A failed Riemann attempt finds a stronger bound

Euler noticed that the sum over all $n$ equals a product over the primes: expanding each factor $1/(1 - p^{-s}) = 1 + p^{-s} + p^{-2s} + \cdots$ and multiplying out produces every $n^{-s}$ exactly once, by unique factorization. Riemann (1859) extended $\zeta$ to all complex $s \neq 1$ and showed that its zeros control how regularly the primes are spaced. Von Koch proved in 1901 that the hypothesis is equivalent to the sharpest natural form of the prime number theorem:

where $\pi(x)$ counts the primes up to $x$ and $\operatorname{Li}(x) = \int_2^x dt / \log t$.

$\zeta$ has trivial zeros at $-2, -4, -6, \dots$. All the others lie in the critical strip $0 < \operatorname{Re} s < 1$, placed symmetrically around the critical line $\operatorname{Re} s = \tfrac12$ because of the functional equation

Their number is known precisely: the zeros with $0 < \operatorname{Im} s \le T$ number

On the critical line, $\zeta(\tfrac12 + it)$ can be turned into a real function, Hardy's $Z(t) = e^{i\theta(t)} \zeta(\tfrac12 + it)$, so every sign change of $Z$ is a zero exactly on the line. Hardy (1914) used this to find infinitely many. Write $N_0(T)$ for the zeros on the line; the question is how large a proportion $\kappa$ one can prove:

Selberg (1942) proved that some $\kappa > 0$ works. Levinson (1974) reached $\kappa > \tfrac13$ and Conrey (1989) $\kappa > \tfrac25$; later refinements moved the record to about 41%. These methods multiply $\zeta$ by a mollifier, a short sum $M(s) = \sum_{n \le y} b_n n^{-s}$ chosen to cancel its large values; longer mollifiers give better proportions but are much harder to control.

A bound of the form $\kappa \ge 0.672$ proves that at least 67.2% of the zeros, counted up to height $T$, lie on the line (and, as the paper's title says, are simple). The rest are not shown to be off the line; they are simply not proved to be on it. The Riemann hypothesis is the statement $N_0(T) = N(T)$ for every $T$, and no proportion short of all of them, even one approaching 100%, rules out infinitely many exceptions.