A335089 Decimal expansion of log(Pi^2/6).

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

Offset: 0

Author

Terry D. Grant, Sep 11 2020

Keywords

• nonn
• cons

Examples

			Equals 1/(2^2) + 1/(3^2) + (1/(4^2))*(1/2) + 1/(5^2) + + 1/(7^2) + (1/(8^2))*(1/3) + ... = 0.4977003024707...
		

Links

• Mathematics Stack Exchange, Infinite series involving Von Mangoldt's function.
• Grant Sanderson, What makes the natural log "natural"?, 3Blue1Brown video (2020).
• Eric Weisstein's World of Mathematics, Mangold Function.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.

Crossrefs

Cf. A000796, A013661, A053510, A131659, A016629.

Cf. A100995, A014963, A025474, A246655.

Programs

Mathematica

RealDigits[Log[Pi^2/6], 10, 120][[1]]
RealDigits[Sum[PrimeZetaP[2 k]/k, {k, 1, inf}], 10, 120][[1]]

PARI

log(Pi^2/6) \\ Michel Marcus, Sep 15 2020

Formula

Equals Sum_{k>=2} MangoldtLambda(k) / ((k^2)*log(k)).

Equals Sum_{k>=1} (1/k)*(1/(A246655(n)^2)) where k is the exponent of the prime power, A025474(n+1).

Equals Sum_{k>=1} primezeta(2*k)/k.

Equals 2*log(Pi) - log(6).

Equals log(zeta(2)) = log(A013661).