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## Fractional integral

Let denote the fractional part. Prove that

for the different values of the integer number .

Solution

Let denote the integral,

since if whereas if . Therefore,

except of a countable set whose measure is .

## Multiple logarithmic integral

Let denote the Riemann zeta function. Evaluate the integral

Solution

Based on symmetries,

Let . It follows that

Using the recursion we get that

Thus,

## An integral equality

Let denote the trigamma function and let be integrable on . It holds that

Solution

We have successively:

where the interchange between the infinite sum and the integration is allowed by the uniform bound

The result follows.

## Digamma series

Let denote the digamma series. Evaluate the series

Solution

The series telescopes;

## Continuous and periodic

Let be a continuous real-valued function on satisfying

Define a function on by

1. Prove that is continuous and periodic with period .
2. Prove that if is continuous and periodic with period then

Solution

1. We note that

2. First of all is bounded on and uniformly. Hence,