Non trivial Zeros of Riemann Zeta knows where primes are

Yasuaki Honda at gmail dot com

[1] Conrey, The Riemann Hypothesis https://www.ams.org/notices/200303/fea-conrey-web.pdf

[2] The LMFDB Collaboration, The L-functions and Modular Forms Database, home page of the Zeros of zeta(s), https://www.lmfdb.org/zeros/zeta/?limit=200&N=1, 2020 , [Online; accessed 10 October 2020].

In [1]:
f(x)='realpart(sum(x^rho[j],j,1,inf));
Out[1]:
\[\tag{${\it \%o}_{1}$}f\left(x\right)={\it realpart}\left(\sum_{j=1}^{\infty }{x^{\rho_{j}}}\right)\]
In [2]:
/*
The LMFDB Collaboration, The L-functions and Modular Forms Database, 
home page of the Zeros of zeta(s),
https://www.lmfdb.org/zeros/zeta/?limit=200&N=1, 2020 , [Online; accessed 10 October 2020].
*/
kill(img_rho);
img_rho:[
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]$
Out[2]:
\[\tag{${\it \%o}_{2}$}\mathbf{done}\]
In [3]:
f(x):=-ev(realpart(sum(x^(1/2+%i*img_rho[j]),j,1,200)),numer);
Out[3]:
\[\tag{${\it \%o}_{4}$}f\left(x\right):=-{\it ev}\left({\it realpart}\left({\it sum}\left(x^{\frac{1}{2}+i\,{\it img\_rho}_{j}} , j , 1 , 200\right)\right) , {\it numer}\right)\]
In [4]:
pi1[n]:=if n<2 then 0 elseif primep(n) then pi1[n-1]+1 else pi1[n-1]$
In [5]:
pi(x):=if integerp(x) and primep(x) then pi1[x]-1/2 else pi1[floor(x)]$
In [6]:
plot2d([f(x),pi(x)*10],[x,1,100]);
Gnuplot Produced by GNUPLOT 5.2 patchlevel 8 x fun1 fun1 fun2 fun2 -200 -150 -100 -50 0 50 100 150 200 250 10 20 30 40 50 60 70 80 90 100
Out[6]:
\[\tag{${\it \%o}_{7}$}\left[ \mbox{ /tmp/maxout614.gnuplot } , \mbox{ /tmp/maxplot.svg } \right] \]
In [ ]: