What the sources say

Answer. The Riemann hypothesis says that every non-trivial zero of the Riemann zeta function has real part 1/2, so all of them lie on one vertical "critical line". The zeros matter for primes because an explicit formula writes the jumps in prime counts as a sum over those zeros, a bit like a sound built from frequencies. If all zeros sit on the line, the gap between the true prime count and its smooth estimate is at most about the square root of x, times a logarithm. The hypothesis has not been proved. I could read only search summaries, not full pages (details under Sources), so the figures below are as those summaries state them.

What the sources say

The statement. The conjecture is that all non-trivial zeros of the zeta function have real part 1/2 and so lie on the critical line. The summary says this controls the error bound in prime-counting estimates (Wikipedia: Riemann hypothesis). The Simons Foundation gives the same statement (Simons Foundation, 6 May 2020).

Zeros and primes are linked by an exact formula. Explicit formulas connect prime-counting functions such as Chebyshev's ψ(x) to sums over the non-trivial zeros (Wikipedia: explicit formulae for L-functions). A MathOverflow thread explains that these formulas express step-like prime sums through sums over zeros (MathOverflow, 2 Dec 2021). This is the source of the "hidden rhythm" picture: each zero contributes one oscillation, and together they reproduce the staircase of primes. A Medium article describes this as the "harmonic frequencies" of the zeros (Medium, 27 Feb 2022). That is a popular account, not a primary source.

What the critical line buys. Lecture notes from Harvard state that the hypothesis holds if and only if π(x) = li(x) + O(x^(1/2) log x) (Elkies, Harvard M259, undated). Here π(x) counts primes up to x and li(x) is the smooth logarithmic-integral estimate. The AIM overview gives the same bound (AIM, Farmer, undated). An explicit version, assuming the hypothesis, is Schoenfeld's: |π(x) − li(x)| < (1/(8π)) √x log x (arXiv 2109.02506, 6 Sep 2021). So if every zero lies on the line, the primes are as regular as this bound allows. If even one zero lay off the line, the error could be larger, which follows from the "if and only if" above.

Status. The Clay Mathematics Institute lists the problem among the Millennium Prize Problems. The page I fetched contained only an introduction to a 2001 lecture series based on those problems (Clay Mathematics Institute). I found no source claiming a proof.

Where sources differ

I found no factual conflict. The summaries differ in wording: some say the zeros "dictate" or "control" the fluctuations, while the precise statement is the one-sided bound and the equivalence above. Read "control" as that bound, not as a formula for individual primes.

Plan for the pictures

These diagrams are my proposal and are not taken from any source.

1. Staircase. An SVG step plot of π(x) for small x, with the smooth li(x) curve over it.
2. Error band. The same plot, with the gap between the two shown as a band of width about √x, which is the size RH allows.
3. Complex plane. A CSS and SVG strip with the vertical line at real part 1/2 and dots for zeros placed on it. Any zero positions I draw must be copied from a published table, which I have not read.
4. Waves. A few sine waves, one per zero, added together to show how oscillations can build a staircase. This is an illustration, not an exact computation, and the page should say so.

Each page should state plainly that the conjecture is unproved.

Sources

• Wikipedia, Riemann hypothesis: search summary only, undated.
• Wikipedia, Explicit formulae for L-functions: search summary only, undated.
• Elkies, Harvard M259 notes: search summary only, undated.
• AIM, Farmer overview: search summary only, undated.
• Simons Foundation, 6 May 2020: search summary only.
• arXiv 2109.02506, 6 Sep 2021: search summary only.
• MathOverflow thread, 2 Dec 2021: search summary only.
• Medium article, 27 Feb 2022: search summary only.
• Clay Mathematics Institute: fetched, but only the lecture-series introduction came back (the page is undated).
• Terence Tao's notes (12 Feb 2021) returned a 404, so I did not use them.

Date looked: I did not record the current date in this session, so I cannot state it. Before publishing, someone should read the Elkies notes and the arXiv paper in full and confirm the two bounds quoted above.