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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.