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Explicit Burgess bound for composite moduli. (English) Zbl 1489.11116

Summary: We prove an explicit version of Burgess’ bound on character sums for composite moduli.

MathOverflow Questions:

Explicit version of the Burgess theorem

MSC:

11L40 Estimates on character sums
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Software:

MathOverflow

References:

[1] Ankeny, N., The least quadratic non residue, Ann. of Math. (2)55(3) (1952) 67-72. · Zbl 0046.04006
[2] Bach, E. and Shallit, J., Efficient Algorithms (MIT Press, 1996). · Zbl 0873.11070
[3] Y. Bilu, Explicit version of the Burgess theorem (2019), https://mathoverflow.net/questions/327455/explicit-version-of-the-burgess-theorem.
[4] Booker, A., Quadratic class numbers, and character sums, Math. Comp.75 (2006) 1481-1493. · Zbl 1095.11057
[5] M. Bordignon, Partial Gaussian sums and the Pólya-Vinogradov inequality for primitive characters, preprint (2020), arXiv:2001.05114v1 [math.NT]. · Zbl 1467.11078
[6] Burgess, D., On character sums and primitive roots, Proc. London Math. Soc.12(3) (1962) 179-192. · Zbl 0106.04003
[7] Burgess, D., On character sums and \(L\)-Series, Proc. London Math. Soc.12(3) (1962) 193-206. · Zbl 0106.04004
[8] Burgess, D., On character sums and \(L\)-Series II, Proc. London Math. Soc.13(3) (1963) 524-536. · Zbl 0123.04404
[9] Burgess, D., The character sum estimate with \(r=3\), J. London Math. Soc.33(2) (1986) 219-226. · Zbl 0593.10033
[10] Conrey, J. and Iwaniec, H., The cubic moment of central values of automorphic \(L\)-functions, Ann. of Math.151(3) (2000) 1175-1216. · Zbl 0973.11056
[11] F. Francis, An investigation into several explicit versions of Burgess’ bound, preprint (2019), arXiv:1910.13669 [math.NT].
[12] Heath-Brown, D., Hybrid bounds for Dirichlet \(L\)-functions, II, Q. J. Math.31(2) (1980) 157-167. · Zbl 0396.10030
[13] Lamzouri, Y., Li, X. and Soundararajan, K., Conditional bounds for the least quadratic non-residue and related problems, Math. Comp.84(295) (2015) 2391-2412. · Zbl 1326.11058
[14] McGown, K., On the constant in Burgess’ bound for the number of consecutive residues or non-residues, Funct. Approx. Comment. Math.46(2) (2012) 273-284. · Zbl 1301.11061
[15] Montgomery, H. and Vaughan, R., Exponential sums with multiplicative coefficients, Invent. Math.43 (1977) 69-82. · Zbl 0362.10036
[16] Montgomery, H. and Vaughan, R., Multiplicative Number Theory I: Classical Theory (Cambridge University Press, 2006). · Zbl 1142.11001
[17] Nicolas, J. and Robin, G., Majorations explicites pour le nombre de diviseurs de \(n\), Canad. Math. Bull.26(4) (1983) 485-492. · Zbl 0497.10034
[18] Paley, R., A theorem on characters, J. London Math. Soc.7(3) (1932) 28-32. · JFM 58.0192.01
[19] I. Petrow and M. Young, The fourth moment of Dirichlet \(L\)-functions along a coset and the Weyl bound, preprint (2019), arXiv:1908.10346 [math.NT].
[20] Petrow, I. and Young, M., The Weyl bound for Dirichlet \(L\)-functions of cube-free conductor, Ann. of Math.192(2) (2020) 437-486. · Zbl 1460.11111
[21] Pólya, G., Über die Verteilung der quadratischen Reste und Nichtreste, Nachr. Ges. Wiss. Göttingen Math-Phys. Kl.1948 (1918) 21-29. · JFM 46.0265.02
[22] Robin, G., Estimation de la fonction de Tchebychef \(\theta\) sur le k-iéme nombres premiers et grandes valeurs de la fonction \(\omega(n)\) nombre de diviseurs premiers de n, Acta Arith.42 (1983) 367-389. · Zbl 0475.10034
[23] Shoup, V., Searching for primitive roots in finite fields, Math. Comp.58(97) (1992) 369-380. · Zbl 0747.11060
[24] Treviño, E., The least \(k\) th power non-residue, J. Number Theory149 (2015) 201-224. · Zbl 1325.11102
[25] Vinogradov, I., Perm. Univ. Fiz.-Mat. ob.-vo Zh.1 (1918) 18-24 and 94-98.
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