A070888 Numerator of Sum_{k=1..n} mu(k)/k.

1, 1, 1, 1, -1, 2, -1, -1, -1, 19, -1, -1, -2323, -89, 304, 304, 163, 163, -81988, -81988, -15019, 410857, -249979, -249979, -249979, 4165258, 4165258, 4165258, 9246047, -65721449, -4193929329, -4193929329, -6504197377, -302679716, 2562470143

Offset: 1

Author

Donald S. McDonald, May 17 2002

Keywords

• frac
• sign

Comments

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

Examples

			a(6) = 2 because 1-1/2-1/3-1/5+1/6 = 4/30 = 2/15.
		

References

• Harold M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0), p. 92.
• Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 568.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2376 (terms 1..1906 from Robert Israel)
• F. K. Richter, A new elementary proof of the Prime Number Theorem, arXiv preprint arXiv:2002.03255 [math.NT], 2020-2021.
• Harold N. Shapiro, Some assertions equivalent to the prime number theorem for arithmetic progressions, Communications on Pure and Applied Mathematics 2.2-3 (1949): 293-308.

Crossrefs

Cf. A008683, A068337, A070889 (denominators).

Programs

Maple

T:= 0:
for n from 1 to 100 do
  T:= T + numtheory:-mobius(n)/n;
  A[n]:= numer(T)
od:
seq(A[n],n=1..100); # Robert Israel, Aug 04 2014

Mathematica

Table[ Numerator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]

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

Python

from functools import lru_cache
from sympy import harmonic
@lru_cache(maxsize=None)
def f(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
        j, k1 = j2, n//j2
    return c+harmonic(j-1)-harmonic(n)
def A070888(n): return f(n).numerator # Chai Wah Wu, Nov 03 2023

Extensions

Edited by Robert G. Wilson v, Jun 10 2002