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A lower bound for a remainder term associated with the sum of digits function

Published online by Cambridge University Press:  20 January 2009

D. M. E. Foster
D. M. E. Foster
Affiliation:
Mathematical InstituteUniversity Of St AndrewsNorth HaughSt Andrews, KYI6 9SS
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Abstract

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For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 121 - 142
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

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