# What is the original source for Gelfand's problem on leading digits of the powers of 2?

The question of the distribution of the leading digits of the sequence $2^n$ is called Gelfand's problem or Gelfand's question. Is there any source that indicates Gelfand's own work on this, or the exact context where it arose with him?

References I have looked at say the first place connecting Gelfand's name with this question is on p. 37 of Avez's book "Ergodic Theory of Dynamical Systems, Vol. 1," but this is not available to me. I tried searching the internet with some Russian terms that seemed relevant (вопрос гельфанда, распределение цифр) and nothing special came up.

I assume that one of the sources is MathWorld. But the question they claim Avez attributes to Gelfand is not the distribution of the leading digits generally, but specifically "will the digit 9 ever occur" as the leading digit in $2^n$ (the answer is yes, but the smallest $n$ is 54). They also link to Avez's 1966 book, which is their source. I was unable to access the book so far, but Eising, Radcliffe and Top in Simple Answer to Gelfand’s Question confirm that the problem is attributed in it to Gelfand, and add that Arnold-Avez's book published two years later replaced 9 by 7.

So it appears that Avez is everybody's source for the attribution to Gelfand. What about Avez? Here is from Arnold's obituary:"Arnold spent 1965 in Paris as a postdoctorate at the Sorbonne. At the request of his supervisor, J. Leray, Arnold delivered a one-semester course on dynamical systems. The audience included many renowned mathematicians (Cartan, Douady, Fréchet, Godement, Leray, Schwarz, Serre, Thom). One of the participants, Andre Avez, recorded the lectures and then published them as a book".

I am going to speculate that Gelfand never published the problem, that Avez got it from Arnold's 1965 lectures, and Arnold, who was a student at the Moscow State University and took classes with Gelfand, did not need a publication to get it from Gelfand directly. In particular, Arnold gave multiple talks at Gelfand's seminar in 1964-65, handwritten notes (in Russian) are available online. On the other hand, Arnold-Avez do not attribute the problem (with 7) to anybody, so either Arnold forgot, or Avez misinterpreted his French in 1965, and the problem was Arnold's own.

• I found a copy of the 1966 book by Avez and it appears your speculation may correct. Avez poses "Gelfand's problem" on p. 37, and in the solution on pp. 63-64 he says in a footnote on p. 63 that the solution was "kindly communicated to me by V. I. Arnold." Avez also directs the reader to volume 16 (1964) of Compositio Mathematica, in which nearly all the articles are about uniform distribution. I looked at your link to a page of notes from Gelfand's seminar, and none of the talks by Gelfand or Arnold up through 1966, when Avez's book appeared, mention leading digits of powers of a number. – KCd Mar 14 '16 at 2:20
• I see in Arnold's book "Mathematical Methods of Classical Mechanics" that he poses the question of how often 7 and 8 appear as leading digits of powers of 2 at the end of the section on Liouville's theorem (p. 74, 2nd edition). – KCd Mar 28 '16 at 0:43

The book of Avez you refer to is indeed rare: it is not listed in the common databases Mathscinet and Zentralblatt (which is very strange). So I cannot say anything about the relation of Gelfand to this problem.

A better known book Arnold and Avez, Ergodic Problems of Classical Mechanics, has this problem as a problem 3.2 in Ch.1.3, with solution in Appendix 12.5 (pp. 134-135 of the Russian edition), but it never mentions Gelfand in connection with this problem.

But what concerns the problem itself, its origin is clear and the problem has been completely solved. For more general problems of this sort, the keywords are "Benford law", and "distribution of first digits", and a solution is given in this paper:

P. Diaconis, The Distribution of Leading Digits and Uniform Distribution Mod 1, Ann. Prob., Volume 5, Number 1 (1977), 72-81.

• I already knew the solution to the problem and Benford's Law. The only thing that baffled me was why Gelfand's name got attached to it. – KCd Feb 28 '16 at 21:27