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

Author

Amiram Eldar, Nov 28 2020

Keywords

• nonn
• cons

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.

Examples

			-0.65862675430016392241347283057950164594093279622043...
		

References

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

Links

• 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.

Crossrefs

Cf. A007404, A036987.

Programs

Mathematica

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