Let denote the golden ratio , the Möbius function and Euler’s totient function. Prove that:

**Solution**

We are using the following facts:

(1)

(2)

Hence,

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Let denote the golden ratio , the Möbius function and Euler’s totient function. Prove that:

**Solution**

We are using the following facts:

(1)

(2)

Hence,

Let denote one of Jacobi’s theta functions and a function such that

Prove that

Suffice to compute .

Combining with the above we get the result, since