MY MATEX NOTES

In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.

          Today we will evaluate the following infinite sum which corresponds to the derivative of Dirichlet eta function @ 1 As a bonus we will compute the following integral Lets first introduce a lemma: Lemma 1: (1) Proof: Claim: (2) If we let     in (1) we get (3) Lets now recall the Euler Maclaurin Formula (proved here ) to estimate the last two sums above (4) Choosing    and         in (4), we get for the first sum : (5) And for the second sum (6) Recall also the integral representation of the Stiltjies constant (shown here ): (7) letting in (7) we obtain (8) Now, plugging (5) and (6) back in (3) and letting we obtain We can now use (2) to calculate the following integral Proof:

          Let´s prove today the following beautiful  relations of the derivative of the Riemann Zeta function: We showed previously ( here ) the functional equation of the Hurwitz Zeta function (1) If we set h=k=1 in (1) we obtain (2) Which is the functional equation of the Riemann zeta function. Let´s now differentiate equation (2) w.r. to s to obtain: (3) Letting , n a positive integer and noting that   and     we may obtain (4) Now plugging in (4) we obtain the desired relations:

In this post we will show that (1) In order to prove (1) we can use the Euler Maclaurin formula derived previously in this post . The first order Euler Maclaurin Formula is given by: (2) Where , is the first Bernoulli Polynomial. Letting   and   , in (2) we obtain (3) (4) Lets now use (4) to prove (1) Proof:

                    In today´s post we want to prove this beautiful integral representation of the Stieltjes constants The Stieltjes constants are given by the limit We start by proving the following result: (1) Proof: With the aid of (1) we now prove the following result (2) Proof:  With the result (2), we can now prove our goal integral:

In this post we will proof the following two integrals related to the derivative of the Riemann Zeta function at Recall Kummer´s fourier expansion for LogGamma    (1) Integrating (1) from 0 to (2) Letting in (2) Where we used that  Proved here , and that For the second integral, lets first evalute the following integral                                                Then, differentiating both sides w.r. to s Setting s=2 Setting Where we again made use of the fact that