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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 first-order 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 first-order derivative $\zeta'(s)$ of the Riemann zeta function. $$HypergeometricPFQ[\{1-s,1-s\},\{2-s,2-s\},-2in\pi x]+HypergeometricPFQ[\{1-s,1-s\},\{2-s,2-s\},2in\pi x]$$ $$MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},-2in\pi x]-MeijerG[\{\{\},\{1+s,1+s\}\},\{\{1,s,s\},\{\}\},2in\pi x]$$ 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 Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post rarian Calculus 1 October 14th, 2014 08:56 AM n3rdwannab3 Calculus 1 January 17th, 2014 03:41 PM nubshat Calculus 2 November 13th, 2012 05:04 PM julian21 Real Analysis 1 November 27th, 2010 03:07 PM rmas Algebra 1 November 12th, 2010 07:11 PM

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