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On a theorem in the additive theory of numbers due to evelyn and linfoot

Published online by Cambridge University Press:  24 October 2008

L. Mirsky
L. Mirsky
Affiliation:
Department of MathematicsUniversity of Sheffield
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Extract

Some years ago Evelyn and Linfoot found an asymptotic formula, with estimation of remainder, for the number of representations of a large integer n as the sum of s r-free integers. For s ≥ 4 their formula was subsequently sharpened by Barham and Estermann, and for s = 3 recently by me.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 3 , July 1948 , pp. 305 - 312
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

* Evelyn, C. J. A. and Linfoot, E. H., ‘On a problem in the additive theory of numbers’, (I), Math. Z. 30 (1929), 433–48;Google Scholar(II) J. Math. 164 (1931), 131–40;Google Scholar(III) Math. Z. 34 (1932), 637–44;Google Scholar(IV), Ann. Math. 32 (1931), 261–70;Google Scholar(V), Quart. J. Math. 3 (1932), 152–60.Google Scholar

† If r ≥ 2, then an integer is called r-free if it is not divisible by the rth power of any prime.

‡ Barham, C. L. and Estermann, T., ‘On the representations of a number as the sum of four or more N-numbers’, Proc. London Math. Soc. (2), 38 (1935), 340–53.Google Scholar

§ Mirsky, L., ‘On the number of representations of an integer as the sum of three r-free int, egers’, Proc. Cambridge Phil. Soc. 43 (1947), 433–41.CrossRefGoogle Scholar I take this opportunity for correcting a misprint in that paper. The inequality in equation (15) on p. 440 should read

∥ Evelyn, C. J. A. and Linfoot, E. H., ‘On a problem in the additive theory of numbers’, (VI), Quart. J. Math. 4 (1933), 309–14.CrossRefGoogle Scholar

We may note at once that the problem is trivial if n≢sk (mod q), or if the highest common factor of q and k is not r-free, since in either case Q r, s(n, q, k) = 0.

* See, for example, Scholz, A., Einführung in die Zahlentheorie (Göschen), Satz31.Google Scholar