A355437 a(n) is the sign of Maslanka's constant A(n).

1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset

0

Comments

In 1997 Krzysztof Maslanka published a rapidly converging series giving values of the zeta function which uses coefficients A(k), involving even powers of Pi.

The sum of all positive A(n) equals B=2.143505674422168409234504232295468835062643090089323...

The sum of all negative A(n) equals C=-1.643505674422168409234504232295468835062643090089323...

B+C = 1/2.

References

K. Maslanka, The Beauty of Nothingness: Essay on the Zeta Function of Riemann, Acta Cosmologica XXIII, pp. 13-18 (1998).

Links

Krzysztof Maslanka, Table of n, a(n) for n = 0..60000

K. Maslanka, Hypergeometric-like Representation of Zeta function of Riemann, arXiv:math-ph/0105007, pp. 1-6 (1997).

K. Maslanka, Báez Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals, arXiv:math/0603713 [math.NT], pp. 1-16 (1997).

K. Maslanka and A. Kolezynski, The High Precision Numerical Calculation of Stieltjes Constants. Simple and Fast Algorithm, Computational Methods in Science and Technology, Volume 28 (2) 2022, 47-59.

Formula

a(n) = sign(Sum_{j=0..n} (-1)^j*binomial(n, j)*(2*j + 1)*zeta(2*j + 2)).

Example

A(0) = +1.644934067 so a(0)=+1,
A(1) = -1.602035634 so a(1)=-1,
A(2) = +0.2377099745 so a(2)=+1 etc.

Mathematica

A[k_] := Sum[(-1)^j Binomial[k, j] (2 j + 1) Zeta[2 j + 2], {j, 0, k}]; Table[Sign[N[A[n], 100]], {n, 0, 115}]

Crossrefs

Cf. A114523, A114524, A354835.

Sequence in context: A096809 A131561 A110515 * A071936 A005088 A076142

Adjacent sequences: A355434 A355435 A355436 * A355438 A355439 A355440

Keyword

sign

Author

Artur Jasinski, Jul 02 2022

Status

approved