On a kind of Dirichlet character sums. (English) Zbl 1470.11246

Ma, Rong; Zhang, Yulong; Zhang, Guohe

Abstr. Appl. Anal. 2013, Article ID 750964, 8 p. (2013).

Summary: Let \(p \geq 3\) be a prime and let \(\chi\) denote the Dirichlet character modulo \(p\). For any prime \(q\) with \(q < p\), define the set \(E \left(q, p\right) = \left\{a \mid 1 \leq a, \overline{a} \leq p, a \overline{a} \equiv 1 \pmod{p} \text{ and } a \equiv \overline{a} \pmod{q}\right\}\). In this paper, we study a kind of mean value of Dirichlet character sums \(\sum a \leq p \, a \in E \left(q, p\right) \chi(a)\), and use the properties of the Dirichlet \(L\)-functions and generalized Kloosterman sums to obtain an interesting estimate.

MSC:

11N37	Asymptotic results on arithmetic functions
11A07	Congruences; primitive roots; residue systems
11L05	Gauss and Kloosterman sums; generalizations

References:

[1]	Pólya, G., Über die Verteilung der quadratische Reste und Nichtreste, Quadratischer Nichtrest, 21-29 (1918) · JFM 46.0265.02
[2]	Hua, L. K.; Min, S. H., On a double exponential sum, Science Record, 1, 23-25 (1942) · Zbl 0060.10910
[3]	Burgess, D. A., On character sums and L-series. II, Proceedings of the London Mathematical Society, 13, 524-536 (1963) · Zbl 0123.04404
[4]	Burgess, D. A., The character sum estimate with \(r = 3\), Journal of the London Mathematical Society, 33, 219-226 (1986) · Zbl 0593.10033
[5]	Xi, P.; Yi, Y., On character sums over flat numbers, Journal of Number Theory, 130, 5, 1234-1240 (2010) · Zbl 1241.11099 · doi:10.1016/j.jnt.2009.10.011
[6]	Wenpeng, Z., On the distribution of inverses modulo \(n\), Journal of Number Theory, 61, 2, 301-310 (1996) · Zbl 0874.11006 · doi:10.1006/jnth.1996.0151
[7]	Richard, K. G., Unsolved Problems in Number Theory (1981), Springer · Zbl 0474.10001
[8]	Xu, Z.; Zhang, W., On a problem of D. H. Lehmer over short intervals, Journal of Mathematical Analysis and Applications, 320, 2, 756-770 (2006) · Zbl 1098.11050 · doi:10.1016/j.jmaa.2005.07.054
[9]	Zhang, W.; Zongben, X.; Yuan, Y., A problem of D. H. Lehmer and its mean square value formula, Journal of Number Theory, 103, 2, 197-213 (2003) · Zbl 1046.11070 · doi:10.1016/S0022-314X(03)00113-6
[10]	Zhang, W., A problem of D. H. Lehmer and its generalization, Compositio Mathematica, 86, 307-316 (1993) · Zbl 0783.11002
[11]	Ma, R.; Zhang, Y., On a kind of generalized Lehmer problem, Czechoslovak Mathematical Jounal, 62, 137, 1135-1146 (2012) · Zbl 1259.11090
[12]	Zhang, W., On the difference between a D. H. Lehmer number and its inverse modulo \(q\), Acta Arithmetica, 68, 3, 255-263 (1994) · Zbl 0826.11003
[13]	Lu, Y. M.; Yi, Y., On the generalization of the D. H. Lehmer problem, Acta Mathematica Sinica, 25, 8, 1269-1274 (2009) · Zbl 1170.11022 · doi:10.1007/s10114-009-7652-3
[14]	Chengdong, P.; Chengbiao, P., Elements of the Analytic Number Theory (1991), Beijing, China: Science Press, Beijing, China
[15]	Malyshev, A. V., A generalization of Kloosterman sums and their estimates, Vestnik Leningrad University, 15, 13, 59-75 (1960) · Zbl 0102.03801