Ramanujan got convoluted

Number Theory Level 5

Let $\mu$ and $\phi$ be two number theoretic functions. Then

evaluate $(\mu * \phi) (1729)$

Notations :-

$\phi$ denotes Euler's Totient function.

$\mu$ denotes Möbius function.

$*$ denotes Dirichlet's convolution of two arithmetic functions.

The answer is 935.

1 solution

Otto Bretscher
Apr 10, 2016

For a prime $p$ we have $(\mu*\phi)(p)=\mu(p)\phi(1)+\mu(1)\phi(p)=p-2$ . Since the function is multiplicative, we find $(\mu*\phi)(1729)=(\mu*\phi)(7)\times(\mu*\phi)(13)\times (\mu*\phi)(19)=5\times 11\times 17=\boxed{935}$

Yes this is the shortest wayto deal with this problem :) :) I did it by converting the convolution to standard form and then with divisors...that was long way....nice solution upvoted..:) :)

A Former Brilliant Member - 5 years, 2 months ago

Ramanujan got convoluted

The answer is 935.

1 solution

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