Cancellation in a short exponential sum
type: Item
instance of: scholarly article
title: Cancellation in a short exponential sum
zbMATH Open document ID: 1198.11074
published in: Journal of Number Theory
publication date: 2010-07-22T00:00:00Z
review text: Let \(q\) be an odd integer, \(\tau\) be the order of 2 modulo \(q\) and let \(\xi\) be a primitive \(q\)th root of unity. In this paper bounds for the sum \[ s(\xi)=\sum\limits_{r=1}^{\tau}\xi^{2^r} \] are obtained. The behavior of \(s(\xi)\) when \(q\) diverges along a sequence for which the quotient \(\tau/\mathcal{L}\) belongs to the range \([\mu,\,\nu]\) with \(1\leq \mu <\nu\), \(\nu-1\) is small enough and \(\mathcal{L}=[\log_2q]\) is studied. If \(U(l,\,k)\) is the number of representation of \(l\) as sum of \(k\) powers of two and \(\varrho\) is a constant such that \(\max\limits_{l}\{U(l,\,k)\}\leq (\varrho k)^k.k^{o(1)}\) holds for enough large \(k\), the upper bound \[ \max\limits_{\xi :\xi ^q=1\atop{\xi \,\text{primitive}}} |s(\xi)|\leq \tau-c_{\mu,\nu}(\varrho)+o_{\varrho,\mu,\nu}(1), \] where \[ c_{\mu,\nu}(\varrho)=\max\limits_{x\geq 1}\bigg\{-\frac{\nu-1/2}{\mu}x^2\log 2+x\log(x/\varrho) \bigg\} \] is obtained. An explicit value for \(\varrho\) is obtained and thereby some improvements are mentioned.
reviewed by: Tatyana L. Todorova
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 11L99
zbMATH DE Number: 5759069
zbMATH Keywords: exponential sums
zbMATH Keywords: k-representations
author: Giuseppe Molteni
describes a project that uses: PARI/GP
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.03.002
OpenAlex ID: W2071609643
cites work: Trigonometric sums in number theory and analysis. Transl. from the Russian
cites work: New bounds on exponential sums related to the Diffie-Hellman distributions
cites work: Estimates on exponential sums related to the Diffie-Hellman distributions
cites work: On an exponential sum related to the Diffie-Hellman cryptosystem
cites work: Character sums with exponential functions
cites work: Primes and powers of 2
cites work: Representation of a 2-power as sum of \(k\) 2-powers: a recursive formula
cites work: New bounds for Gauss sums derived from KTH powers, and for Heilbronn's exponential sum
cites work: Integers represented as a sum of primes and powers of two.
cites work: Q4328069
cites work: Q4264395
cites work: Q4003239
cites work: The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes (II)
cites work: The number of powers of 2 in a representation of large even integers. I
cites work: The number of powers of 2 in a representation of large even integers. II
cites work: On the almost Goldbach problem of Linnik
cites work: On Linnik's approximation to Goldbach's problem, I
cites work: On the distribution of the Diffie-Hellman pairs
DOI: 10.1016/J.JNT.2010.03.002
description: scientific article; zbMATH DE number 5759069