The Riemann hypothesis, dating back to 1859, states that the zeta function ζ(s), with s = σ + it has zeros in the critical strip 0 < σ < 1 only for σ = 1/2. If proved, it would have a profound impact not just in number theory, but in many other areas of mathematics and beyond. In layman's terms, it can be re-formulated as follows.
We could introduce a parametric family of real-valued functions, defined as follows:
with 0 < σ < 1, t a real number, α, β, γ three real parameters, and λ(⋅) a real-valued function with logarithmic growth. Elementary computations show that s = σ + it is a complex root (also called zero) of ζ(s), with 0 < σ < 1, if and only if
- ϕ(σ, t; 0, 1, 0) = 0,
- ϕ(σ, t; 0, 1, −π/2) = 0,
- λ(n) = log(n).
Moving forward, we will focus on Riemann Hypothesis as being a problem of finding the zeroes (or lack of) of a bivariate function in the standard plane: σ is the first variable, attached to the X-axis, and t is the second variable, attached o the Y-axis. A generalized version of Riemann Hypothesis seems to also be true: it corresponds to arbitrary values for α, β, γ. However we focus here on the classical Riemann Hypothesis. For ease of presentation, we use the following notation:
- ϕ1(σ, t) = ϕ(σ, t; 0, 1, 0)
- ϕ2(σ, t) = ϕ(σ, t; 0, 1,−π/2 )
Much of the discussion has to do with the orbit of (ϕ1, ϕ2) when σ is fixed but arbitrary, and only t is allowed to vary. In short, we are dealing with a bivariate time series in continuous time. Without loss of generality, we assume that t is positive. The spectacular plot shown below is just a scatterplot of the orbit, computed for σ = 0.75. It easily generalizes to other values of σ that are strictly greater than 0.5.
The plot below is called the Eye of the Zeta Function. It is the first time that such a plot was created for the Riemann zeta function. It corresponds to σ = 0.75, with t between 0 and 3,000, with t increments equal to 0.01. Thus 300,000 points of the orbit are displayed here.
The spectacular feature in that plot is the hole around (0, 0). It has deep implications. It suggests that if σ = 0.75, not only ϕ1(σ, t) and ϕ2(σ, t) can not be simultaneously equal to zero. This is a particular case of Riemann Hypothesis. But most importantly, that it never jointly gets very close to zero. This is new and suggests that proving Riemann Hypothesis might be a little less challenging than initially thought. The same plot features a similar "eye" if one tries various values of σ. In particular, the hole gets smaller and smaller as σ gets closer to 0.5. At σ = 0.5, the hole is entirely gone, and infinitely many values of t yield ϕ1(σ, t) = ϕ2(σ, t) = 0. The same is true for a generalized version of Riemann Hypothesis.
Note that it is very tricky to get the scatterplot right. The series for ϕ1 and ϕ2 converge very slowly, and in chaotic, unpredictable way. This can result in false positives: points very close to zero due to approximation errors, artificially obfuscating the hole. Convergence boosting techniques are required. In addition, the frequency of oscillations in ϕ1 and ϕ2 increases more and more as t gets larger, and thus t increments should be made smaller and smaller accordingly, as t grows, in order to get a good coverage of the orbit and not miss potential true zeroes.
QUESTION I : How one can generalize Riemann Hypothesis?
QUESTION II : Could the Scatter Plot supply a proof of Riemann Hypothesis?
REFERRENCE : Spectacular Visualization: The Eye of the Riemann Zeta Function