A242301 Decimal expansion of C(2), where C(x) = -Sum{k>=1} (-1)^k/prime(k)^x.

1, 6, 2, 8, 1, 6, 2, 4, 6, 6, 6, 3, 6, 0, 1, 4

Offset: 0

Author

Stanislav Sykora, May 14 2014

Keywords

• nonn
• cons
• hard
• more

Comments

The alternating series of reciprocal powers of prime numbers converges for any x > 0 (absolutely so if x > 1) but is hard to compute.

The next digits of C(2), after ...6014, seem to converge to a(16)=1, a(17)=5.

Examples

			0.1628162466636014...
		

Links

• Stanislav Sykora, PARI/GP scripts for primes-related functions, see function AltSum1DivPrimePwr(x,eps), with instructions.
• Eric Weisstein's World of Mathematics, Prime Sums
• Eric Weisstein's World of Mathematics, Prime Zeta Function
• Wikipedia, Prime Zeta Function

Crossrefs

Cf. A078437 (x=1), A242302 (x=3), A242303 (x=4), A242304 (x=5).

Cf. A085548.

Programs

Mathematica

k = 1; p = 2; s = 0; While[p < 1000000000, s = N[s + (-1)^k/p^2, 40]; k = Mod[++k, 2]; p = NextPrime@ p]; s (* takes ~30 minutes on an average laptop to 18 decimal digits *)(* Robert G. Wilson v, Dec 30 2017 *)

PARI

See Sykora link.