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Primes make difference in Summatory Liouville function

Source: IMEO 2020 Problem 5

July 15, 2020

IMEOnumber theoryfunction

Problem Statement

For a positive integer

n

with prime factorization

n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}

let's define

\lambda(n) = (-1)^{\alpha_1 + \alpha_2 + \dots + \alpha_k}

.
Define

L(n)

as sum of

\lambda(x)

over all integers from

1

to

n

.
Define

K(n)

as sum of

\lambda(x)

over all composite integers from

1

to

n

.
For some

N>1

, we know, that for every

2\le n \le N

,

L(n)\le 0

.
Prove that for this

N

, for every

2\le n \le N

,

K(n)\ge 0

.
Mykhailo Shtandenko

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