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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

REFERENCES

1.Bellman, R. and Shapiro, H. N.On a problem in additive number theory, Ann. of Math. (2) 49 (1948), 333–340.CrossRefGoogle Scholar
2.Bush, L. E.An asymptotic formula for the average sums of the digits of integers, Amer. Math. Monthly 47 (1940), 154–156.CrossRefGoogle Scholar
3.Coquet, J., Power sums of digital sums, J. Number Theory 22 (1986), 161–176.Google Scholar
4.Davenport, H., The Higher Arithmetic (Hutchinson's University Library, 1952).Google Scholar
5.Delange, H., Sur la fonction sommatoire de la fonction “somme des chiffres”, Enseign. Math. 21 (1975), 31–47.Google Scholar
6.Drazin, M. P. and Griffiths, J. S., On the decimal representation of integers, Proc. Cambridge Philos. Soc. 48 (1952), 555–565.CrossRefGoogle Scholar
7.Foster, D. M. E., Estimates for a remainder term associated with the sum of digits function, Glasgow Math. J. 29 (1987), 109–129.CrossRefGoogle Scholar
8.Kirschenhofer, P. and Tichy, R. F., On the distribution of digits in Cantor representations of integers, J. Number Theory 18 (1984), 121–134.CrossRefGoogle Scholar
9.Larcher, G. and Tichy, R. F., Some number-theoretical properties of generalized sum-of-digits functions, Acta Arith. 52 (1989), 183–196.Google Scholar
10.Mcilroy, M. D., The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput. 3 (1974), 255–261.CrossRefGoogle Scholar
11.Mirsky, L., A theorem on representations of integers in the scale of r, Scripta Math. 15 (1949), 11–12.Google Scholar
12.Shiokawa, I., On a problem in additive number theory, Math. J. Okayama Univ. 16 (1974), 167–176.Google Scholar
13.Stolarsky, K. B., Power and exponential sums of digital sums related to binomial coefficient parity, SIAM J. Appl. Math. 32 (1977), 717–730.CrossRefGoogle Scholar
14.Trollope, J. R., An explicit expression for binary digital sums, Math. Mag. 41 (1968), 21–25.CrossRefGoogle Scholar
15.Trollope, J. R., Generalized bases and digital sums, Amer. Math. Monthly 74 (1967), 690–694.Google Scholar