Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-15T12:44:07.314Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimates for a remainder term associated with the sum of digits function

Published online by Cambridge University Press:  18 May 2009

D. M. E. Foster
D. M. E. Foster
Affiliation:
Mathematical Institute University of St Andrews, North Haugh, St Andrews, KY16 9SS
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If q(≥2) is a fixed integer it is well known that every positive integer k may be expressed uniquely in the form

We introduce the ‘sum of digits’ function

Both the above sums are of course finite. Although the behaviour of α(q, k) is somewhat erratic, its average behaviour is more regular and has been widely studied.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 1 , January 1987 , pp. 109 - 129
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

REFERENCES

1.Bellman, R., and Shapiro, H. N., On a problem in additive number theory, Ann. of Math. (2), 49 (1948), 333–40.CrossRefGoogle Scholar
2.Bush, L. E., An asymptotic formula for the average sums of the digits of integers, Amer. Math. Monthly 47 (1940), 154–6.CrossRefGoogle Scholar
3.Delange, H., Sur la fonction sommatoire de la fonction ‘somme des chiffres’, Enseignment Math. 21 (1975) 31–47.Google Scholar
4.Drazin, M. P., and Griffiths, J. S., On the decimal representation of integers, Proc. Cambridge Philos. Soc. 48 (1952), 555–565.CrossRefGoogle Scholar
5.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
6.McIlroy, M. D., The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput. 3 (1974), 255–261.CrossRefGoogle Scholar
7.Mirsky, L., A theorem on representations of integers in the scale of r, Scripta Math. 15 (1949), 11–12.Google Scholar
8.Shiokawa, I., On a problem in additive number theory, Math. J. Okayama Univ. 16 (1974), 167–176.Google Scholar
9.Stolarsky, K. B., Power and exponential sums of digital sums related to binomial coefficient parity, SIAMJ. Appl. Math. 32 (1977), 717–730.CrossRefGoogle Scholar
10.Trollope, J. R., An explicit expression for binary digital sums, Math. Mag. 41 (1968), 21–25.CrossRefGoogle Scholar
11.Trollope, J. R., Generalized bases and digital sums, Amer Math. Monthly 74 (1967), 690–694.CrossRefGoogle Scholar