A346202 a(n) = L(n)^2, where L is Liouville's function.

1, 0, 1, 0, 1, 0, 1, 4, 1, 0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 16, 9, 4, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 9, 4, 1, 0, 1, 4, 9, 16, 25, 16, 25, 36, 25, 36, 25, 36, 49, 36, 25, 16, 9, 4, 9, 4, 9, 4, 9, 4, 1, 4, 9, 16, 9, 16, 25, 36, 49, 36, 49, 64, 49

Offset: 1

Author

Mats Granvik, Jul 10 2021

Keywords

• nonn

Comments

The Riemann Hypothesis is equivalent to the statement that, for every fixed eps > 0, lim_{n->oo} (L(n) / n^(eps + 1/2)) = 0.

References

• Peter Borwein, Stephen Choi, Brendan Rooney, and Andrea Weirathmueller, The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, 2007, page 6, Theorem 1.2.

Crossrefs

Cf. A002819, A008836, A028488 (positions of zeros).

Programs

Maple

L:= proc(n) option remember; `if`(n<1, 0,
     (-1)^numtheory[bigomega](n)+L(n-1))
    end:
a:= n-> L(n)^2:
seq(a(n), n=1..77);  # Alois P. Heinz, Jul 28 2021

Mathematica

Table[Sum[LiouvilleLambda[n], {n, 1, nn}]^2, {nn, 1, 77}]

PARI

a008836(n) = (-1)^bigomega(n) \\ after Charles R Greathouse IV in A008836
a(n) = sum(i=1, n, sum(j=1, n, a008836(i)*a008836(j))) \\ Felix Fröhlich, Jul 10 2021

Python

from functools import reduce
from operator import ixor
from sympy import factorint
def A346202(n): return sum(-1 if reduce(ixor, factorint(i).values(),0)&1 else 1 for i in range(1,n+1))**2 # Chai Wah Wu, Dec 20 2022

Formula

a(n) = Sum_{i=1..n} Sum_{j=1..n} A008836(i)*A008836(j).

a(n) = A002819(n)^2. - Ilya Gutkovskiy, Jul 10 2021