[Scilab-users] Riemann Zeta update

[Lester Anderson]

Finally got to the end of the problem and replicated the plot of the
Riemann Zeta function on the critical line (s=0.5 + %i*t)
Looks pretty close to that shown on the Wikipedia page for the Riemann Zeta
Function!

function zs1=zeta_0_1(s, n)

// Vectorised version    zs1=0k=linspace(1,n,n);
          zs1 = sum((-1).^(k+ 1)./(k.^s ));
          zs1 = 1./(1 - 2.^( 1-s )).*zs1;endfunction
k=linspace(1,50,1000)
s_list=0.5+%i*k;
for i = 1:length(k)
    s2(i) = zeta_0_1(s_list(i),1e6);end
S_real=real(s2);S_imag=imag(s2);
plot(S_real, S_imag)

Note: the function for the "Critical strip" uses vectorisation to improve
speed.

Lester

On Sun, 22 May 2022 at 07:31, Lester Anderson <arctica1963 at gmail.com> wrote:

> Hi all,
>
> After a lot of trial and error, I have managed to get a set of functions
> to compute the approximations of Riemann's Zeta for negative and positive
> real values; values of n > 1e6 seem to give better results:
>
> function zs=zeta_s(z, n)
>     // Summation loop
>     zs=1;
>     if z == 0
>        zs = -0.5
>     elseif z == 1
>        zs = %inf
>     else
>         for i = 2: n-1
>             zs = zs + i.^-z;
>         end
>     endendfunction
> function zfn=zeta_functional_eqn(s)// Riemann's functional equation// Analytic continuation for negative values
>     zfn = 2.^s .* %pi.^(s - 1) .* sin(%pi.*s./2) .* gamma(1 - s) .* zeta_s((1 - s),n)endfunction
>
> For even values of s < -20 the values of Zeta(s) increase in value and are
> not as close to zero as expected e.g. zeta_functional_eqn(-40) gives
> 7.5221382. At small even values e.g. -10, the result is of the order of
> ~1e-18 (close enough to zero). Any ideas why the even zeta values increase
> or how to reduce that response?
>
> The solution over the critical strip (zero to one) is not so efficient
> unless n is very large( > 1e8), and there seems to be a performance issue
> when using a for-loop compared to vectorisation. Vectorised n speeds things
> up quite a bit.
>
> function zs2=zeta_0_1(s, n)zs2=0for i = 1: n
>          zs2 = zs2 + (-1).^(i + 1)./(i.^s );end
>          zs2 = 1./(1 - 2.^( 1-s )).*zs2;    endfunction
> function zs1=zeta_0_1(s, n)// Vectorised version    zs1=0k=linspace(1,n,n);
>           zs1 = sum((-1).^(k+ 1)./(k.^s ));
>           zs1 = 1./(1 - 2.^( 1-s )).*zs1;endfunction
>
> For example, calculating the approximation of Zeta(0.5) using a for-loop
> takes ~150s to give a value of -1.4602337981325388 (quite close),
> whereas the vectorised version does the computation in under 20s, both
> tested using n=1e8. Can the functions be optimised to improve speed and
> accuracy?
>
> Using Scilab 6.1.1 on Windows 10 (16 Gb RAM).
>
> Thanks
> Lester
>
>
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