MY MATEX NOTES

In this entry we will prove the following result Click here to read it. Some previous established results were used, i.e.: Lerch´s formula Click here for the proof. Relation between Bernoulli´s polynomials and Hurwitz zeta function click here for the proof (equations 8.22 and 8.30)

          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:

          The goal of today´s post is to prove the following Fourier expansion for the Hurwitz zeta Function Recall the Series definition of the Hurwitz Zeta function (1) valid for . Where Integral representation of the Hurwitz zeta function (2) To prove (2), we start from the Gamma function Contour integral representation of the Hurwitz zeta function We now derive a contour integral representation for the Hurwitz Zeta function. The contour is the classic Hankel contour which is a loop around the negative real axis. It starts at −∞, encircles the origin once in the positive direction without enclosing any of the points ±2πi,±4πi,⋯ and returns to −∞ acording to the picture below For the function defined by the contour integral (3) is entire. For we have: (4) Proof: Over the bottom edge of the contour Letting On the upper edge we have Letting To show the boundedness of        we proceed as following Taking l...