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An extension of a theorem of Heilbronn
Published online by Cambridge University Press: 26 February 2010
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It was proved by Heilbronn that if ε > 0, N > 1 and ϑ is any real number then there exists an integer n satisfying 1 ≤ n ≤ N such that
where C depends only on ε. Here ║α║ denotes the difference between α and the nearest integer, taken positively. Professor Heilbronn has remarked (in conversation) that the exponent of N cannot be decreased beyond -1, since if p is an odd prime and a is not divisible by p then
for 1 ≤ n ≤ p—1. He has also remarked that if one could improve the exponent of N to -1 + η, say, it would follow that the absolutely least quadratic non-residue (mod p) is less than Cpn. For if a is a quadratic non-residue (mod p) then so is each of the numbers an2 (1 ≤ n ≤ p—1) and ║an2/p║<Cp-1+η implies that an2 is congruent (mod p) to a number of absolute value less than Cpη.
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- Copyright © University College London 1958
References
page 30 note ‡ Quart. J. of Math., 19 (1948), 249–256.Google Scholar
page 30 note § Davenport, H., “Indefinite quadratic forms in many variables”, Mathematika, 3 (1956), 81–101.CrossRefGoogle Scholar
page 31 note † A more detailed proof of a similar result has appeared in Davenport, H., “Indefinite quadratic forms in many variables (II)”, Proc. London Math. Soc. (3), 8 (1958), 109–126.CrossRefGoogle Scholar This paper will be referred to as D.
page 31 note ‡ See D., §2.
page 32 note † See D., equation (5).
page 33 note † Translated Roth and Davenport (Interscience Publishers, 1954).
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