On character sums and power residues

Norton, Karl K.

doi:10.1090/S0002-9947-1972-0296034-8

On character sums and power residues
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by Karl K. Norton

Trans. Amer. Math. Soc. 167 (1972), 203-226

DOI: https://doi.org/10.1090/S0002-9947-1972-0296034-8

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Erratum: Trans. Amer. Math. Soc. 174 (1972), 507.

Abstract:

Sharp estimates are given for a double sum involving Dirichlet characters. These are applied to the problem of estimating certain sums whose values give a measure of the average distance between successive power residues to an arbitrary modulus. A particularly good result of the latter type is obtained when the modulus is prime.

References

D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc. (3) 12 (1962), 179–192. MR 132732, DOI 10.1112/plms/s3-12.1.179
D. A. Burgess, On character sums and $L$-series, Proc. London Math. Soc. (3) 12 (1962), 193–206. MR 132733, DOI 10.1112/plms/s3-12.1.193
D. A. Burgess, On character sums and $L$-series. II, Proc. London Math. Soc. (3) 13 (1963), 524–536. MR 148626, DOI 10.1112/plms/s3-13.1.524
D. A. Burgess, A note on $L$-functions, J. London Math. Soc. 39 (1964), 103–108. MR 160762, DOI 10.1112/jlms/s1-39.1.103
H. Davenport and P. Erdös, The distribution of quadratic and higher residues, Publ. Math. Debrecen 2 (1952), 252–265. MR 55368, DOI 10.5486/pmd.1952.2.3-4.18
P. Erdös, The difference of consecutive primes, Duke Math. J. 6 (1940), 438–441. MR 1759, DOI 10.1215/S0012-7094-40-00635-4
Paul Erdös, On the coefficients of the cyclotomic polynomial, Bull. Amer. Math. Soc. 52 (1946), 179–184. MR 14110, DOI 10.1090/S0002-9904-1946-08538-9
P. Erdős, On the integers relatively prime to $n$ and on a number-theoretic function considered by Jacobsthal, Math. Scand. 10 (1962), 163–170. MR 146125, DOI 10.7146/math.scand.a-10523
Paul Erdős, Some recent advances and current problems in number theory, Lectures on Modern Mathematics, Vol. III, Wiley, New York, 1965, pp. 196–244. MR 0177933

An introduction to the theory of numbers

C. Hooley, On the difference of consecutive numbers prime to $n$, Acta Arith. 8 (1962/63), 343–347. MR 155807, DOI 10.4064/aa-8-3-343-347
Christopher Hooley, On the difference between consecutive numbers prime to $n$. II, Publ. Math. Debrecen 12 (1965), 39–49. MR 186641, DOI 10.5486/pmd.1965.12.1-4.06
Christopher Hooley, On the difference between consecutive numbers prime to $n$. III, Math. Z. 90 (1965), 355–364. MR 183702, DOI 10.1007/BF01112354
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände, Chelsea Publishing Co., New York, 1953 (German). 2d ed; With an appendix by Paul T. Bateman. MR 0068565
L. Moser and R. A. MacLeod, The error term for the squarefree integers, Canad. Math. Bull. 9 (1966), 303–306. MR 200251, DOI 10.4153/CMB-1966-039-4
Karl K. Norton, Upper bounds for $k\textrm {th}$ power coset representatives modulo $n$, Acta Arith. 15 (1968/69), 161–179. MR 240065, DOI 10.4064/aa-15-2-161-179
Karl K. Norton, On the distribution of $k\textrm {th}$ power residues and non-residues modulo $n$, J. Number Theory 1 (1969), 398–418. MR 250992, DOI 10.1016/0022-314X(69)90003-1

On the distribution of power residues and nonresidues

T. Vijayaraghavan, On a problem in elementary number theory, J. Indian Math. Soc. (N.S.) 15 (1951), 51–56. MR 43837
I. M. Vinogradov, Osnovy teorii čisel, Gosudarstvennoe Izdatel′stvo Tehniko-Teoretičeskoĭ Literatury, Moscow-Leningrad, 1949 (Russian). 5th ed.]. MR 0035779