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Prime Numbers and the Riemann Hypothesis by Mazur and Stein makes use of an interesting function: $$\hat{\Phi}_{\le C}(\theta)=2\sum_{prime\:powers\:p^n\le C}p^{-n/2}\cdot log(p)\cdot cos(n\cdot log(p)\cdot \theta)$$

The function is basically a Fourier transform of symmetrized Dirac $\delta$ functions at the prime powers. Per the authors, the "spikes" of $\hat{\Phi}_{\le C}(\theta)$ "for large $C$ pinpoints the spectrum" $\{\theta_1,\theta_2,\theta_3,... \}$, i.e., the sequence in order of imaginary parts of the non-trivial $\zeta$-function zeros.

As an example, here's what I get with a C program and MATLAB to replicate Figure 32.7 in the book: enter image description here

That matches the figure very well so I'm pretty sure the code is right. The red lines are at the locations of the the first six $\theta$'s

The first few $\theta_i$ are more or less at the negative peaks. However, increasing $C$ further doesn't really improve the situation:

enter image description hereenter image description here enter image description here

Shouldn't the $\theta_i$ corresponding to the imaginary parts of the non-trivial $\zeta$-function zeros get more distinct as $C$ increases, not less? Or maybe there's a bug?

EDIT:

Per reuns suggestion of regularizing each term in the sum by $p^{−a^2θ^2n/2}$, here's a comparison of the result at $\theta_1$ using $a=0$ vs. $a=0.1$:

$a=0$: enter image description here

$a=0.1$: enter image description here

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    $\begingroup$ The series $f(\theta) = 2\sum_{prime\:powers\:p^n}p^{-n/2}\cdot log(p)\cdot cos(n\cdot log(p)\cdot \theta)$ converges only in the sense of distributions, you need to look at the regularized version $f_a(\theta)= 2\sum_{prime\:powers\:p^n\le C}p^{-n/2}\cdot log(p)\cdot cos(n\cdot log(p)\cdot \theta) p^{- a^2 \theta^2 n/2}$ with $a$ small to make appear the convolution of the gaussian $|a|^{-1} e^{- \theta^2/(2a^2)}$ with the distribution $\sum_{t \text{ imaginary part of non trivial zeros}} \delta(\theta-t)$ en.wikipedia.org/wiki/Explicit_formulae_(L-function) $\endgroup$
    – reuns
    Feb 22, 2019 at 0:37
  • $\begingroup$ That regularization term is interesting--I'll try to grok what it's doing, but as a magic trick at the moment, given the right choice of $a$ it does seem to make the $\theta_i$ stand out (see edit above). Does $a$ need to be tuned empirically for each $\theta_i$? $\endgroup$
    – Joe Knapp
    Feb 22, 2019 at 11:15

1 Answer 1

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I made this code

    N = 10^5; mu = zeros(1,N); mu(1)= 1; for n = 1:N,  mu(n+n:n:end) = mu(n+n:n:end)-mu(n); end; 
    Lambda = zeros(1,N); logn = log(1:N); for n = 1:N, Lambda(n:n:end) =  Lambda(n:n:end) + log(n)*mu(1:floor(N/n)); end; 


    Lambda12 = Lambda ./ (1:N).^(1/2);         % Lambda(p^k) = log(p)

    % the raw Fourier transform of the OP
    dx = 0.01; X = [0:dx:50]; f = zeros(1,length(X)); 
    for l = 1:length(X), x = X(l); f(l) = sum(Lambda12 .* exp(-i*logn * x));end; 

    % substract the contribution of the pole at s=1/2 and the trivial zeros at s = -2k-1/2
    h = (exp(log(N)*(1/2-i*X))-1)./(1/2-i*X); for k = 1:500, h = h - (exp(log(N)*(-2*k-1/2-i*X))- 1)./(-2*k-1/2-i*X); end; g = f-h; 

    % remove oscillations due to finite N
    G = [g(end:-1:1),g]; K = 50; filter = ones(1,K)/K; filter = conv(filter,filter); filter = conv(filter,filter); F = conv(real(G),2*filter); F = F-F(5000); 

    plot(-F);

You can run it in https://octave-online.net/ (click 2 times on add 15 seconds during the execution)

The peaks are the non-trivial zeros of $\zeta(s)$, each one integrates to $2\pi$

enter image description here

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  • $\begingroup$ Neat--the code runs nicely on MATLAB. I gather you first get the Mobius function, then Mangoldt, and then the graph above is of $$f(\theta)=\sum_{c=1}^C \frac{\Lambda(c)-1}{\sqrt c}\cdot r(c)\cdot cos(\theta \log c)$$ Where $r$ is the regularization term. Is that it? Is this closely related to the $\Phi$ function above somehow? $\endgroup$
    – Joe Knapp
    Feb 22, 2019 at 20:23
  • $\begingroup$ One thing I noticed: the regularization term in your program doesn't do much. If it's set to 1 the output is pretty much the same. But if 1 isn't subtracted from $\Lambda$ the result looks pretty much like the $\Phi$ function above. So maybe that's really the key (removing the effect of the pole at 1 like the comment states)? $\endgroup$
    – Joe Knapp
    Feb 23, 2019 at 1:43
  • $\begingroup$ @JoeKnapp I cleaned up the code, can't do much better. $\endgroup$
    – reuns
    Feb 23, 2019 at 5:24

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