cp's OEIS Frontend

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

%I A345208 #10 Mar 26 2022 10:18:42
%S A345208 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,
%T A345208 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,
%U A345208 3,6,6,5,0,7,6,8,5,6,1,2,4,4,5,2,5,3,9
%N A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).
%C A345208 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).
%C A345208 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...
%D A345208 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.42, pages 145 and 195.
%H A345208 Ovidiu Furdui, <a href="https://lovetoan.files.wordpress.com/2017/06/k2pi-net-vn-2006_6_solutions.pdf">Solution to Problem U27</a>, Mathematical Reflections, Vol. 6 (2006), pp. 27-28.
%H A345208 Ovidiu Furdui, <a href="https://doi.org/10.1524/anly.2011.1131">Exotic fractional part integrals and Euler's constant</a>, Analysis, Vol. 31, No. 3 (2011), pp. 249-257.
%H A345208 Mircea Ivan and Alexandru Lupaş, <a href="https://www.jstor.org/stable/27641877">Problem 11206</a>, The American Mathematical Monthly, Vol. 113, No. 2 (2006), p. 180; <a href="https://www.jstor.org/stable/27642376">A Limit Involving Euler's Constant</a>, Solution to problem 11206 by Richard A. Stong, ibid., Vol. 114, No. 10 (2007), pp. 928-929.
%H A345208 Albert F. S. Wong, <a href="https://www.jstor.org/stable/10.4169/002557010x482925">Problem 1845</a>, Mathematics Magazine, Vol. 83, No. 2 (2010), p. 150; <a href="https://www.jstor.org/stable/10.4169/math.mag.84.2.150">Integrating a square-fractional-reciprocal function</a>, Solution to problem 1845 by Allen Stenger, ibid., Vol. 84, No. 2 (2011), pp. 155-156.
%F A345208 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.
%F A345208 Equals Integral_{x=0..1} frac(1/x)^2 dx.
%F A345208 Equals 2 * Sum_{k>=2} (zeta(k)-1)/(k*(k+1)).
%F A345208 Equals A061444 - A001620 - 1.
%F A345208 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
%e A345208 0.26066140150781262295414738272883284868063561133564...
%t A345208 RealDigits[Log[2*Pi] - EulerGamma - 1, 10, 100][[1]]
%Y A345208 Cf. A001008, A001620, A002805, A061444, A153810.
%K A345208 nonn,cons
%O A345208 0,1
%A A345208 _Amiram Eldar_, Jun 10 2021