Claims

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1995_09_18_duverney: Duverney's 1995 note proves that (q-1)^2 times the sum of sigma(n) over q^n is irrational for every integer q other than -1, 0 and 1; the case q = 2 is the series of Problem 250, so the answer is yes; refereed and reviewed.

1996_03_07_nesterenko: Nesterenko's 1996 theorem that at least three of q, P(q), Q(q), R(q) are algebraically independent for 0 < |q| < 1 makes P(1/2) = 1 - 24 times the sum of sigma(n) over 2^n transcendental, hence the sum irrational.

2001_11_08_zudilin: Zudilin (Mat. Sb. 193 (2002)) proves the q-analog of zeta(2) irrational at q = 1/p with irrationality measure at most 4.0787; at q = 1/2 this is the sum of sigma(n) over 2^n; refereed.

2006_04_13_postelmans_van_assche: Postelmans and Van Assche (J. Number Theory 126 (2007)) prove 1, zeta_q(1) and zeta_q(2) linearly independent over Q for q = 1/p, p at least 2; at q = 1/2 this makes the sum of sigma(n) over 2^n irrational; refereed.

2008_09_15_smet_van_assche: Smet and Van Assche (Acta Arith. 138 (2009)) prove zeta_q(2) irrational at q = 1/p with measure at most 10 pi^2/(5 pi^2 - 24); at q = 1/2 this is the sum of sigma(n) over 2^n; refereed.

2025_10_21_romanlelan: A note linked in the site's thread claims the sum of sigma(n) q^n transcendental for every algebraic q with 0 < |q| < 1; the curator showed the same day that the argument does not reach the series itself.

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