OFFSET
0,1
COMMENTS
Average fractional part of a random (large) integer when divided by all numbers up to it. The result remains true if primes or numbers from particular (fixed) congruence classes are used instead. The result is due to Vallée Poussin. - Charles R Greathouse IV, Apr 11 2012
Expected value of the fractional part of 1/x where x is chosen uniformly at random from (0, 1]. - Charles R Greathouse IV, Apr 11 2012
Value of digamma function psi(x) for x=2. - Stanislav Sykora, Apr 30 2012
The asymptotic evaluation of the counting function of A064052 ("jagged" numbers) is j(n) ~ log(2)*n - (1-gamma)*n/log(n) + ... - Jean-François Alcover, May 16 2014, after Steven Finch.
Letting eta denote the Dirichlet eta function, and letting zeta denote the Riemann zeta function, we have that 1-gamma is equal to lim x -> infinity 2^x+(4/3)^x-zeta(2-eta(x)). - John M. Campbell, Jan 28 2016
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, chapter 2.21, p. 166.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2000
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.2)
Friedrich Pillichshammer, Euler's constant and averages of fractional parts.
Charles Jean de la Vallée Poussin, Sur les valeurs moyennes de certaines fonctions arithmétiques, Annales de la société scientifique de Bruxelles 22 (1898), pp. 84-90.
Wikipedia, Digamma function.
FORMULA
Equals Integral_{x>=1} {x}dx/x^2 dx, where {x} is the fractional part of x. - Charles R Greathouse IV, Apr 11 2012
Equals Integral_{x>=0} x*log(x)*exp(-x) dx. - Jean-François Alcover, Jun 17 2013
Equals Sum_{n>=2} (zeta(n)-1)/n. - Vaclav Kotesovec, Dec 11 2015
Equals Sum_{k>=1} zeta(2*k+1)/((k+1)*(2*k+1)). - Amiram Eldar, May 24 2021
Equals Sum_{j>=2} Sum_{k>=2} (1/(k * j^k)). - Mike Tryczak, Apr 07 2023
Equals Integral_{x=0..1} {1/x} dx, where {x} is the fractional part of x. From this expression we have 1 - gamma = Sum_{k>=1} Integral_{x=1/(k+1)..1/k} (1/x - k) dx = Sum_{k>=1} (log(1+1/k) - 1/(k+1)). - Jianing Song, Mar 24 2024
EXAMPLE
0.422784335...
MATHEMATICA
RealDigits[N[PolyGamma[2], 105]][[1]] (* Arkadiusz Wesolowski, Jan 10 2013 *)
RealDigits[1 - EulerGamma, 10, 50][[1]] (* G. C. Greubel, Aug 29 2016 *)
PROG
(PARI) 1-Euler \\ Charles R Greathouse IV, Apr 11 2012
CROSSREFS
KEYWORD
AUTHOR
Omar E. Pol, Jan 28 2009
STATUS
approved