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

@article{Granville2005LargeCS, 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}, year={2005}, volume={20}, pages={357-384} }

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…

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## References

SHOWING 1-10 OF 19 REFERENCES

### THE SPECTRUM OF MULTIPLICATIVE FUNCTIONS

- Mathematics
- 2007

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

- Mathematics
- 1967

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…

### UPPER BOUNDS FOR |L(1, χ)|

- Philosophy, Mathematics
- 2001

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

- Mathematics
- 1999

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

- Philosophy
- 1977

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

- Mathematics
- 1984

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

- Mathematics
- 1999

We provide estimates for the exponential sum