×

On the mean values of certain character sums. (English) Zbl 1303.11093

Let \(q\geq 5\) be an odd number, this paper gives asymptotic formulae to the character sums: \[ \mathop{\sum\nolimits^{*}}_{\substack{\chi \mod q,\\ \chi(-1)=-1}}\left|\sum_{1\leq a\leq q/4}a\chi(a)\right|^{4} \] and \[ \mathop{\sum\nolimits^{*}}_{\substack{\chi \mod q,\\ \chi(-1)=1}} \left|\sum_{1\leq a\leq q/4}a\chi(a)\right|^{4}, \] where \(\sum^{*}\) denotes the summation over primitive characters modulo \(q\).
The authors express the character sum in terms of Gauss sums and \(L\)-functions, then the mean values of \(L\)-functions give the asymptotic formulae.

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11L40 Estimates on character sums
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ayoub, R.; Chowla, S.; Walum, H., On sums involving quadratic characters, Journal of the London Mathematical Society, 42, 152-154 (1967) · Zbl 0146.26804
[2] Fine, N. J., On a question of Ayoub, Chowla and Walum concerning character sums, Illinois Journal of Mathematics, 14, 88-90 (1970) · Zbl 0227.10027
[3] Williams, K. S., A class of character sums, Journal of the London Mathematical Society, 3, 2, 67-72 (1971) · Zbl 0213.32804
[4] Toyoizumi, M., On certain character sums, Acta Arithmetica, 55, 3, 229-232 (1990) · Zbl 0702.11054
[5] Peral, J. C., Character sums and explicit estimates for \(L\)-functions, Contemporary Mathematics, 189, 4, 449-459 (1995) · Zbl 0837.11045
[6] Liu, H.; Zhang, W., A note on certain character sums, Soochow Journal of Mathematics, 32, 2, 295-300 (2006) · Zbl 1230.11100
[7] Burgess, D. A., Mean values of character sums II, Mathematika, 34, 1, 1-7 (1987) · Zbl 0626.10036
[8] Xu, Z.; Zhang, W., On the \(2k\)-th power mean of the character sums over short intervals, Acta Arithmetica, 121, 2, 149-160 (2006) · Zbl 1153.11046
[9] Xu, Z.; Zhang, W., Fourth power mean of character sums, Acta Arithmetica, 135, 1, 31-49 (2008) · Zbl 1229.11111
[10] Berndt, B. C., Classical theorems on quadratic residues, L’Enseignement Mathématique, 22, 3-4, 261-304 (1976) · Zbl 0337.10031
[11] Pólya, G., Über die Verteilung der quadratische Reste und Nichtreste, Göttingen Nachrichten, 21-29 (1918) · JFM 46.0265.02
[12] Zhang, W., On a Cochrane sum and its hybrid mean value formula, Journal of Mathematical Analysis and Applications, 267, 1, 89-96 (2002) · Zbl 1106.11303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.