Beyond a Ninth: Sieving Zeros on the Golden Line ✛
Context
In the infinite river of integers, where eddies of primes swirl, mathematicians sieve the sand in search of golden nuggets — the zeros of L-functions. These enigmatic objects, discovered by Riemann in 1859, govern the distribution of primes like an invisible hand. The famous Riemann Hypothesis asserts that all non-trivial zeros of the zeta function lie on a single vertical line, the critical line. Despite generations of effort, a proof remains elusive. But for entire families of L-functions, we can catch the statistics: what proportion of zeros lies exactly on this mother lode? That's the question this new work answers, using the sophisticated technique of spectroscopy — analyzing the distribution of zeros through their averaged behavior.
Methods
The tool of the catch is the Levinson method, an ingenious way of 'weighing' zeros by adding a special factor (mollifier) to the L-function that smooths out fluctuations near the critical line. But the real breakthrough lies in the construction of a colossal sieve. The researchers average L-functions not individually but over a family of Dirichlet characters (twisting the function like a Doppler shift). This gives rise to the asymptotic large sieve: repeated application of interferometry of characters, akin to combining radio waves from many antennas, allows the weak signal of the critical line to be isolated. The subtlety is that the arithmetic coefficients of L-functions are highly irregular, and without unproven hypotheses (the Generalized Ramanujan Conjecture), the classical approach collapses. The authors overcame this by manually analyzing the detailed structure of the coefficients using computer algebra (numerical simulation) in Mathematica — a true telescope into the world of automorphic representations.
Results
The result is staggering: for self-dual representations of PGL(3), at least 1/9 of all zeros lie on the critical line. Mathematicians may not have caught all the gold ore, but a ninth is already a huge haul! For PGL(2), the proportion is higher — 1/3, exactly matching Levinson's classic result for the zeta function. Moreover, an asymptotic formula is proved for the twisted mean square of L-functions with an unprecedented error term of order O($Q^{7/4+ε}$). It holds in the range Q^ε ≤ T ≤ $Q^{1/3–ε}$ and for mollifier polynomials of length up to 1/2. It's as if, in a river of varying depth, we learned to sieve sand of different coarseness without losing any flakes of gold.
Implications
The result is the first unconditional evidence that even for degree-3 L-functions, the Riemann Hypothesis is 'almost true' in a statistical sense. It breathes new life into the vast research program on families of automorphic L-functions, connecting them to moments, subconvexity, and prime distribution. The developed asymptotic large sieve technique becomes a universal tool — now applicable to other problems where hypotheses about Fourier coefficients were once needed. And the hidden mass of off-critical zeros (dark matter of the number world) becomes slightly less mysterious.
Future development
The proportion 1/9 can be improved by optimizing the mollifier and accounting more precisely for arithmetic. The next goal is to remove entirely the technical hypothesis (Π4) on coefficient moments, proving it unconditionally. Pushing beyond T = $Q^{1/3}$ will require overcoming monstrous difficulties with convolution sums, but will open the path to proving the Lindelöf Hypothesis on average. The computer algebra algorithms honed in this work will form the basis for automated analysis of arithmetic factors in future projects.
Impact
Number theory receives a powerful boost: from now on, one can prove the existence of zeros on the critical line for a wide class of L-functions without the crucial unproven hypotheses. Cryptography, which relies on the distribution of primes, gains greater confidence in the predictability of 'randomness.' Chaos physics and quantum systems modeled by automorphic spectra see here new food for analogies.
Next steps
The immediate task is to remove Hypothesis (Π4) for all PGL(3) representations and push the proportion above 1/9. Then — apply the sieve to high-order moments and joint distributions of several L-functions, paving the way for the Montgomery pair correlation.
Key open problems
The work is directly connected to the Generalized Riemann Hypothesis (GRH) and the Lindelöf Hypothesis — it gives the first unconditional estimates in a situation where GRH was previously required. The sieve technique touches on the CFKRS moment conjectures, which predict exact constants in mean values, and subconvexity bounds for L-functions. Overcoming the barrier linked to the Generalized Ramanujan Conjecture is important for all of analytic number theory.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper applies Levinson's method with a large sieve to families of Dirichlet L-functions twisted by characters, proving that at least 1/9 of zeros for PGL(3) and 1/3 for PGL(2) lie on the critical line. An asymptotic formula is obtained for the twisted second moment with an error of O(Q^{7/4+ε}) in the range Q^ε ≤ T ≤ Q^{1/3–ε} and for smoothing polynomials of length up to 1/2.
