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A Polya-Vinogradov-type inequality on Z[i] [article]

Stephan Baier
2017 arXiv   pre-print
We establish a Polya-Vinogradov-type bound for finite periodic multipicative characters on the Gaussian integers.  ...  Classical Polya-Vinogradov over Z The classical Polya-Vinogradov inequality is the following bound. Theorem 1.1. Assume that y > 0 and χ is a non-principal Dirichlet character modulo q ≥ 2.  ...  A version of Polya-Vinogradov over Z[i] Next, we give a first version of the Polya-Vinogradov inequality for Z[i] whose proof is an easy extension of that for the classical case.  ... 
arXiv:1703.06498v2 fatcat:qtstux74rjfq7ikaalroe6skmy

An upper bound for the sum $\sum\sp {a+H}\sb {n=a+1}f(n)$ for a certain class of functions $f$

Edward Dobrowolski, Kenneth S. Williams
1992 Proceedings of the American Mathematical Society  
For a certain class of functions /: Z -> C an upper bound is obtained for the sum 5ZnÍa+i /(") • This bound is used to give a proof of a classical inequality due to Pólya and Vinogradov that does not require  ...  the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols J2x=i {  ...  and Vinogradov [10] What is particularly interesting about our proof of Theorem 2 is that it does not require the value of the modulus of the Gauss sum for a primitive character X modulo k ; namely,  ... 
doi:10.1090/s0002-9939-1992-1068118-6 fatcat:5qigss2dnbd4rnplgp2gwl35xe

Pólya-Vinogradov and the least quadratic nonresidue [article]

Jonathan Bober, Leo Goldmakher
2015 arXiv   pre-print
Among other results, we demonstrate that improving the constant in the P\'{o}lya-Vinogradov inequality would lead to significant progress on nonresidues.  ...  It is well-known that cancellation in short character sums (e.g. Burgess' estimates) yields bounds on the least quadratic nonresidue.  ...  The celebrated Pólya-Vinogradov bound on character sums immediately implies the bound n p ≪ √ p log p, and an innovation due to Vinogradov reduced this to n p ≪ p 1/(2 √ e) log 2 p.  ... 
arXiv:1311.7556v2 fatcat:3llf7eyr6vavbpbkkat4m7lsfa

Pólya–Vinogradov and the least quadratic nonresidue

Jonathan W. Bober, Leo Goldmakher
2015 Mathematische Annalen  
The celebrated Pólya-Vinogradov bound on character sums immediately implies the bound n p ≪ √ p log p, and an innovation due to Vinogradov reduced this to n p ≪ p 1/(2 √ e) log 2 p.  ...  To get a feel for the strength of Theorem 1, we explore several consequences. First, we consider the constant in the Pólya-Vinogradov inequality.  ... 
doi:10.1007/s00208-015-1353-2 fatcat:izivplwx7valflfeqcyw5fvbie

A Pólya–Vinogradov inequality for short character sums [article]

Matteo Bordignon
2021 arXiv   pre-print
In this paper we obtain a variation of the P\'{o}lya--Vinogradov inequality with the sum restricted to a certain height.  ...  Assume $\chi$ to be a primitive character modulo $q$, $\epsilon > 0$ and $N\le q^{1-\gamma}$, with $0\le \gamma \le 1/3$.  ...  Acknowledgements I would like to thank my supervisor Tim Trudgian for his kind help and his suggestions in developing this paper.  ... 
arXiv:2002.02640v2 fatcat:moswvjgepzgzdc5xj3sepa57dq

Singular Gauss sums, Polya-Vinogradov inequality for GL(2) and growth of primitive elements [article]

Satadal Ganguly, C. S. Rajan
2021 arXiv   pre-print
We establish an analogue of the classical Polya-Vinogradov inequality for $GL(2, \F_p)$, where $p$ is a prime. In the process, we compute the 'singular' Gauss sums for $GL(2, \F_p)$.  ...  As an application, we show that the collection of elements in $GL(2,\Z)$ whose reduction modulo $p$ are of maximal order in $GL(2, \F_p)$ and whose matrix entries are bounded by $x$ has the expected size  ...  Main ideas behind the proofs and the structure of the paper. The proof of Theorem 1.4 follows the usual approach for proving the classical Polya-Vinogradov inequality.  ... 
arXiv:1912.01310v2 fatcat:h4bbkrji3bawfkf2lrf4gvsp3e

On the constant in the Polya-Vinogradov inequality [article]

Bryce Kerr
2018 arXiv   pre-print
In this paper we obtain a new constant in the P\'{o}lya-Vinogradov inequality.  ...  Our argument follows previously established techniques which use the Fourier expansion of an interval to reduce to Gauss sums.  ...  ] have shown how improvements on the constant in the Polya-Vinogradov inequality may be used to obtain new estimates for short character sums.  ... 
arXiv:1807.09573v1 fatcat:3kol2quhuzbpdcaau357eqel7q

