MathDB

Liouville function on two certain primes

Source: 2021 CIIM, P5

November 2, 2021

number theoryprime factorizationfunction

Problem Statement

For every positive integer

n

, let

s(n)

be the sum of the exponents of

71

and

97

in the prime factorization of

n

; for example,

s(2021) = s(43 \cdot 47) = 0

and

s(488977) = s(71^2 \cdot 97) = 3

. If we define

f(n)=(-1)^{s(n)}

, prove that the limit

\lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n}

exists and determine its value.

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