ON CANTOR’S SINGULAR MOMENTS

HELMUT PRODINGER

Abstract. We evaluate a constant explicitly, thereby answering a question raised in [1].

In problem 10621 of the American Mathematical Monthly, Cantor’s singular moments J_{n} were to be computed. In the published answers [1] they come out as

J_{n} = \frac{2}{3(n + 1)} ∑_{j=0}^{n} \binom{n + 1}{j}\frac{B_{j}}{3 · 2^{j−1} − 1} for n ≥ 1

and J_{0} = 1, with Bernoulli numbers B_{n}.

The editor asked, whether it is possible to compute

J_{−1} = ∑_{n≥0} J_{n}

exactly.

The purpose of this note is to do that. In [5] we considered a similar problem, and the gentle reader is invited to consult this paper for more background about the technique, as well as [6, 4] for more information about the Cantor distribution.

Following the method described in [3], we can write J_{n} as a contour integral viz.

J_{n} = \frac{2}{3(n + 1)} · \frac{1}{2πi} ∫_{− \frac{1}{2} −i∞}^{− \frac{1}{2} +i∞} \frac{Γ(n + 2)Γ(1 − s)}{Γ(n + 2 − s)} \frac{ζ(1 − s)}{3 · 2^{s−1} − 1} ds

= \frac{2}{3} · \frac{1}{2πi} ∫_{− \frac{1}{2} −i∞}^{− \frac{1}{2} +i∞} \frac{Γ(n + 1)Γ(1 − s)}{Γ(n + 2 − s)} \frac{ζ(1 − s)}{3 · 2^{s−1} − 1} ds.

Therefore

∑_{n=0}^{N} J_{n} = J_{0} + \frac{2}{3} · \frac{1}{2πi} ∫_{− \frac{1}{2} −i∞}^{− \frac{1}{2} +i∞} \frac{Γ(N + 2)Γ(1 − s)}{Γ(N + 2 − s)s} \frac{ζ(1 − s)}{3 · 2^{s−1} − 1} ds

− \frac{2}{3} · \frac{1}{2πi} ∫_{− \frac{1}{2} −i∞}^{− \frac{1}{2} +i∞} \frac{Γ(2)Γ(1 − s)}{Γ(2 − s)s} \frac{ζ(1 − s)}{3 · 2^{s−1} − 1} ds.

From this form, one could even compute the asympotics as N → ∞. However, here, we only have to note that the first integral is of order N^{1−log_{2} 3}, which means that it goes to zero. Consequently

Date: April 14, 1999.

∑_{n≥0} J_{n} = 1 − \frac{2}{3} · \frac{1}{2πi} ∫_{− \frac{1}{2} −i∞}^{− \frac{1}{2} +i∞} \frac{1}{(1 − s)s} \frac{ζ(1 − s)}{3 · 2^{s−1} − 1} ds

= \frac{4}{3} + \frac{2}{3} · \frac{1}{2πi} ∫_{\frac{3}{2} −i∞}^{\frac{3}{2} +i∞} \frac{1}{s(s − 1)} \frac{ζ(s)}{3 · 2^{−s} − 1} ds

= \frac{4}{3} + \frac{2}{3} ∑_{k,m≥1} 3^{−k} · \frac{1}{2πi} ∫_{\frac{3}{2} −i∞}^{\frac{3}{2} +i∞} \frac{1}{s(s − 1)}(\frac{2^{k}}{m} )^{s} ds.

The last step was by using the Dirichlet series for ζ(s) and the geometric series, both valid for ℜs = \frac{3}{2}. A simple application of residue calculus, as it is often used in the context of the Mellin–Perron summation formula (see [7, 2]) evaluates the integrals inside the summation:

\frac{1}{2πi} ∫_{\frac{3}{2} −i∞}^{\frac{3}{2} +i∞} \frac{1}{s(s − 1)}t^{s}ds = \begin{cases}t − 1 & for t ≥ 1 \\ 0 & for t < 1 \end{cases} .

Therefore

∑_{n≥0} J_{n} = 1 + \frac{2}{3} ∑_{k≥1} ∑_{1≤m≤2^{k}} 3^{−k}(\frac{2^{k}}{m} − 1)

= 1 + \frac{2}{3} ∑_{k≥1} (\frac{2}{3})^{k}H_{2^{k}} − \frac{2}{3} ∑_{k≥1} (\frac{2}{3})^{k}

= −\frac{1}{3} + \frac{2}{3} ∑_{k≥1} (\frac{2}{3})^{k}H_{2^{k}} = 3.36465 07281 00925 16083 89349 6289 …,

with harmonic numbers H_{n} = ∑_{1≤k≤n} \frac{1}{k}.

References

[1] H. G. Diamond and B. Reznick. Problems and solutions. American Mathematical Monthly, 106:175–176, 1999.

[2] P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R.F. Tichy. Mellin transforms and asymptotics: Digital sums. Theoretical Computer Science, 123:291–314, 1994.

[3] P. Flajolet and R. Sedgewick. Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoretical Computer Science, 144:101–124, 1995.

[4] P. Grabner and H. Prodinger. Asymptotic analysis of the moments of the Cantor distribution. Statistics and Probability Letters, 26:243, 1996.

[5] P. Grabner and H. Prodinger. An asymptotic study of a recursion occurring in the analysis of an algorithm on broadcast communication. Information Processing Letters, 65:89–93, 1998.

[6] A. Knopfmacher and H. Prodinger. Exact and asymptotic formulæ for average values of order statistics of the Cantor distribution. Statistics and Probability Letters, 27:189, 1996.

[7] G. Tenenbaum. Introduction to analytic and probabilistic number theory. Cambridge University Press, 1995.

Helmut Prodinger, Centre for Applicable Analysis and Number Theory, Department of Mathematics, University of the Witwatersrand, P. O. Wits, 2050 Johannesburg, South Africa

E-mail address: helmut@gauss.cam.wits.ac.za