On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

Guohui Chen, Han Zhang
2014 The Scientific World Journal  
In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties ofN(k,r;p)and give two interesting asymptotic formulae for it.  ...  LetN(k,r;p)denote the number of alla1, a2, ⋯aksuch thata1a2⋯ak≡r mod pand 2†a1+a2+⋯ + ak.  ...  Acknowledgments The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.  ... 
doi:10.1155/2014/726053 pmid:25133256 pmcid:PMC4124714 fatcat:wnbyqdabdberxhczq4gwebo3uy

Remarks on the Pólya–Vinogradov Inequality

Carl Pomerance
2011 Integers  
AbstractWe establish a numerically explicit version of the PólyaVinogradov inequality for the sum of values of a Dirichlet character on an interval.  ...  While the technique of proof is essentially that of Landau from 1918, the result we obtain has better constants than in other numerically explicit versions that have been found more recently.  ...  And I thank Ke Gong for informing me of [14] .  ... 
doi:10.1515/integ.2011.039 fatcat:5r5mk2pcorgd7k34anmcmmdgqm

On two lattice points problems about the parabola [article]

Jing-Jing Huang, Huixi Li
2019 arXiv   pre-print
We also obtain essentially sharp upper bounds for the latter lattice points problem. Our proofs utilize techniques in Fourier analysis, quadratic Gauss sums and character sums.  ...  These results can be regarded as achieving the square root cancellation in the context of the parabola, whereas its analogues are wide open conjectures for the circle and the hyperbola.  ...  Therefore, we may apply the Pólya-Vinogradov inequality (Lemma 3) to the character sum h 1 ≤N χ(h 1 ) after another application of Lemma 2, and obtain S(N ) ≪ a 1 log a 1 .  ... 
arXiv:1902.06047v1 fatcat:szyufo7kmnbb5ncj2g4xfmovty

On the Sphere Problem

Fernando Chamizo, Henryk Iwaniec
1995 Revista matemática iberoamericana  
For the rst author, this visit was nanced by a NATO Collaborative Research Grant (CRG921184) and a CICYT grant, and for the second author by NSERC Grant A5123.  ...  The second author is grateful for the hospitality and support. This work was nished during our visits to the University of Toronto, we are both grateful for this opportunity.  ...  Applying Polya-Vinogradov inequality for each character sum in k gives W K M M 1=2+" : On the other hand by Lemma 4.1 W K M (K 7=8 + K M ;1=4 N 3=64 ) N " : 426 F. Chamizo and H.  ... 
doi:10.4171/rmi/178 fatcat:qk5zoqidfjddjb5m4nxh4v5m5m

A Pólya-Vinogradov inequality for short character sums

Matteo Bordignon
2020 Canadian mathematical bulletin  
In this paper we obtain a variation of the PólyaVinogradov inequality with the sum restricted to a certain height. Assume χ to be a primitive character modulo q, > 0 and N ≤ q 1−γ , with 0 ≤ γ ≤ 1/3.  ...  We prove that with c = 2/π 2 if χ is even and c = 1/π if χ is odd. The result is based on the work of Hildebrand [7] and Kerr [4] .  ...  Acknowledgements I would like to thank my supervisor Tim Trudgian for his kind help and his suggestions in developing this paper.  ... 
doi:10.4153/s0008439520000934 fatcat:w25ensu7sbatrd4gtzdpxgmj54

Character Sums Over The Prime Numbers [article]

N. A. Carella
2012 arXiv   pre-print
In addition, an omega result for character sums over the primes is also included.  ...  A few elementary estimates of a basic character sum over the prime numbers are derived here. These estimates are nontrivial for character sums modulo large q.  ...  The estimates of character sums over the primes derived here, by elementary methods, are simpler, and similar to the Polya-Vinogradov inequality, and the Paley inequality in forms and the theoretical frameworks  ... 
arXiv:1205.5454v1 fatcat:nwthnjvqn5aijdhj6qcgd42tou

An explicit Pólya-Vinogradov inequality via Partial Gaussian sums [article]

Matteo Bordignon, Bryce Kerr
2019 arXiv   pre-print
In this paper we obtain a new fully explicit constant for the P\'olya-Vinogradov inequality for squarefree modulus.  ...  We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients.  ...  limit of previous approaches to an explicit Pólya-Vinogradov inequality.  ... 
arXiv:1909.01052v1 fatcat:empmfvammbckhoypmmvxi4gd7i

Mean values of divisors twisted by quadratic characters

Peng Gao
2020 Acta Arithmetica  
A special case of the well-known Pólya-Vinogradov inequality (see [3, Chap. 23]) asserts that for any positive m = (where represents the square of a rational integer), and any Y > 0,  ...  As estimations for character sums have wide applications in analytic number theory, many important results have been obtained in this direction.  ...  The author is grateful to the referee for a very careful reading of the paper and many valuable suggestions. The author is supported in part by NSFC grant 11871082.  ... 
doi:10.4064/aa190417-23-7 fatcat:75oz5sy2q5dehg26ablec4u3oa
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