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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?

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