MathDB

5

Part of 2021 CIIM

Problems(1)

Liouville function on two certain primes

Source: 2021 CIIM, P5

11/2/2021
For every positive integer nn, let s(n)s(n) be the sum of the exponents of 7171 and 9797 in the prime factorization of nn; for example, s(2021)=s(43⋅47)=0s(2021) = s(43 \cdot 47) = 0 and s(488977)=s(712⋅97)=3s(488977) = s(71^2 \cdot 97) = 3. If we define f(n)=(−1)s(n)f(n)=(-1)^{s(n)}, prove that the limit lim⁡n→+∞f(1)+f(2)+⋯+f(n)n \lim_{n \to +\infty} \frac{f(1) + f(2) + \cdots+ f(n)}{n} exists and determine its value.
number theoryprime factorizationfunction