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On a Congruence Related to Character Sums

Published online by Cambridge University Press:  20 November 2018

J. H. H. Chalk
J. H. H. Chalk*
Affiliation:
University of TorontoToronto, Canada
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Abstract

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If χ is a Dirichlet character to a prime-power modulus pα, then the problem of estimating an incomplete character sum of the form ∑1≤x≤h χ (x) by the method of D. A. Burgess leads to a consideration of congruences of the type

f(x)g'(x) - f'(x)g(x) ≡ 0(pα),

where fg(x) ≢ 0(p) and f, g are monic polynomials of equal degree with coefficients in Ζ. Here, a characterization of the solution-set for cubics is given in terms of explicit arithmetic progressions.

Keywords

10A1010G05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 4 , 01 December 1985 , pp. 431 - 439
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Burgess, D.A., On Character Sums and L-series, Proc. London Math. Soc, (3), 12 (1962), pp. 193–196.Google Scholar
2. Chalk, J.H.H., A New Proof of Burgess’ Theorem on Character Sums, C-R Math, Rep. Acad. Sci. Canada, No. 4 V (1983), pp. 163–168, (see Math. Reviews for a revised statement of the result).Google Scholar
3. Chalk, J.H.H. and Smith, R.A., Sándor's Theorem on Polynomial Congruences and Hensel's Lemma, C-R Math. Rep. Acad. Sci. Canada, No. 1, II (1982), pp. 49–54.Google Scholar
4. Davenport, H. and P. Erdös, The Distribution of Quadratic and Higher Residues, Publications Math ematicae, T-2, fasc, 3-4 (1952), pp. 252–265.Google Scholar