Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

  title={Large character sums: Pretentious characters and the P{\'o}lya-Vinogradov theorem},
  author={Andrew Granville and Kannan Soundararajan},
  journal={Journal of the American Mathematical Society},
In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for… 

Pólya–Vinogradov and the least quadratic nonresidue

It is well-known that cancellation in short character sums (e.g. Burgess’ estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since

Pólya–Vinogradov and the least quadratic

. It is well-known that cancellation in short character sums (e.g. Burgess’ es-timates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since

Improving the Burgess bound via Pólya-Vinogradov

We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bound on certain

On the constant in the Pólya-Vinogradov inequality

Lower bounds on odd order character sums

A classical result of Paley shows that there are infinitely many quadratic characters $\chi\mod{q}$ whose character sums get as large as $\sqrt{q}\log \log q$; this implies that a conditional upper

On character sums and exponential sums over generalized arithmetic progressions

Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum ∑a∈Aχ(a) over all proper generalized arithmetic progressions A⊂ℤ/q ℤ of rank r:

Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters

In this paper we obtain a new fully explicit constant for the P\'olya-Vinogradov inequality for primitive characters. Given a primitive character $\chi$ modulo $q$, we prove the following upper bound

A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the

Long large character sums

This paper proves a lower bound for maxχ≠χ0|∑n⩽xχ(n) so that the expected maximal value of character sums for most characters is recovered.



Large values of character sums


Dedicated to Richard Guy on his 80th birthday, for all the inspiring problems that he has posed Contents 1 Introduction: Deenitions and properties of the spectrum 2 The natural and logarithmic

Multiplicative Number Theory

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The


Given a non-principal Dirichlet character χ (mod q), an important problem in number theory is to obtain good estimates for the size of L(1, χ). The best bounds known give that q−ǫ ≪ǫ |L(1, χ)| ≪ log

Large character sums

Assuming the Generalized Riemann Hypothesis, the authors study when a character sum over all n infinity and q -> infinity (q is the size of the finite field).

Applications de la formule des traces aux sommes trigonométrigues

Dans cet expose, j’explique comment la formule des traces permet de calculer ou d’etudier diverses sommes trigonometriques et comment, jointe a la conjecture de Weil, elle peut permettre de les

Lecture Notes in Math

Une notion très importante pour la géometrie algébrique est celle de fibré projectif. Si f : X → S est un morphisme lisse entre variétés algébriques lisses, dont toute fibre est isomorphe à P, on se

Exponential sums with multiplicative coefficients

We provide estimates for the exponential sum