Riemann hypothesis

Circle the prime numbers up to 100: there are 25. A prime is a whole number greater than 1 whose only positive divisors are 1 and itself. Count up to 1,000 and there are 168. The count jumps whenever another prime appears.

A smooth estimate captures the broad trend. The Riemann hypothesis predicts a tight limit on how far the actual count can stray from a refined estimate as we count further.[3] [6]

Prime counts rise in uneven steps, although a smooth estimate describes their overall growth. Can we tightly bound the departures from that estimate? The Riemann hypothesis gives an equivalent target in another setting: every nontrivial zero of the zeta function must have real part one-half. A complex input has two coordinates; its real part is the horizontal one. A zero is an input where the function gives zero.

The arithmetic connection starts by adding one weighted contribution for each positive integer. For a real input of two, the sum begins $1+1/4+1/9+1/16+\cdots$. More generally, for $\operatorname{Re}(s)>1$,

$$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p\text{ prime}}(1-p^{-s})^{-1}.$$

Each prime's factor expands into contributions from its powers. Unique prime factorization means their product contains each integer's term exactly once: $12=2^2\times3$, for example. The series and product converge absolutely here, so these rearrangements are valid.

Analytic continuation uniquely extends the function beyond this domain, with a single pole at $s=1$, where its values become unbounded as the input approaches one. This extension, rather than the original divergent series, is used in the critical strip $0<\operatorname{Re}(s)<1$. The negative even integers are the trivial zeros. All other zeros lie in that strip; RH asks whether every one lies on the critical line $\operatorname{Re}(s)=1/2$. Generalized hypotheses for other functions are separate questions.[2] [3] [4] [5]

The exact claim concerns a function called zeta. Its inputs are complex numbers, which can be pictured as points with horizontal and vertical coordinates. A zero is an input where the function gives zero.

The relevant zeros lie in a band whose horizontal coordinates are between 0 and 1. The hypothesis puts every one on the middle line, at 1/2. Infinitely many zeros must be covered; checking a finite number leaves more beyond them.[5]

The nontrivial zeros are symmetric about the critical line and the real axis. Symmetry permits an off-line zero with reflected companions, so it does not settle the hypothesis. Even one exception would disprove RH; the zeros need not all be simple.

Platt and Trudgian's rigorous computation covers a finite range of heights. A different kind of progress is Lamzouri's September 2026 version-2 preprint: let $N(T)$ count all nontrivial zeros with $0<\operatorname{Im}(s)\le T$, including multiplicities, and $N_0^{\mathrm{s}}(T)$ count the simple ones on the line, which are zeros without repetition. It reports

$$\liminf_{T\to\infty}\frac{N_0^{\mathrm{s}}(T)}{N(T)}\ge 0.6725007\ldots.$$

This is an asymptotic lower proportion. It does not guarantee that percentage in every finite range, and it leaves room for off-line zeros. Neither kind of progress proves the universal claim. Clay continues to list the classical problem as unsolved.[1] [5] [7] [8] [9]

Averages can hide large errors. A proof would limit the error in estimates of prime counts, even at scales where listing every prime is impractical. It would turn a powerful assumption into a theorem. The gaps between individual primes would still be uneven.[2] [6]

A proof would turn a conjectural limit on prime-count error into a theorem. The prime-counting function $\pi(x)$ counts primes up to $x$; the refined smooth estimate $\operatorname{li}(x)$ is the principal-value integral of $1/\log t$ from zero to $x$. Here logarithms are natural, and the principal value handles the singularity at $t=1$. RH is equivalent to

$$\pi(x)=\operatorname{li}(x)+O(\sqrt{x}\log x),\qquad x\to\infty.$$

The error is bounded by a fixed multiple of $\sqrt{x}\log x$ for all sufficiently large $x$. This preserves the irregular staircase while restricting how far it can stray from the curve.

Why should zeros control that error? An explicit formula for a weighted count of primes and prime powers contains a contribution $-x^{\rho}/\rho$ for each nontrivial zero $\rho=\beta+i\gamma$. Since $x^\rho=x^\beta\exp(i\gamma\log x)$, the horizontal coordinate sets the growth scale and the vertical coordinate sets oscillation. On the critical line, $x^\beta=\sqrt{x}$. Controlling the combined contributions still requires estimates; the size of one term alone is not the prime-count error.[2] [3] [6]