A note on Dirichlet characters

Hudson, Richard H.

doi:10.1090/S0025-5718-1973-0337850-8

A note on Dirichlet characters
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Math. Comp. 27 (1973), 973-975 Request permission

Abstract:

Denoting by $r(k,m,p)$ the first occurrence of m consecutive kth power residues of a prime $p \equiv 1 \pmod k$, we show that $r(k,m,p) > c\log p$ for infinitely many p (c is an absolute constant) provided that k is even and $m \geqq 3$.

References

S.-B. Preuss. Akad. Wiss. Phys. Math. Kl.

D. A. Burgess, A note on the distribution of residues and non-residues, J. London Math. Soc. 38 (1963), 253–256. MR 148628, DOI 10.1112/jlms/s1-38.1.253
P. D. T. A. Elliott, Some notes on $k$-th power residues, Acta Arith. 14 (1967/68), 153–162. MR 228419, DOI 10.4064/aa-14-2-153-162
Richard H. Hudson, A bound for the first occurrence of three consecutive integers with equal quadratic character, Duke Math. J. 40 (1973), 33–39. MR 314743
D. H. Lehmer and Emma Lehmer, On runs of residues, Proc. Amer. Math. Soc. 13 (1962), 102–106. MR 138582, DOI 10.1090/S0002-9939-1962-0138582-6
D. H. Lehmer, E. Lehmer, W. H. Mills, and J. L. Selfridge, Machine proof of a theorem on cubic residues, Math. Comp. 16 (1962), 407–415. MR 162379, DOI 10.1090/S0025-5718-1962-0162379-2
U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S. 15(57) (1944), 139–178 (English, with Russian summary). MR 0012111
W. H. Mills, Bounded consecutive residues and related problems, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 170–174. MR 0176958
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689
Hans Salié, Über den kleinsten positiven quadratischen Nichtrest nach einer Primzahl, Math. Nachr. 3 (1949), 7–8 (German). MR 33844, DOI 10.1002/mana.19490030104

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