cp's OEIS Frontend

A339253 Decimal expansion of the unique real nontrivial zero of the Fredholm series, i.e., the complex equation Sum_{k>=0} z^(2^k) = 0 (negated).

%I A339253 #6 Apr 05 2024 10:05:36
%S A339253 6,5,8,6,2,6,7,5,4,3,0,0,1,6,3,9,2,2,4,1,3,4,7,2,8,3,0,5,7,9,5,0,1,6,
%T A339253 4,5,9,4,0,9,3,2,7,9,6,2,2,0,4,3,6,5,8,7,0,6,2,8,0,4,7,7,7,7,3,7,4,5,
%U A339253 8,6,8,2,9,9,9,7,5,1,3,0,2,2,4,0,7,5,9
%N A339253 Decimal expansion of the unique real nontrivial zero of the Fredholm series, i.e., the complex equation Sum_{k>=0} z^(2^k) = 0 (negated).
%C A339253 The trivial zero is z = 0.
%C A339253 This constant was found by Mahler (1980), who also found 3 pairs of conjugate complex zeros, and later (1982) 5 more pairs.
%C A339253 Zannier and Veneziano (2020) proved that there are infinitely many complex zeros in the complex unit disk.
%D A339253 David Masser, Auxiliary Polynomials in Number Theory, Cambridge University Press, 2016. See pp. 27-29.
%H A339253 Kurt Mahler, <a href="https://doi.org/10.1016/0022-314X(80)90069-4">On a special function</a>, Journal of Number Theory, Vol. 12, No. 1 (1980), pp. 20-26; <a href="https://core.ac.uk/download/pdf/82725617.pdf">alternative link</a>.
%H A339253 Kurt Mahler, <a href="https://doi.org/10.1090/S0025-5718-1982-0658225-3">On the zeros of a special sequence of polynomials</a>, Mathematics of Computation, Vol. 39, No. 159 (1982), pp. 207-212; <a href="https://carmamaths.org/resources/mahler/docs/211.pdf">alternative link</a>.
%H A339253 Umberto Zannier and Francesco Veneziano, <a href="https://arxiv.org/abs/2006.11922">A note on the zeroes of the Fredholm series</a>, arXiv:2006.11922 [math.CV], 2020.
%e A339253 -0.65862675430016392241347283057950164594093279622043...
%t A339253 m = 10; RealDigits[x /. FindRoot[Sum[x^(2^k), {k, 0, m}] == 0, {x, -0.65}, WorkingPrecision -> 120], 10, 100][[1]]
%Y A339253 Cf. A007404, A036987.
%K A339253 nonn,cons
%O A339253 0,1
%A A339253 _Amiram Eldar_, Nov 28 2020