Problem 250

\sum \frac{\sigma(n)}{2^n}

irrational? (Here $\sigma(n)$ is the sum of divisors function.)

Source: erdosproblems.com/250

Number theory
Irrationality

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Proved. The sum is irrational for every integer base $q$ with $|q|\ge2$ (Duverney 1995, Théorème, p. 1287, proof p. 1289) and transcendental (Nesterenko 1996, Theorem 1 with Corollary 2 at $q=1/2$ , pp. 66--67 of the Russian original). Both proofs are refereed publications. The site's "(LEAN)" badge is explained under Formal status below and adds nothing to this standing.

ErGr80: Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980), p. 61.
Er88c: Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986) (1988), 102-109, p. 102.
Ne96: Nesterenko, Yuri, Modular functions and transcendence problems. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 10, 909-914. (The site's entry omits the volume.)
Er48: Erdős, P., On arithmetical properties of Lambert series. J. Indian Math. Soc. (N.S.) 12 (1948), 63-66, p. 66.
Er57: Erdős, P., On the irrationality of certain series. Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 212-219, p. 212.
Du93: Duverney, D., Propriétés arithmétiques d'une série liée aux fonctions thêta. Acta Arith. 64 (1993), no. 2, 175-188.
Du95: Duverney, D., Irrationalité d'un q-analogue de ζ(2). C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 10, 1287-1289.
Ne96b: Nesterenko, Yu. V., Modular functions and transcendence questions. Mat. Sb. 187 (1996), no. 9, 65-96; Sb. Math. 187 (1996), no. 9, 1319-1348.
Wa97: Waldschmidt, M., Sur la nature arithmétique des valeurs de fonctions modulaires. Séminaire Bourbaki 1996/97, exp. 824, Astérisque 245 (1997), 105-140.
Zu02: Zudilin, W., On the irrationality measure for a q-analogue of ζ(2). Mat. Sb. 193 (2002), no. 8, 49-70; Sb. Math. 193 (2002), no. 8, 1151-1172.
PV07: Postelmans, K. and Van Assche, W., Irrationality of ζ_q(1) and ζ_q(2). J. Number Theory 126 (2007), no. 1, 119-154.
SV09: Smet, C. and Van Assche, W., Irrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials. Acta Arith. 138 (2009), no. 2, 165-178.

Problem 250

Status

Proof claims

Discussion