Might there be a good survey paper on the application of maximum entropy inference for non-trivial problems in probabilistic number theory?
So far I am aware of the work of Ioannis Kontoyiannis, an information theorist at Cambridge, who referred me to two of his publications [1,2]. There are also two relevant MathOverflow posts:
- Is there a Kolmogorov complexity proof of the prime number theorem?
- An information-theoretic derivation of the prime number theorem
I think the incompressibility method based on algorithmic information theory and the probabilistic method pioneered by Erdős are related methods. However, I have yet to find a comprehensive theory for applying maximum entropy inference to problems in probabilistic number theory. For concreteness, there are two specific applications I have in mind.
I suspect that such methods may provide us with new insights into the distribution of prime numbers and that they might help us determine whether Archimedes' constant is absolutely normal.
References:
I. Kontoyiannis. "Some information-theoretic computations related to the distribution of prime numbers." In Festschrift in Honor of Jorma Rissanen, (P. Grunwald, P. Myllymaki, I. Tabus, M. Weinberger, B. Yu, eds.), pp. 135-143, Tampere University Press, May 2008.
I. Kontoyiannis. "Counting the primes using entropy." IEEE Information Theory Society Newsletter, 58, no. 2, pp. 6-9, June 2008. [pdf] [pdf] Slides from a talk on this work at ITW 2008 in Porto, May 2008.
E. Kowalski. Arithmetic Randonnée: An introduction to probabilistic number theory. 2021.
Peter Grünwald and Paul Vitányi. Shannon Information and Kolmogorov Complexity. 2010.
E.T. Jaynes. Information Theory and Statistical Mechanics. The Physical Review. 1957.
