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%I #40 Apr 27 2025 08:02:02
%S 1,1,1,1,-1,2,-1,-1,-1,19,-1,-1,-2323,-89,304,304,163,163,-81988,
%T -81988,-15019,410857,-249979,-249979,-249979,4165258,4165258,4165258,
%U 9246047,-65721449,-4193929329,-4193929329,-6504197377,-302679716,2562470143
%N Numerator of Sum_{k=1..n} mu(k)/k.
%C Sum_{k>0} mu(k)/k = limit_{n->oo} A070888(n)/A070889(n) = 0. - _Jean-François Alcover_, Apr 18 2013. This is equivalent to the Prime Number Theorem! - _N. J. A. Sloane_, Feb 04 2022
%D Harold M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0), p. 92.
%D Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 568.
%H Amiram Eldar, <a href="/A070888/b070888.txt">Table of n, a(n) for n = 1..2376</a> (terms 1..1906 from Robert Israel)
%H F. K. Richter, <a href="https://arxiv.org/abs/2002.03255">A new elementary proof of the Prime Number Theorem</a>, arXiv preprint arXiv:2002.03255 [math.NT], 2020-2021.
%H Harold N. Shapiro, <a href="https://doi.org/10.1002/cpa.3160020208">Some assertions equivalent to the prime number theorem for arithmetic progressions</a>, Communications on Pure and Applied Mathematics 2.2-3 (1949): 293-308.
%e a(6) = 2 because 1-1/2-1/3-1/5+1/6 = 4/30 = 2/15.
%p T:= 0:
%p for n from 1 to 100 do
%p T:= T + numtheory:-mobius(n)/n;
%p A[n]:= numer(T)
%p od:
%p seq(A[n],n=1..100); # _Robert Israel_, Aug 04 2014
%t Table[ Numerator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
%o (PARI) t = 0; v = []; for( n = 1, 60, t= t + moebius( n) / n; v = concat( v, numerator( t))); v \\ adapted to latest PARI version by _Michel Marcus_, Aug 04 2014
%o (Python)
%o from functools import lru_cache
%o from sympy import harmonic
%o @lru_cache(maxsize=None)
%o def f(n):
%o if n <= 1:
%o return 1
%o c, j = 1, 2
%o k1 = n//j
%o while k1 > 1:
%o j2 = n//k1 + 1
%o c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
%o j, k1 = j2, n//j2
%o return c+harmonic(j-1)-harmonic(n)
%o def A070888(n): return f(n).numerator # _Chai Wah Wu_, Nov 03 2023
%Y Cf. A008683, A068337, A070889 (denominators).
%K frac,sign
%O 1,6
%A _Donald S. McDonald_, May 17 2002
%E Edited by _Robert G. Wilson v_, Jun 10 2002