize the result obtained by Zhang and Feng and contain theirs as a special case. The parameters of these codebooks are new.

A Polynomial Sieve and Sums of Deligne Type

Let $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we present a new bound for $N(f,F,B):=|\{\textbf{x}\in \mathbb{Z}^{n+1}:\max _{0\leq i\leq n}|x_{i}|\leq B,\exists t\in \mathbb{Z}\textrm{ such that}\ f(t)=F(\textbf{x})\}|.$ To do this, we introduce a generalization of the power sieve [14, 28] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

Some Riemann Hypotheses from random walks over primes

The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes, we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is [Formula: see text], then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this [Formula: see text] growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results, we obtain an equation which relates every individual non-trivial zero of the [Formula: see text]-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet [Formula: see text]-functions due to the existence of the pole at [Formula: see text], in which the Riemann [Formula: see text]-function is a particular case.

On the distribution of Jacobi sums

Let \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters \chi_{1},\ldots,\chi_{m} of \mathbf{F}_{q}^{\times} , we let J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2 , the normalized Jacobi sum q^{-1/2}J(\chi_{1},\chi_{2}) ( \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q\to\infty , when \chi_{1} and \chi_{2} run through all nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} . In this paper, we show a similar property for m\geq 2 . More generally, we show that the normalized Jacobi sum q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m=2 answers a question of Shparlinski.

Stratification for Multiplicative Character Sums

We prove a stratification result for certain families of n-dimensional (complete algebraic) multiplicative character sums. The character sums we consider are sums of products of r multiplicative characters evaluated at rational functions, and the families (with nr parameters) are obtained by allowing each of the r rational functions to be replaced by an “offset”, that is, a translate, of itself. For very general such families, we show that the stratum of the parameter space on which the character sum has maximum weight $n+j$ has codimension at least j⌊(r − 1)/2(n − 1)⌋ for 1 ≤ j ≤ n − 1 and ⌈nr/2⌉ for j = n. This result is used to obtain multivariate Burgess bounds in joint work with Lillian Pierce.