Central Binomial representation for zeta(2)
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.
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RELATIONS OF THE DERIVATIVES OF THE RIEMANN ZETA FUNCTION
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:
RELATION BETWEEN ZETA FUNCTION AND EULER MASCHEORNI CONSTANT
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:
INTEGRAL REPRESENTATION STILTJIES CONSTANT
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:
Hjortnes series for zeta(3)
In this post We will derive the remarkable beatiful Hjortnes series used by Apery to prove the irrationality of zeta(3) Recall (proved here ) (1) Letting in (1) and using the fact that we obtain (2) Proof: Dividing (2) by x and integrating from 0 to 1/2 we obtain (3) Let´s now focus on the R.H.S. of (3) Plugging the result of the integral back in (3) we obtain the remarkable result Appendix 1 Recall the following relations regarding the Golden Ratio (A.1) We have proven the following relation in this post (A.2) Also, recall the Trilogarithm identity proved here (A.3) And the Polylogarithm relation proved here (A.4) Example, letting in (A.4) we obtain (A.5) Claim: (A.6) Proof: If we let in (1) we obtain Appendix 2 And we get (A.7) Now we focus on the integral on the L.H.S. Plugging this result back in (A.7) we obtain (A.8) Example, letting x=1 in (A.8) we obtain Reference POLYLOGAR...