The quasi-Riemann hypothesis

The explanation

Add 1 + 1/4 + 1/9 + 1/16 and keep going, using one over every square number. The total settles at π²/6, about 1.6449. Replace the power 2 by any number s and you get Riemann's zeta function, written ζ(s). A standard extension lets s be a complex number, with a real part and an imaginary part. Call the imaginary part the height. The first zero of ζ, a point where it equals 0, has real part 1/2 and height about 14.1347. The next two also have real part 1/2, at heights about 21.0220 and 25.0109.

The real part of each zero is what matters here. The zeros of interest all have real part between 0 and 1, a band called the critical strip. The Riemann hypothesis says every one of them has real part exactly 1/2. It is still open, and this work does not change thatSource 1. The quasi-Riemann hypothesis is a much weaker wish. It asks for one fixed wall below 1, such as 0.99. No zero may lie to the right of that wall, at any height. It says nothing about where the zeros sit, only that they keep a fixed distance from 1.

Zeros near real part 1 matter. In 1859, Riemann tied the zeros of ζ to how primes are spread out. In 1896, Hadamard and de la Vallée Poussin showed that no zero has real part exactly 1. That gave the prime number theorem, which counts primes up to a given size. Later work cleared a thin zone just left of the line at real part 1. But every such zone narrows as the height grows. In 1958, Vinogradov and Korobov found a different zone, and it narrows too. No fixed wall was known.

The same question covers a wider family. Take 1 − 1/3 + 1/5 − 1/7 and so on. Each odd number gets + or − by its remainder after dividing by 4. The total is π/4, about 0.7854. That sign pattern is a Dirichlet character: a rule that repeats with some period and respects multiplication. Each character gives its own Dirichlet L-function, and ζ is the case where every sign is +. Dirichlet built these in 1837 to find primes in arithmetic progressions, such as 1, 5, 9, 13 and so on. For them the classical zones narrow as the height or the period grows. They also allow one possible exception. That is a single real zero very close to 1. It can occur only for a character with values 0, 1 and −1. Such a zero is called a Landau–Siegel zero, and nobody could rule it out.

The first paper, dated 30 September 2026, proves that the wall exists and puts it at 7/8, or 0.875Source 2. No Dirichlet L-function, ζ included, has a zero with real part above 7/8. The same wall holds for every character, every period and every heightSource 3. The line at 7/8 itself is not covered. Any Landau–Siegel zero above 7/8 is ruled out as wellSource 4. For the basic, so-called primitive L-functions, a mirror symmetry then keeps every such zero between 1/8 and 7/8. That still allows zeros at real part 0.6 or 0.8, which the Riemann hypothesis forbids.

The one obstacle was that classical methods lose strength as the height and the period grow. A fixed wall needs an argument whose margin is the same for every function. The paper gets this by working in a larger number system, the Eisenstein integers. These are numbers a + bω, with a and b whole numbers. Here ω is a cube root of 1 other than 1. They are the whole numbers of the number field Q(√−3). The paper proves the wall for a whole family of functions over that field, the finite-order Hecke L-functionsSource 2. The whole family is needed even for ζ, because the method keeps producing twisted relatives of the function under studySource 5. It draws on a cubic version of the theta series, from S. J. Patterson in 1977Source 6.

The core move bounds one carefully weighted sum directly. It also compares that sum with an integral containing 1 divided by the L-functionSource 7. If some zero lay past the wall, take the rightmost possible zeros. The two computations would show that this reciprocal stays well behaved a fixed distance left of them. A reciprocal cannot stay well behaved at a zero, so no such zero exists. A first pass gives a wall at 11/12, about 0.9167, and a refined pass reaches 7/8Source 8. Each Dirichlet L-function, times a partner, matches a Hecke L-function up to finitely many harmless factorsSource 9. So the wall passes down to every Dirichlet L-function.

Two companion papers sit beside it. One, dated 5 October 2026, claims to prove the 11/12 wall by itself, as a shorter intermediate stepSource 10. The other, dated 1 October 2026, takes a separate, mostly algebraic route aimed only at Landau–Siegel zerosSource 11. It proves that one constant c > 0 works for every such character with period q of at least 3Source 12. Each real zero β of such a function then satisfies (1 − β) × log q ≥ c. It gives no value for c. The first paper also draws consequences. One concerns the smallest remainder that is not a perfect square, modulo a prime. Modulo 7, squares leave remainders 1, 2 and 4, so that smallest non-square is 3. The paper claims this number is at most a fixed power of log p for every odd prime pSource 13.

How sure is this? A computer proof checker called Lean has checked formal versions of the 7/8 wall. They cover ζ and every Dirichlet L-function, with one wall for all of them. They also cover the finite-order Hecke L-functions over Q(√−3). Lean has also checked the uniform gap for real zeros from the 1 October paper, with no explicit c. The paper's later applications, such as the non-square bound, are outside that check. The 11/12 companion is not in the Lean scope note. No specialist review has been published.

What it doesn't show. This does not prove the Riemann hypothesis: zeros may still lie anywhere from 1/8 to 7/8, not only at 1/2.