A257134 Decimal expansion of Pi^4/45.
2, 1, 6, 4, 6, 4, 6, 4, 6, 7, 4, 2, 2, 2, 7, 6, 3, 8, 3, 0, 3, 2, 0, 0, 7, 3, 9, 3, 0, 8, 2, 3, 3, 5, 8, 0, 5, 5, 4, 9, 5, 0, 1, 9, 0, 3, 8, 3, 7, 4, 5, 3, 8, 1, 5, 3, 6, 5, 9, 5, 2, 4, 3, 0, 8, 8, 8, 2, 4, 1, 2, 3, 2, 3, 7, 3, 9, 3, 7, 6, 9, 3, 1, 1, 3, 8, 1, 9, 2, 7, 1, 8, 8, 3, 3, 9, 9, 8, 3, 4, 4, 6, 5, 9, 8
Offset: 1
Examples
2.16464646742227638303200739308233580554950190383745381536595243...
References
- L. J. P. Kilford, Modular Forms: A Classical and Computational Introduction, Imperial College Press, 2008, p. 15.
Links
- Alain Tissier, Apéry's Constant, Solution to Problem 10635, The American Mathematical Monthly, Vol. 106, No. 10 (1999), pp. 965-966.
- Eric Weisstein's World of Mathematics, Eisenstein Series.
- Index entries for transcendental numbers.
Programs
-
Mathematica
RealDigits[Pi^4/45, 10, 105] // First
-
PARI
Pi^4/45 \\ Charles R Greathouse IV, Oct 01 2022
Formula
Pi^4/45 = 2*zeta(4) = G_4(oo), where the function G_k(z) is the Eisenstein nonzero modular form of weight k.
Equals -Integral_{x=0..1} log(x)^2 * log(1 - x)/x dx. - Amiram Eldar, Jul 21 2020
Equals Sum_{n,m>=1} (Pi^2/6 - Sum_{k=1..n+m} 1/k^2)/(n*m) (Tissier, 1999). - Amiram Eldar, Jan 27 2024
Equals Integral_{x=0..1} Li(3,sqrt(x))/x dx, where Li(n,x) is the polylogarithm function. - Kritsada Moomuang, Jun 18 2025