Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form
$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $
where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [
Citation: Wenjia Guo, Xiaoge Liu, Tianping Zhang. Dirichlet characters of the rational polynomials[J]. AIMS Mathematics, 2022, 7(3): 3494-3508. doi: 10.3934/math.2022194
Denote by $ \chi $ a Dirichlet character modulo $ q\geq 3 $, and $ \overline{a} $ means $ a\cdot\overline{a} \equiv 1 \bmod q $. In this paper, we study Dirichlet characters of the rational polynomials in the form
$ \sum\limits^{q}_{a = 1}'\chi(ma+\overline{a}), $
where $ \sum\limits_{a = 1}^{q}' $ denotes the summation over all $ 1\le a\le q $ with $ (a, q) = 1 $. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity [
| [1] | G. Pólya, Über die Verteilung der quadratischen Reste und Nichtreste, Göttingen Nachr., 167 (1918), 21–29. |
| [2] | I. M. Vinogradov, On the distribution of residues and non-residues of powers, J. Phys. Math. Soc. Perm., 1 (1918), 94–96. |
| [3] | A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. U.S.A., 34 (1948), 204–207. doi: 10.1073/pnas.34.5.204. |
| [4] |
W. P. Zhang, Y. Yi, On Dirichlet characters of polynomials, Bull. Lond. Math. Soc., 34 (2002), 469–473. doi: 10.1112/S0024609302001030. doi: 10.1112/S0024609302001030
|
| [5] |
W. P. Zhang, W. L. Yao, A note on the Dirichlet characters of polynomials, Acta Arith., 115 (2004), 225–229. doi: 10.4064/aa115-3-3. doi: 10.4064/aa115-3-3
|
| [6] |
W. P. Zhang, T. T. Wang, A note on the Dirichlet characters of polynomials, Math. Slovaca, 64 (2014), 301–310. doi: 10.2478/s12175-014-0204-z. doi: 10.2478/s12175-014-0204-z
|
| [7] | D. A. Burgess, Dirichlet characters and polynomials, Tr. Mat. Inst. Steklova, 132 (1973), 203–205. |
| [8] |
E. A. Grechnikov, An estimate for the sum of Legendre symbols, Math. Notes, 88 (2010), 819–826. doi: 10.1134/S0001434610110222. doi: 10.1134/S0001434610110222
|
| [9] |
V. Pigno, C. Pinner, Binomial character sums modulo prime powers, J. Théor. Nombres Bordeaux, 28 (2016), 39–53. doi: 10.5802/jtnb.927. doi: 10.5802/jtnb.927
|
| [10] |
X. X. Lv, W. P. Zhang, A new hybrid power mean involving the generalized quadratic Gauss sums and sums analogous to Kloosterman sums, Lith. Math. J., 57 (2017), 359–366. doi: 10.1007/s10986-017-9366-z. doi: 10.1007/s10986-017-9366-z
|
| [11] | J. F. Zhang, X. X. Lv, On the character sums of polynomials and L-functions, (Chinese), Acta Math. Sinica (Chin. Ser.), 62 (2019), 903–912. |
| [12] |
X. X. Lv, W. P. Zhang, On the character sum of polynomials and the two-term exponential sums, Acta. Math. Sin.-English Ser., 36 (2020), 196–206. doi: 10.1007/s10114-020-9255-y. doi: 10.1007/s10114-020-9255-y
|
| [13] |
A. P. Mangerel, Short character sums and the Pólya-Vinogradov inequality, The Quarterly Journal of Mathematics, 71 (2020), 1281–1308. doi: 10.1093/qmath/haaa031. doi: 10.1093/qmath/haaa031
|
| [14] | P. Xi, Moments of certain character sums that are unnamed, 2021, arXiv: 2105.15051. |
| [15] | A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Scientifiques et Industrielles, 1948. |
| [16] | A. Weil, Variétés abéliennes et courbes algébriques, Actualités Scientifiques et Industrielles, 1948. |
| [17] | L. K. Hua, Introduction to number theory, (Chinese), Peking: Science Press, 1964. |
Wenjia Guo, Xiaoge Liu, Tianping Zhang. Dirichlet characters of the rational polynomials[J]. AIMS Mathematics, 2022, 7(3): 3494-3508. doi: 10.3934/math.2022194
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