A307715 Decimal expansion of Sum_{t>0} log((t + 1)/t)^2.
9, 7, 7, 1, 8, 9, 1, 8, 3, 2, 6, 8, 9, 3, 6, 5, 5, 4, 4, 5, 7, 8, 8, 5, 7, 4, 9, 4, 7, 6, 4, 3, 4, 7, 4, 8, 0, 7, 7, 3, 9, 2, 5, 0, 6, 4, 7, 4, 7, 2, 3, 9, 0, 1, 7, 7, 0, 2, 0, 9, 8, 9, 7, 5, 5, 3, 1, 8, 4, 4, 5, 2, 9, 3, 9, 2, 3, 9, 3, 3, 5, 6, 2, 9, 0, 1, 2, 3, 2, 1, 0, 7, 9, 7, 4, 3, 2, 0, 3, 3, 5, 9, 2, 3, 2
Offset: 0
Examples
0.9771891832689365544578857494764347480773925064747239017702...
Links
- P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe and M. Tiba, The structure and number of Erdős covering systems, arXiv:1904.04806 [math.CO], 2019. J. Eur. Math. Soc. 26, 75-109 (2024).
- Y. Buttkewitz, C. Elsholtz, K. Ford, and J.-C. Schlage-Puchta, The maximal order of iterated multiplicative functions, arxiv:1108.1815 [math.NT], 2011. IMRN issue 17, 4051-4061 (2012).
- C. Elsholtz, M. Technau, and N. Technau, The maximal order of iterated multiplicative functions, arxiv:1709.04799 [math.NT], 2017. Mathematika, 65 (4) 2019, 990-1009.
- P. Erdős, Egy kongruenciarendszerekrol szóló problémáról, (On a problem concerning congruence-systems, in Hungarian), Mat. Lapok, 4 (1952), 122-128.
Programs
-
Mathematica
First[RealDigits[NSum[(Log[(t + 1)/t])^2, {t, 1, Infinity}, NSumTerms -> 100, Method -> {"NIntegrate", "MaxRecursion" -> 10}, WorkingPrecision -> 100]]] -
PARI
sumpos(t=1, log((t + 1)/t)^2) \\ Michel Marcus, Apr 26 2019
Formula
From Amiram Eldar, Jun 17 2023: (Start)
Equals 2 * Sum_{k>=1} H(k) * (zeta(k+1)-1) / (k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
Equals -Sum_{k>=1} zeta'(2*k) / k. (End)
Comments