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# Tag Archives: Analytic Number Theory

## On the factorial

Let denote the Möbius function and denote the floor function. Prove that:

**Solution**

The RHS equals

since for .

## On Euler’s totient function series

Let denote Euler’s totient function. Prove that for it holds that:

where stands for the Riemann zeta function.

**Solution**

Well by Euler’s product we have,

thus,

(1)

and

(2)

Combining we get the result.

**Note: **It also holds that

## Inverse zeta(3) limit

Evaluate the limit

**Solution**

We are proving that the limit is . Indeed , one has:

If then

whereas if then the above limit is . Thus:

## On a strange Möbius series

Let denote the Möbius function. Evaluate the series

where .

**Solution**

we deduce that

## An analytic logarithmic sum

Evaluate the sum

*(Seraphim Tsipelis)*

**Solution **[pprime]

We are invoking Kummer’s formula for the evaluation of the last sum. Evaluating the Fourier series that appear for we get that