Let denote one of the Jacobi Theta functions. Prove that
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,
and the exercise is complete.
Let denote the Euler’s constant. Prove that:
First of all by making the substitution we get that
For let us consider the function
where is the Euler’s Gamma function. Differentiating once we get:
Thus, the desired integral is obtained by setting . Hence,
Let denote the Riemann zeta function. Evaluate the series:
Let . We are proving the more general result.
where denotes the digamma function.
First of all, we note that:
due to the reflection formula .
Side note: If then the sum equals .
Evaluate the integral
using Riemann sums.
Partition the interval into subintervals of length . This produces,
On the other hand, assuming is even:
since it holds that
Euler’s Gamma reflection formula was used at line . Letting we get that
If is odd we work similarly.
Let denote the – th harmonic number. Prove that
We note that
At the last integral we apply the substitution . Thus,
After all these we have that
proving the result.