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# Tag Archives: Real Analysis

## 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

## Logarithmic mean inequality

Let such that . Prove that

Solution

We are invoking the Hermite – Hadamard Inequality for the convex function . Thus,

## On an integral

Evaluate the integral

Solution

Successively we have:

We used the simple observation that

## Limit of a sequence

Let be a function such that and is differentiable at . Let us set

Evaluate the limit .

Solution

Since is differentiable at , there is some such that

and is of course continuous.

Thus,

Let . There exists such that which in return means that . Hence , for larger than it holds that

On the other hand , the sum is a Riemann sum and converges to .

In conclusion,

## Integral of Jacobi Theta function

Let denote one of the Jacobi Theta functions. Prove that

Solution

We have successively,

The sum is evaluated as follows. Consider the function

and integrate it around a square with vertices . The function has poles at every integer with residue as well as at with residues . We also note that as the contour integral of tends to . Thus,

Hence,

and the exercise is complete.