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November 3rd, 2016, 11:05 AM  #1 
Newbie Joined: Sep 2016 From: Tempe, Arizona Posts: 6 Thanks: 1  How do I Simplify HypergeometricPFQ and MeijerG Sums?
I've derived a formula for the Riemann zeta function $\zeta(s)$ which is illustrated in the plot below and on the following website, and am now attempting to derive a formula for the firstorder derivative $\zeta'(s)$, which is the context of this question. Illustration of $\zeta(s)$ Formula The following plot illustrates the absolute value of my formula for $\zeta(s)$ (blue curve) evaluated along the critical line. The $Zeta[s]$ function defined in the Wolfram Language (orange curve) is shown as a reference. The red dots represent the location of the zeta zeros. I'm having problems with evaluation times of the following two expressions where $s$ is complex, $x$ is real, $n$ is a positive integer, and the $HypergeometricPFQ$ and $MeijerG$ functions are defined in the Wolfram Language. I believe these two expressions are related as they're terms from two different formula's I'm investigating related to the firstorder derivative $\zeta'(s)$ of the Riemann zeta function. $$HypergeometricPFQ[\{1s,1s\},\{2s,2s\},2in\pi x]+HypergeometricPFQ[\{1s,1s\},\{2s,2s\},2in\pi x]$$ $$MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},2in\pi x]MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},2in\pi x]$$ Does anyone have any ideas on how I might simplify either of these two expressions to improve evaluation times? 

Tags 
hypergeometricpfq, meijerg, number theory, riemann hypothesis, riemannzeta, simplify, sums 
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