Where its strength is
A strong point is the unconditional proof (without the generalized Ramanujan conjecture) of a positive proportion of zeros on the critical line for degree 3 L-functions, achieved through a delicate analysis of coefficients using computer algebra.
Where it could go next
- It might be interesting to compare the obtained proportion 1/9 for PGL(3) with results on zero distribution for other families of L-functions, such as those related to modular forms, as in the work on the tuning fork for primes.based on: λ∞: The Tuning Fork for Prime Numbers arXiv:2607.01265
- The techniques of quantum chaos and spectral statistics are nearby; perhaps they could be applied to refine the statistics of zeros near the critical line, much like quantum billiards model eigenvalue distributions.based on: A Quantum Instrument Plays the Music of Primes arXiv:2510.16759
- There is a paper in the archive on a geometric approach to L-functions; its methods might help interpret the contribution of the main term in the second moment asymptotics via geometric invariants.based on: Geometry Beats Guesswork arXiv:2605.23669
Worth trying
- Try applying the developed large sieve to other families of L-functions, for example, to symmetric square L-functions.
- Use numerical simulations to test hypotheses about the proportion of zeros on the critical line for higher degrees.
- Investigate the possibility of improving the error term in the asymptotic formula by refining the coefficient analysis.
Nearby works in our archive
- λ∞: The Tuning Fork for Prime Numberssimilarity 0.53
- A Quantum Instrument Plays the Music of Primessimilarity 0.53
- Numbers Hide Patternssimilarity 0.53
- Geometry Beats Guessworksimilarity 0.52
- The Master Key to the Secrets of Spinning Black Holessimilarity 0.52
- Neutrino Rebellion Decoder: How AI Reads Chaos in Stellar Coressimilarity 0.52
The archive is sparse around this work: the closest review is only 0.53, against a usual 0.64. It looks like few have drilled nearby.
Original abstract
as written by the authors arXiv:2607.00282
Let $$\Pi_{0}$$ be a cuspidal automorphic representation of $$\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$$. In this paper, we use Levinson's method to prove that, as $$Q\to \infty$$, at least $$1/9$$ of the zeros of the $$L$$-functions $$L(s, \Pi_{0}\,\times\, \chi)$$ lie on the critical line, where $$\chi$$ ranges over the family of primitive Dirichlet characters of conductor up to $$Q$$. This result is unconditional when $$\Pi_{0}$$ is self-dual, and otherwise holds under a mild condition. The key technical input is a new asymptotic formula with a power-saving error term for the mean square of the product of $$L(s, \Pi_{0}\times \chi)$$ and a Dirichlet polynomial with arbitrary coefficients in both the $$T$$- and $$Q$$-aspects for the range $$Q^{\epsilon}\le T \le Q^{1/3-\epsilon}$$. When $$T=Q^{\epsilon}$$, our asymptotic formula allows Dirichlet polynomials of length $$\theta <1/2-\epsilon$$; when $$\theta=0$$, it gives a strong error term of size $$O_{\epsilon}(Q^{7/4+\epsilon})$$. Furthermore, our result provides evidence for the CFKRS conjectures for large twists and large vertical shifts. We also obtain corresponding results for $$\mathrm{PGL}_{2}(\mathbb{A}_{\mathbb{Q}})$$, which are fully unconditional, quantitatively stronger, and also appear to be new. This work develops a refined, flexible, and uniform version of the Asymptotic Large Sieve for $$L$$-functions that does not require any unproven progress toward the Generalized Ramanujan Conjecture. The arithmetic of $$\Pi_{0}$$ plays a crucial and delicate role in our argument. This work also makes extensive use of Mathematica to handle various elaborate Hecke algebra computations. Our mean value theorem is readily applicable to many other problems in analytic number theory.
The proportion 1/9, at first glance, seems modest. But it was obtained with the simplest mollifier from Levinson's original 1974 work and is the rounded value of 0.1147. In other words, even a coarse sieve captures more than a tenth of the gold flakes! And for PGL(2), the proportion 1/3 is exactly the same as for the Riemann zeta function — a harmonious echo across ranks.
Key numbers
- proportion of zeros on the critical line for PGL(3): ≥1/9 (~11.1%)
- proportion for PGL(2): ≥1/3
- range of parameter T: Q^ε ≤ T ≤ Q^{1/3–ε}
- length of mollifier polynomial: θ < 1/2