# Question:How to plot the approximation graph for the primes staircase

## Question:How to plot the approximation graph for the primes staircase

Maple

I must feed the not trivial zeros numbers into this aproximation formula ?

Riemann hypothese and staircase of primes

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The prime counting function is approximated by  and .

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The staircase of primes approximated by two functions
Interesting is the video: How i learned to love and fear the Riemann Hypothesis

https://www.quantamagazine.org/how-i-learned-to-love-and-fear-the-riemann-hypothesis-20210104/

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 > ps:=Array(1..30): y:=0: for n from 1 to 30 do  if is(n,prime)      then ps[n]:=plot([[n,y],[n,y+1],[n+1,y+1]]):      y:=y+1;      else ps[n]:=plot([[n,y],[n+1,y]]):  end if ; od; with(plots): display({seq(ps[n],n=1..30)}):
 > plot([PrimeCounting(x)] ,x = 1 .. 35, legend = [pi(x)]):
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Prime counting function
What found RIEMANN for the prime counting function in relation to the zeta function after he defined the zeta function?

He found further a function what follows exactly the shape of the prime counting function

Final discovery v. Riemann.

- step in the omhoog in de priemtelfunctie = log(p) (zie video)

Using the logarithmic primecount function( from Chebyshev) (approximation)
Further  analyse with this Chebyshev approximation formula in relation to the not trivial zero points from Riemann zeta function ( zeros) gives another real function for approximating the primecounting function what uses the non trivial zeros from Riemanns zeta function  in this function:

Number now all nottrivial zeros in the upperhalfplane from down to bottom,  as

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Its only  that must be calculated out of the not trivial zeros and i must have a list of  serie of not trivial zeroes from the zeta function. => see Hardy's Z(t) ? from this .....   can be calculated ?
All not trivial zeros are complex numbers laying on a line ,but  orginating from (0,0) in the complex plane as  vectors to the points

 > varphi(x):= x - ln(2*Pi) - 1/2*ln(1 - 1/x^2) - sum(2*x^v[k]*cos(w[k]*ln(x) - alpha[k])/abs(u[k]), k = 1 .. infinity);
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This formula seems to be correct .
Now how to make a plot ?
Hardy's Z(t) function shows the not trivial zeros in the upperhalfplane of the critical strip of the Riemann zeta function  as zeros in this Z(t) real function : derived from a alternating serie ?