Quasi-Riemann Hypothesis

A prime is a whole number above 1 that only 1 and itself divide, such as 2, 3, 5, 7 and 11. Primes become rarer as numbers grow. There are 25 primes up to 100 and 168 up to 1,000. There are 78,498 up to one million.

A good formula estimates how many primes lie up to any number $x$ . It is called $li (x)$ . For one million it gives 78,628, so it is off by only 130. The question is how far off this estimate can be for huge numbers.

The answer depends on the Riemann zeta function. A function is a rule that gives an output for each input. The zeta function starts with this sum, which has one term for each whole number:

ζ (s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + \dots

The symbol $ζ$ is the Greek letter zeta, and the dots mean that the sum never ends. For $s = 2$ , the terms are 1, $1 / 4$ , $1 / 9$ , $1 / 16$ and so on, and their sum approaches about 1.645.

Mathematicians also know how to give this function a meaning for other inputs, including complex numbers. A complex number has the form $a + b i$ , where $i^{2} = - 1$ . It is a point on a flat map, with $a$ across and $b$ up. The number $a$ is its real part.

A zero of the function is an input that gives the output 0. There are easy zeros at $- 2$ , $- 4$ , $- 6$ and so on. All the others lie in a narrow band, where the real part is between 0 and 1. The figure shows this band.

The band that holds the zeros. The dots are the first five zeros, and each number says how high up the dot sits. The label Re s means the real part. The new result clears the shaded strips. The lower half of the map is a mirror image.

Bernhard Riemann showed that these zeros control the error in the prime count. Each zero adds a wave to the error. The farther right the zero sits, the bigger its wave. Call the real part of a zero $β$ , the Greek letter beta. That zero gives a wave of size about $x^{β}$ . So the zero farthest to the right sets the size of the error.

Here is the scale when $x$ is one million. If $β = 1 / 2$ , then $x^{β}$ is the square root of $x$ , which is 1,000. If $β = 7 / 8$ , then $x^{β}$ is about 178,000. If $β$ is close to 1, the wave is nearly one million, about as large as $x$ itself.

The Riemann hypothesis says that every zero in the band has real part exactly $1 / 2$ . That would make the error about as small as possible. Computers have confirmed this for the first 12 trillion zeros, but nobody has a proof for all of them.

The quasi-Riemann hypothesis asks for much less. It says only that the zeros stay away from the right edge. Some fixed line left of real part 1 has no zeros to its right. Even this was unknown. Zeros very high up in the band could still sit closer and closer to the edge.

OpenAI reports a proof with the line at $7 / 8$ . A symmetry of the function then also clears the left side, below $1 / 8$ . The error in the prime count is then not much larger than $x^{7 / 8}$ .

The same limit holds for related functions that count primes in groups, such as primes that end in 7. Up to one million, the counts for the endings 1, 3, 7 and 9 are 19,617, 19,665, 19,621 and 19,593. The new result limits how uneven such counts can become. The Riemann hypothesis itself stays open.