A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.

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

Offset: 0

Author

Amiram Eldar, Jul 31 2025

Keywords

• nonn
• cons

Examples

			0.01461170261653269568427203547387356507606811502683...
		

References

• Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.

Links

• Ovidiu Furdui, A class of fractional part integrals and zeta function values, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.

Crossrefs

Cf. A001620 (gamma), A013661, A061444, A074962 (A), A225746.

Cf. A147533 (m=1), A386713 (m=2), this constant (m=3).

Programs

Mathematica

RealDigits[-Zeta[2] + 3*EulerGamma + 36*Log[Glaisher] - 6*Log[2*Pi] + 2, 10, 120, -1][[1]]

PARI

-zeta(2) + 3*Euler + 36*(1/12-zeta'(-1)) - 6*log(2*Pi) + 2

Formula

Equals -zeta(2) + 3*gamma + 36*log(A) - 6*log(2*Pi) + 2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant.

In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).