Abstract
It is well-known that cancellation in short character sums (e.g. Burgess’ estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess’ work, so it is desirable to find another approach to nonresidues. In this note we formulate a new line of attack on the least nonresidue via long character sums, an active area of research. Among other results, we demonstrate that improving the constant in the Pólya–Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod k) is bounded by \((\log k)^{1.4}\).
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References
Ankeny, N C.: The least quadratic non residue. Ann. Math. 55(2), 65–72 (1952)
Balog, A., Granville, A., Soundararajan, K.: Multiplicative functions in arithmetic progressions, Ann. Math. Qué. 37(1), 3–30 (2013)
Bober, J.: Averages of character sums, Preprint available at arXiv:1409.1840
Goldmakher, L.: Character sums to smooth moduli are small, Canad. J. Math. 62(5), 1099–1115 (2010)
Goldmakher, L.: Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality. Algebra Num.b Theory 6(1), 123–163 (2012)
Goldmakher, Leo, Lamzouri, Youness: Lower bounds on odd order character sums. Int. Math. Res. Not. IMRN 21, 5006–5013 (2012)
Goldmakher, L., Lamzouri, Y.: Large even order character sums. Proc. Am. Math. Soc. 142(8), 2609–2614 (2014)
Granville, A.: Smooth numbers: computational number theory and beyond. Math. Sci. Res. Inst. Publ., 44, pp 267–323, Cambridge Univ. Press, Cambridge (2008)
Granville, A., Soundararajan, K.: Large character sums: pretentious characters and the Pólya–Vinogradov theorem. J. Am. Math. Soc. 20(2), 357–384 (2007)
Hildebrand, A.: A note on Burgess’ character sum esimate. C. R. Math. Rep. Acad. Sci. Canada. VIII(1), 35–37 (1986)
Hildebrand, A.: Large values of character sums. J. Numb. Theory 29(3), 271–296 (1988)
Stephens, P.J.: Optimizing the size of \(L(1,\,\chi )\). Proc. Lond. Math. Soc. 24(3), 1–14 (1972)
Acknowledgments
We are grateful to John Friedlander and Soundararajan for encouragement and some helpful suggestions. We would also like to thank Enrique Treviño for correcting a misunderstanding about Hildebrand’s result, and the anonymous referee for meticulously reading the paper and making useful suggestions. The second author was partially supported by an NSERC Discovery Grant.
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Bober, J.W., Goldmakher, L. Pólya–Vinogradov and the least quadratic nonresidue. Math. Ann. 366, 853–863 (2016). https://doi.org/10.1007/s00208-015-1353-2
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DOI: https://doi.org/10.1007/s00208-015-1353-2