Multiplicative Dilemma Part 1

Calculus Level 4

n=1ϕ(n)τ(n)μ(n)\Large{ \sum_{n=1}^\infty \frac{\phi(n)}{\tau(n)^{\mu(n)}} }

True or False: The above sum converges.

Definitions:

  • ϕ(n)\phi(n) is the number of positive integers not greater than nn that are relatively prime to nn (Euler's totient function).
  • τ(n)\tau(n) is the number of positive divisors of nn.
  • μ(n)\mu(n) is 0 if nn is not squarefree (divisible by a positive square number greater than 1), and otherwise is equal to (1)Ω(n)(-1)^{\Omega(n)} where Ω(n)\Omega(n) is the number of prime divisors of nn.

Try Part 2

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