Abstract
Delange and Trollope proved that the average value of the sum of digits in base 2 representation of the integers 0, 1, ..., N − 1 is given by ½log2 N + δ(log2 N), where δ(x) is a continuous periodic function of period 1.
In this paper we prove that the variance fulfills 
where δ
1(x) is again continuous and periodic with period 1.
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References
H. Delange, Sur la fonction sommatoire de la fonction “somme de chiffres”, Enseign. Math. 21 (1975), 31–47.
Ph. Flajolet, L. Ramshaw, A note on Gray Code and Odd-Even Merge, SIAM J. Comput. 9 (1980) 142–158.
P. Kirschenhofer, Subblock occurrences in the q-ary representation of n, SIAM J. Alg. Disc. Meth. 4 (1983), 231–236.
P. Kirschenhofer, R. F. Tichy, On the distribution of digits in Cantor representations of integers, J. Number Th. 18 (1984), 121–134.
H. Prodinger, Generalizing the sum of digits function, SIAM J. Alg. Disc. Meth. 3 (1982), 35–42.
H. Trollope, An explicit expression for binary digital sums, Meth. Mag. 41 (1968), 21–25.
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© 1990 Springer-Verlag
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Kirschenhofer, P. (1990). On the variance of the sum of digits function. In: Hlawka, E., Tichy, R.F. (eds) Number-Theoretic Analysis. Lecture Notes in Mathematics, vol 1452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096983
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DOI: https://doi.org/10.1007/BFb0096983
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