A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).

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

Offset: 0

Author

Amiram Eldar, Jun 10 2021

Keywords

• nonn
• cons

Comments

The first two formulae (in the Formula section) are similar to the sum and integral lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k) = Integral_{x=0..1} frac(1/x) dx = 1 - gamma (A153810).

The second raw moment of the distribution of the fractional part of 1/x, where x is chosen uniformly at random from (0, 1]. Since the expected value is 1 - gamma, the second central moment, or variance, is log(2*Pi) - gamma - 1 - (1 - gamma)^2 = log(2*Pi) - gamma^2 + gamma - 2 = 0.081914807503... and the standard deviation is sqrt(log(2*Pi) - gamma^2 + gamma - 2) = 0.2862076300...

Examples

			0.26066140150781262295414738272883284868063561133564...
		

References

• Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.42, pages 145 and 195.

Links

• Ovidiu Furdui, Solution to Problem U27, Mathematical Reflections, Vol. 6 (2006), pp. 27-28.
• Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31, No. 3 (2011), pp. 249-257.
• Mircea Ivan and Alexandru Lupaş, Problem 11206, The American Mathematical Monthly, Vol. 113, No. 2 (2006), p. 180; A Limit Involving Euler's Constant, Solution to problem 11206 by Richard A. Stong, ibid., Vol. 114, No. 10 (2007), pp. 928-929.
• Albert F. S. Wong, Problem 1845, Mathematics Magazine, Vol. 83, No. 2 (2010), p. 150; Integrating a square-fractional-reciprocal function, Solution to problem 1845 by Allen Stenger, ibid., Vol. 84, No. 2 (2011), pp. 155-156.

Crossrefs

Cf. A001008, A001620, A002805, A061444, A153810.

Programs

Mathematica

RealDigits[Log[2*Pi] - EulerGamma - 1, 10, 100][[1]]

Formula

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k)^2, where frac(x) = x - floor(x) is the fractional part of x.

Equals Integral_{x=0..1} frac(1/x)^2 dx.

Equals 2 * Sum_{k>=2} (zeta(k)-1)/(k*(k+1)).

Equals A061444 - A001620 - 1.

Equals -2 * Sum_{k>=1} (H(k) - log(k) - gamma - 1/(2*k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2013). - Amiram Eldar, Mar 26 2022