Abstract
We give a simple explicit formula of the exponential sum of digital sums by use of the distribution function of the binomial measure. We also apply our formula to the study of the power sums of digital sums.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. E. Bush, An asymptotic formula for the average sums of the digits of integers. Amer. Math. Monthly,47(1940), 154–156.
J. Coquet, A summation formula related to the binary digits. Invent. Math.,73(1983), 107–115.
J. Coquet, Power sums of digital sums. J. Number Theory,22(1986), 161–176.
H. Delange, Sur la fonction sommatoire de la fonction “somme des chiffres”. Enseign. Math., (2),21(1975), 31–47.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function. Fibonacci Numbers and Applications, Vol.5, Kluwer, 1993.
H. Harborth, Number of odd binomial coefficients. Proc. Amer. Math. Soc.,62(1977), 19–22.
M. Hata and M. Yamaguti, The Takagi function and its generalization. Japan J. Appl. Math.,1(1984), 183–199.
L. Mirsky, A theorem on representations of integers in the scale, ofr. Scripta Math.,15(1949) 11–12.
T. Okada, T. Sekiguchi and Y. Shiota, Applications of binomial measures to power sums of digital sums. J. Number Theory,52(1995), 256–266.
T. Okada, T. Sekiguchi and Y. Shiota, A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case. Preprint.
A. H. Osbaldestin, Digital sum problems. Fractals in the Fundamental and Applied Sciences, Elsevier Science Publishers B. V., North-Holland, 1991.
R. Salem, On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc.,53(1943), 427–439.
T. Sekiguchi and Y. Shiota, A generalization of Hata-Yamaguti’s results on the Takagi function. Japan J. Indust. Appl. Math.,8(1991), 203–219.
A. H. Stein, Exponential sums of sum-of-digit functions. Illinois J. Math.,30(1986), 660–675.
K. B. Stolarsky, Power and exponential sums of digital sums related to binomial coefficient parity. SIAM J. Appl. Math.,32(1977), 717–730.
J. R. Trollope, An explicit expression for binary digital sums. Math. Mag.,41(1968), 21–25.
Author information
Authors and Affiliations
About this article
Cite this article
Okada, T., Sekiguchi, T. & Shiota, Y. An explicit formula of the exponential sums of digital sums. Japan J. Indust. Appl. Math. 12, 425–438 (1995). https://doi.org/10.1007/BF03167237
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03167237