arXiv subjects

Bryce Kerr

Publications and source records attributed to Bryce Kerr.

45 records · Page 3Linked to original sources

arXiv2017

On the exponential large sieve inequality for sparse sequences modulo primes

We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $λ^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.

math.NT↗

arXiv2017

Incomplete Gauss sums modulo primes

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.

math.NT↗

arXiv2014

Some 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 circumstances we may improve on some results of Pierce for multidimensional mixed sums and we extend a bound for character sums with products of linear forms to the setting of mixed sums.

math.NT↗

arXiv2013

On a bound of Heath-Brown for Dirichlet $L$-functions on the critical line

Let $χ$ a primitive character$\pmod q$ and consider the Dirichlet $L$-function $$L(s,χ)=\sum_{n=1}^{\infty}\frac{χ(n)}{n^s}.$$ We give a new proof of an upper bound of Heath-Brown for $|L(s,χ)|$ on the critical line $s=1/2+it$

math.NT↗

arXiv2013

Character sums over shifted primes

For integer $q$, let $χ$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}Λ(n)χ(n+a),$$ where $Λ(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski

math.NT↗

arXiv2013

Rational exponential sums over the divisor function

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

math.NT↗

arXiv2013

Solutions to polynomial congruences in well shaped sets

We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when the solutions lie in a very general class of sets, including all convex sets.

math.NT↗

arXiv2013

Incomplete exponential sums over exponential functions

We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2πiag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.

math.NT↗

arXiv2012

On the Distribution of Values and Zeros of Polynomial Systems over Arbitrary Sets

Let $G_1,..., G_n \in \Fp[X_1,...,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\Fp$ of $p$ elements. A result of {É}. Fouvry and N. M. Katz shows that under some natural condition, for any fixed $\varepsilon$ and sufficiently large prime $p$ the vectors of fractional parts $$ (\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in Γ, $$ are uniformly distributed in the unit cube $[0,1]^n$ for any cube $Γ\in [0, p-1]^m$ with the side length $h \ge p^{1/2} (\log p)^{1 + \varepsilon}$. Here we use this result to show the above vectors remain uniformly distributed, when $\vec{x}$ runs through a rather general set. We also obtain new results about the distribution of solutions to system of polynomial congruences.

math.NT↗