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).

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, 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, 8, 6, 8, 2, 9, 9, 9, 7, 5, 1, 3, 0, 2, 2, 4, 0, 7, 5, 9

Offset

0,1

Comments

The trivial zero is z = 0.

This constant was found by Mahler (1980), who also found 3 pairs of conjugate complex zeros, and later (1982) 5 more pairs.

Zannier and Veneziano (2020) proved that there are infinitely many complex zeros in the complex unit disk.

References

David Masser, Auxiliary Polynomials in Number Theory, Cambridge University Press, 2016. See pp. 27-29.

Links

Table of n, a(n) for n=0..86.

Kurt Mahler, On a special function, Journal of Number Theory, Vol. 12, No. 1 (1980), pp. 20-26; alternative link.

Kurt Mahler, On the zeros of a special sequence of polynomials, Mathematics of Computation, Vol. 39, No. 159 (1982), pp. 207-212; alternative link.

Umberto Zannier and Francesco Veneziano, A note on the zeroes of the Fredholm series, arXiv:2006.11922 [math.CV], 2020.

Example

-0.65862675430016392241347283057950164594093279622043...

Mathematica

m = 10; RealDigits[x /. FindRoot[Sum[x^(2^k), {k, 0, m}] == 0, {x, -0.65}, WorkingPrecision -> 120], 10, 100][[1]]

Crossrefs

Cf. A007404, A036987.

Sequence in context: A133618 A194599 A253300 * A335165 A080799 A262512

Adjacent sequences: A339250 A339251 A339252 * A339254 A339255 A339256

Keyword

nonn,cons,changed

Author

Amiram Eldar, Nov 28 2020

Status

approved