A345412 Decimal expansion of Sum_{k>=1} 1/(2^k*zeta(2*k)).

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

Offset

0,1

Comments

Equals the alternating sum of a sequence of real numbers c(k) (see the Formula section). The Riemann Hypothesis is equivalent to c(k) ~ O(k^(-3/4+eps)) for all eps>0 (Báez-Duarte, 2005).

The sum without the alternating signs is Sum_{k>=0} c(k) = 1/zeta(0) = -2.

Links

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

Luis Báez-Duarte, A sequential Riesz-like criterion for the Riemann hypothesis, International Journal of Mathematics and Mathematical Sciences, Vol. 2005, No. 21 (2005), pp. 3527-3537.

Jerzy Cisło and Marek Wolf, Equivalence of Riesz and Baez-Duarte criterion for the Riemann Hypothesis, arXiv:math/0607782 [math.NT], 2006.

Jerzy Cisło and Marek Wolf, Criteria equivalent to the Riemann Hypothesis, AIP Conference Proceedings, Vol. 1079, No. 1. (2008), pp. 268-273; arXiv preprint, arXiv:0808.0640 [math.NT], 2008.

Jerzy Cisło and Marek Wolf, On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis, arXiv:0807.2971 [math.NT], 2008.

Mark W. Coffey, On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions, arXiv:math-ph/0608050, 2006.

Formula

Equals Sum_{k>=0} (-1)^k * c(k), where c(k) = Sum_{n>=1} mu(n)*(1-1/n^2)^k/n^2 = Sum_{j=0..k} (-1)^j * binomial(k,j)/zeta(2*j+2), where mu is the Möbius function (A008683).

Equals 1 + Integral_{x>=2} (1 - 1/2^floor(x/2)) * zeta'(x)/zeta(x) dx.

Example

0.78252798532538423457668847428378407680580097965259...

Maple

evalf(Sum(1/(2^k*Zeta(2*k)), k = 1..infinity), 120); # Vaclav Kotesovec, Jun 19 2021

Mathematica

RealDigits[Sum[1/(2^k*Zeta[2*k]), {k, 1, 1000}], 10, 100][[1]]

Prog

(PARI) suminf(k=1, 1/(2^k*zeta(2*k)))

Crossrefs

Cf. A008683.

Sequence in context: A198938 A244067 A225449 * A021565 A343617 A011103

Adjacent sequences: A345409 A345410 A345411 * A345413 A345414 A345415

Keyword

nonn,cons

Author

Amiram Eldar, Jun 18 2021

Status

approved