A086280 -id:A086280 - cp's OEIS frontend

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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

Offset: 1

Amiram Eldar, Jun 09 2023

			5.82174008504864652889686861550204134315033324319577...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.

Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.

RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]

Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).

A385612 Decimal expansion zeta''''(0) (negated).

Original entry on oeis.org

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

Offset: 2

Artur Jasinski, Jul 04 2025

n-th derivative of zeta function at 0 is close to -n!, which here is the present constant close to 4! = 24.

			23.997103188013707958987219527741...

Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), pp. 223-232.

Cf. A075700, A257549, A261508.

Cf. A001620, A061444, A082633, A086279, A086280.

evalf(-Zeta(4, 0), 120); # Vaclav Kotesovec, Jul 04 2025

RealDigits[-3 EulerGamma^4/2 - EulerGamma^2 Pi^2/4 + 19 Pi^4/480 - 4 EulerGamma^3 Log[2 Pi] - 3 EulerGamma^2 Log[2Pi]^2 +  Pi^2 Log[2 Pi]^2/4 + Log[2 Pi]^4/2 - 6 EulerGamma^2 StieltjesGamma[1] - Pi^2 StieltjesGamma[1]/2 - 12 EulerGamma Log[2 Pi] StieltjesGamma[1] - 6 Log[2 Pi]^2 StieltjesGamma[1] - 6 EulerGamma StieltjesGamma[2] - 6 Log[2Pi] StieltjesGamma[2] - 2 StieltjesGamma[3] + 4 Log[2 Pi] Zeta[3],10,105][[1]]

PARI
```
-zeta''''(0)
```

Equals -3*gamma^4/2 - gamma^2*Pi^2/4 + 19*Pi^4/480 - 4*gamma^3*log(2*Pi) -3*gamma^2*log(2*Pi)^2 + Pi^2*log(2*Pi)^2/4 + log(2*Pi)^4/2 - 6*gamma^2*StieltjesGamma(1) - Pi^2*StieltjesGamma(1)/2 - 12*gamma*log(2*Pi)* StieltjesGamma(1) - 6*log(2*Pi)^2*StieltjesGamma(1) - 6*gamma*StieltjesGamma(2) - 6*log(2*Pi)*StieltjesGamma(2) - 2*StieltjesGamma(3) + 4*log(2*Pi)*zeta(3).

cp's OEIS Frontend

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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