MY MATEX NOTES

          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 today´s post we will derive a functional equation for the Hurwitz Zeta Function, namely: To this end we introduce the periodic zeta function defined by the following expression: (1) Theorem 1: (2) for and Proof: First recall the previously proved fourier expansion (see here ) (3) for and Now lets expand (2) by means of Euler´s formula Lemmma 1: Let h and k be two integers, , then for (4) Proof: To prove this equality used in the proof above Lets expand first it´s L.H.S. We used the fact that for m an integer Now, lets expand it´s R.H.S. and show that they are equal to each other Theorem 2: If k and h are integers with , then for all s we have Proof: Plugging (4) in (2) we obtain:

          In this post we will derive the Laurent expansion for the Riemann Zeta Function Recall the first order Euler Maclaurin expansion for the riemann zeta function shown here (1) We can rewrite (1) as following: Now, define the function: (2)   is an analytic function, therefore, we can represent it by a series expansion around in the following form: (3) Lets calculate the coefficients of (3) by calculating the derivatives of using eq. (2) above. For the first derivative we have: In the point we obtain for we obtain for we obtain for we obtain If we keep this process further we may obtain the following general form : (4) For . Evaluated at we obtain: (5) But we have already proved in a previous post that the integral in (5) is precisely the Stiltjies constants ! (6) Plugging (6) in (5) we get (7) for . To find we need to calculate Which we have also already calculated here and it´s equal to the Euler-Mascheroni constant . (8) H