A337404 - OEIS

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A337404

Decimal expansion of real part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function and i=sqrt(-1).

1

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

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OFFSET

0,5

COMMENTS

For the decimal expansion of the imaginary part of Sum_{m>=1} 1/(1/2 + i*z(m))^4 where z(m) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function see A337365.

See also links in A332645.

LINKS

Table of n, a(n) for n=0..108.

J. B. Keiper, Power series expansions of Riemann's function, Math. Comp. 58 (1992), 765-773.

FORMULA

Re(Sum_{m>=1} 1/(1/2 + i*z(m))^n) where n is a positive integer is equal to Keiper's sigma(n)/2.

For n=4 this equals 1/2 + EulerGamma^4/2 - Pi^4/192 + 2*EulerGamma^2*StieltjesGamma(1) + StieltjesGamma(1)^2 + EulerGamma*StieltjesGamma(2) + StieltjesGamma(3)/3.

EXAMPLE

0.0000368136106308...

MATHEMATICA

Join[{0, 0, 0, 0}, RealDigits[N[1/192 (96 + 96 EulerGamma^4 - Pi^4 + 384 EulerGamma^2 StieltjesGamma[1] + 192 StieltjesGamma[1]^2 + 192 EulerGamma StieltjesGamma[2] + 64 StieltjesGamma[3]), 105]][[1]]]

CROSSREFS

Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A335826, A337365.

Sequence in context: A137128 A256372 A330890 * A133442 A133193 A200131

Adjacent sequences: A337401 A337402 A337403 * A337405 A337406 A337407

KEYWORD

nonn,cons

AUTHOR

Artur Jasinski, Aug 26 2020

STATUS

approved