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At the wikipedia page for the Möbius function there is an expression for the ordinary generating function for the Möbius function.

Is it possible to calculate the radius of convergence for any of the terms in the expression?

For example what would the radius of convergence for the double sum be: $\sum_{a=2}^{\infty} \sum_{b=2}^{\infty} x^{a*b}$

Or the triple sum: $\sum_{a=2}^{\infty} \sum_{b=2}^{\infty} \sum_{c=2}^{\infty} x^{a*b*c}$

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up vote 3 down vote accepted

The radius of convergence for each is $1$. The radius of convergence is at most $1$ since there are infinitely many terms with positive integer coefficients. The radius of convergence is at least $1$ since the coefficient of $x^n$ in the $k$th term is at most $n^k$ which is subexponential in $n$.

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