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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

Since

we deduce that

## An analytic logarithmic sum

Evaluate the sum

(Seraphim Tsipelis)

Solution [pprime]

We have successively:

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

### Who is Tolaso?

Find out more at his Encyclopedia Page.