A070889 -id:A070889 - cp's OEIS frontend

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

Donald S. McDonald, May 17 2002

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

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

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.

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.

Cf. A008683, A068337, A070889 (denominators).

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

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

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

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

Edited by Robert G. Wilson v, Jun 10 2002

A070891 Denominator of Sum_{k=1..n} mu(k)/k when it changes sign.

Original entry on oeis.org

30, 15, 105, 210, 2310, 5005, 1616615, 9699690, 223092870, 111546435, 2156564410, 100280245065, 3710369067405, 7420738134810, 6541380665835015

Offset: 1

Donald S. McDonald, May 17 2002

Cf. A070890, A070888, A070889.

t = 0; v = []; for( n = 1, 80, t1 = t; t = t + moebius( n) / n; if( t * t1 < 0, v = concat( v, denominator( t)), )); v

A068337 a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function.

Original entry on oeis.org

1, 1, 1, 4, -4, 96, -48, -384, -3456, 328320, -17280, -207360, -481697280, -516741120, 79427174400, 1270834790400, 681401548800, 12265227878400, -6169334376038400, -123386687520768000, -158218429759488000, 47610136717000704000

Offset: 1

Leroy Quet, Feb 27 2002

sign

Amiram Eldar, Table of n, a(n) for n = 1..450
Friedrich Roesler, Riemann's hypothesis as an eigenvalue problem, Linear Algebra and its Applications, Vol. 81 (1986), pp. 153-198.
Friedrich Roesler, Riemann's hypothesis as an eigenvalue problem. II, Linear Algebra and its Applications, Vol. 92 (1987), pp. 45-73.

Cf. A000142, A008683, A070888, A070889.

n = 25; Accumulate[Table[MoebiusMu[k]/k, {k, 1, n}]] * Range[n]! (* Amiram Eldar, Oct 22 2020 *)

from math import factorial
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 A068337(n): return factorial(n)*f(n) # Chai Wah Wu, Nov 03 2023

a(n) = (-1)^(n-1)*{determinant of the n X n matrix m(i,j) = i+(j (mod i))} - Benoit Cloitre, May 28 2004
From Amiram Eldar, Oct 22 2020: (Start)
a(n) = A000142(n)*A070888(n)/A070889(n).
a(n) ~ O(n! * n^(-1/2 + eps)), for every eps>0, if and only if Riemann's hypothesis is true (Roesler, 1986). (End)

A070890 Numerator of Sum_{k=1..n} mu(k)/k when it changes sign.

Original entry on oeis.org

-1, 2, -1, 19, -1, 304, -81988, 410857, -249979, 4165258, -65721449, 2562470143, -5468849774, 184344882947, -137190436674212, 10026981687881, -12611493192339623, 519973962150962777, -8549627883788520181, 1874648830674470878723, -200643437220052588790575, 877316785444551755504875

Offset: 1

Donald S. McDonald, May 17 2002

			a(1)= -1 because numerator first sign change is 1-1/2-1/3-1/5= > (-1)/30 <0.
a(2)= 2 because next sign change is -1/30+1/6 = 2/15, reverts to positive.

Cf. A070891, A070888, A070889.

t = 0; v = []; for( n = 1, 120, t1 = t; t = t + moebius(n) / n; if( t * t1 < 0, v = concat( v, numerator( t)), )); v

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

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A070891 Denominator of Sum_{k=1..n} mu(k)/k when it changes sign.

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A068337 a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function.

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A070890 Numerator of Sum_{k=1..n} mu(k)/k when it changes sign.

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