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On the Constant in the Pólya-Vinogradov Inequality

Published online by Cambridge University Press:  20 November 2018

Adolf Hildebrand
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Abstract

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The Pólya-Vinogradov inequality states that for any non-principal character x modulo q and any N ≧ 1,

where c is an absolute constant. We show that (*) holds with c = 2/(3π2) + o(1) in the case x is primitive and x (— 1) =1 with c = 1/(3π) + o(l) in the case x is primitive and x(— 1) = — 1- This improves by a factor 2/3 the previously best-known values for these constants.

Keywords

10G15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 3 , 01 September 1988 , pp. 347 - 352
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Burgess, D. A., The character sum estimate with r = 3. J. London Math. Soc. 33 (1986), pp. 219–226.Google Scholar
2. Landau, E., Abschätzungen von Charaktersummen, Einheiten und Klassenzahlen, Nachrichten Königl. Ges. Wiss. Göttingen (1918), pp. 79–97.Google Scholar
3. Montgomery, H. and Vaughan, R. C., Exponential sums with multiplicative coefficients, Inventiones Math. 43 (1977), pp. 69–82.Google Scholar
4. Polya, G., Uher die Verteilung der quadratischen Reste und Nichtreste, Nachrichten Königl. Ges. Wiss. Göttingen (1918), pp. 21–29.Google Scholar
5. Pólya, G. and Szegö, G., Aufgaben und Lehrsdtze aus der Analysis II, 3rd edition, Springer (1964).Google Scholar
6. Schur, I., Einige Bemerkungen zur vorstehenden Arbeit des Ilerrn G. Polya, Nachrichten Königl. Ges. Wiss. Göttingen (1918), pp. 30–36.Google Scholar
7. Vinogradov, I. M., Uber die Verteilung der quadratischen Reste und Nichtreste, J. Soc. Phys. Math. Univ. Permi 2 (1919), pp. 1–14.Google Scholar
8. Young, W. H., On a certain series of Eourier, Proc. London Math. Soc. (2) 11 (1913), pp. 357–366.Google Scholar