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From: Axel Vogt <&noreply@axelvogt.de>
Newsgroups: comp.soft-sys.math.maple
Subject: Re: A zeta function limit
Date: Tue, 09 Jul 2013 21:00:52 +0200
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Xref: csiph.com comp.soft-sys.math.maple:783

On 09.07.2013 19:28, peter.luschny@gmail.com wrote:
> With Maple V R5 I get
> 
> f := z -> -(-Zeta(1-z)+z*Zeta(1,1-z))/(z*Zeta(1-z));
> limit(f(z), z=0);
> 
> MAPLE> -gamma
> 
> But looking at the plot kindly provided by Wolfram Alpha
> I feel unsure if this is correct.
> 
> Limit[(-Zeta[1 - z] + z D[Zeta[1 - z], z])/(-(z Zeta[1 - z])), z->0]
> 
> Peter
> 

Using M17 also gives me that (the alternative MultiSeries
does it as well), and f(1e-12) is close (even with 100
decimal places. Plotting with y-range fits with that,
plot(f(z), z=0 .. 1, -1 .. 0);

However the Maple equivalent for z D[Zeta[1 - z], z] is
z*diff(Zeta(1-z),z) = - z*Zeta(1,1-z) while in your f
you have a '+' sign, you have D(Zeta)(1-z) there.