How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All:

I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com:

https://mathematica.stackexchange.com/questions/95294/can-anyone-re-produce-this-result-related-to-the-spectrum-of-riemann-zeta-using

Can anyone help ?

It is related to a plot of paper "The Riemann Hypothesis" by J. Brian Conrey: the Fourier transform of the error term in the prime number theorem.

Beside Mathematica problems, I also have questions regarding the math behind it:

A: Do we do Fourier analysis on Chebyshev function itself or the error term, (Chebyshev - x) ?

B: Why do we need to calculate 10^6 points of Chebyshev function for this purpose ? Can we do with some less points, such as: 10^4 ?

C: How is the number of data points for Chebyshev function related to the resolution of frequency domain of zero spectrum ?

D: Can we use built Fast Fourier Transform to do above analysis or do we have to implement discrete sine transform ourselves as in the code.

This is really not a specific answer to the question, but one way to illustrate evolution of the zeta zeros from the primes is to evaluate the real part of formula (1) below along the critical line.

(1) $\int_{2-\epsilon}^{N+\epsilon}\frac{d\,(\text{J}(x)-\text{li}(x))}{dx}x^{-s}dx\approx\sum_{n=1}^N\text{If}\left[\text{PrimePowerQ}[n],\frac{n^{-s}}{\Omega (n)},0\right]-\text{Ei}((1-s)\log (N))+\text{Ei}((1-s)\log(\text{2}))$

The following plot illustrates the real part of formula (1) evaluated along the critical line using an evaluation limit of $N=1000$. Formula (1) is illustrated in blue, and $\Re\left(\log\zeta\left(\frac{1}{2}+i\,t\right)\right)$ is illustrated in orange as a reference. The red discrete portion of the plot illustrates the evaluation of formula (1) at the first 10 zeta zeros.

Another approach which is more closely related to the question is to evaluate formula (2) below.

(2) $\quad 2\sum_{n=1}^N\text{MangoldtLambda}\,[n]\,n^{-\frac{1}{2}}\,\text{Cos}\,[\,\text{Log}\,[n]\,t]$

The following plot illustrates formula (2) with an evaluation limit of $N=1000$. The red discrete portion of the plot illustrates the evaluation of formula (2) at the first 10 zeta zeros.

Formula (2) diverges as $N\to\infty$ (the oscillation in the corresponding plot grows in both magnitude and frequency as $N$ increases), but at finite evaluation limits formula (2) illustrates evolution of the zeta zeros from the primes.

• it diverges as $K \to \infty$ – reuns Mar 9 '17 at 0:31
• A sequence diverges if it doesn't converge. Your plot is a nonsense. – reuns Mar 10 '17 at 4:20
• I told you many times that (assuming the RH) you need to plot $\int_1^N (\psi(x)-x) x^{-s-1}dx$ at $s=1/2+\epsilon+it$. – reuns Mar 11 '17 at 8:39
• @user1952009 Yes, and I initially illustrated this formula in a question (which I've since deleted), and subsequently illustrated this formula in a post on my website. But I was trying to keep things simple in my answer above as my answer above was intended for a recreational demonstration of the evolution of the zeta zeros from the primes. As you've indicated, formula (1) diverges as $N\to\infty$, but for a recreational demonstration I think the presence of the oscillation (which grows in both in magnitude and frequency as $N$ increases) is acceptable. – Steven Clark Mar 11 '17 at 17:17
• No it isn't. Did you try plotting what I said ? – reuns Mar 11 '17 at 18:38