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:

          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 E...