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Power sums and Siegel-type zero-free regions for L-functions

This paper introduces a novel approach using lower bounds for power sums to establish ineffective Siegel-type zero-free regions for standard and Rankin-Selberg $L$ -functions associated with unitary cuspidal automorphic representations, yielding significant improvements to prime number theorems and generalizations of the Brauer-Siegel theorem.

Original authors: Jesse Thorner

Published 2026-08-13

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CC BY 4.0

Original authors: Jesse Thorner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the number line as a vast, infinite highway stretching out in both directions. For centuries, mathematicians have been obsessed with a specific stretch of this road: the "critical line" where the famous Riemann Hypothesis claims all the traffic stops. But before we get to that, we need to understand the cars themselves. In the world of numbers, there are special functions called L-functions. Think of these not as cars, but as complex musical scores or radio signals that encode the hidden patterns of prime numbers—the building blocks of arithmetic. Just as a radio signal can have static or silence, these mathematical functions can hit zero. The "zeros" of these functions are the silent spots where the signal disappears.

The location of these zeros is crucial. If a zero appears in the wrong place (specifically, too close to the number 1 on the real number line), it throws the entire system of prime number distribution into chaos. It's like finding a ghost in the machine that messes up the timing of every clock in the city. For a long time, mathematicians could prove these "ghosts" didn't exist in certain safe zones, but the safety zone was narrow and depended heavily on how complicated the signal was. The bigger the signal (measured by something called an "analytic conductor"), the tighter the safety zone became, making it harder to predict the behavior of the primes. The big question has been: Can we push the safety zone wider, regardless of how complex the signal gets, without having to assume unproven theories?

This paper by Jesse Thorner tackles exactly that problem. It introduces a brand-new method to prove that these dangerous zeros cannot hide too close to the number 1, even for the most complicated signals. The author doesn't just improve the existing safety zone; he creates a much wider, more uniform one that works for a vast family of these L-functions, including the standard ones and the more complex "Rankin–Selberg" ones.

Here is the core of the discovery: The author proves that for any tiny margin of error (let's call it $\epsilon$ ), there is a guaranteed "no-go zone" for these zeros. Specifically, if you look at a point on the number line that is $1 - c(C_\pi(|t| + 3))^{-\epsilon}$ (where $C_\pi$ is the complexity of the signal and $t$ is a time-like variable), the L-function will definitely not be zero there. In fact, the value of the function at that point is guaranteed to be at least $c(C_\pi(|t| + 3))^{-\epsilon}$ . This is a massive improvement over previous results, which required the "no-go zone" to shrink much faster as the signal got more complex.

The paper achieves this by rejecting the old, standard way of solving the problem. Previously, mathematicians tried to build a "safety net" by multiplying several L-functions together to create a new function with only positive numbers (like stacking weights to ensure a scale never tips). This worked well in simple cases but hit a wall for the most complex, self-dual signals. Thorner shows that this old method is insufficient for the general case. Instead, he uses a clever two-step dance involving "power sums." Imagine you have a collection of numbers, and you want to know if they are all small or if at least one is big. The paper uses two different mathematical tools (inspired by work from the 1940s) to check this. One tool checks the total size of the numbers, and the other checks their real parts. By combining these, the author can detect if a "ghost" zero is trying to sneak in. If a ghost is there, the math forces a contradiction, proving the ghost cannot exist in that specific region.

The results are rigorous and unconditional, meaning they don't rely on guessing that other unproven theories are true. The paper establishes that these zero-free regions exist for all $\epsilon > 0$ , with constants that depend on the number field and the dimensions of the representations, though the constants themselves are "ineffective" (meaning we know they exist and are positive, but the paper doesn't give a specific recipe to calculate their exact numerical value).

The implications of this are immediate and practical for number theory. The paper shows that this new, wider safety zone leads to better versions of the "Prime Number Theorem" for these complex signals. Essentially, it allows mathematicians to predict how prime numbers are distributed with much greater accuracy and over a wider range of numbers than ever before. It also leads to new generalizations of the "Brauer–Siegel theorem," which relates the size of a number field to its class number (a measure of how far its arithmetic is from being perfect). In short, Thorner has built a stronger, more flexible fence around the most dangerous part of the number line, giving us a clearer view of the fundamental structure of numbers without needing to lean on unproven assumptions.

Technical Summary: Power Sums and Siegel-Type Zero-Free Regions for L-Functions

Problem Statement
The paper addresses the problem of establishing zero-free regions near the line $\text{Re}(s) = 1$ for standard $L$ -functions $L(s, \pi)$ associated with unitary cuspidal automorphic representations $\pi$ of $\text{GL}(n)$ over a number field $F$ , and for Rankin–Selberg $L$ -functions $L(s, \pi \times \pi')$ .

While the Generalized Riemann Hypothesis (GRH) predicts non-vanishing for $\text{Re}(s) > 1/2$ , unconditional results are limited. Previous work established "standard" zero-free regions of the form $\sigma \geq 1 - c(\log C_\pi)^{-1}$ for non-self-dual cases or under unproven modularity hypotheses (e.g., Langlands functoriality). For self-dual cases or general Rankin–Selberg products, unconditional results were restricted to regions where the lower bound on the $L$ -function value and the width of the zero-free region decayed polynomially with the analytic conductor $C_\pi$ (e.g., $\sigma \geq 1 - c C_\pi^{-\delta}$ ). The goal is to improve these bounds to a "Siegel-type" form, where the decay is arbitrarily small (i.e., $C_\pi^{-\varepsilon}$ ), without relying on unproven hypotheses regarding the modularity of Rankin–Selberg $L$ -functions.

Methodology
The author introduces a new approach that diverges from the classical strategy of constructing auxiliary Dirichlet series with non-negative coefficients to detect poles. Instead, the proof relies on two distinct lower bounds for power sums, inspired by the work of Sós and Turán, applied in complementary roles:

Turán's Second Main Theorem (Proposition 2.1): Used to establish a lower bound for the magnitude of a sum of powers of complex numbers. This is applied to the zeros of an auxiliary function $F(z) = L(z, \pi_\epsilon \times \pi')L(z, \pi_\epsilon \times \tilde{\pi})$ to show that if an exceptional zero $\beta_\epsilon$ exists, a specific derivative of the logarithmic derivative, $G_k(1+\epsilon)$ , must be large.
Real Part Lower Bound (Proposition 2.2): Used to bound the real part of power sums. This is applied via Moreno's version of the Deuring–Heilbronn zero repulsion phenomenon. It allows the author to bound the same quantity $G_k(1+\epsilon)$ from above by a factor involving the distance of a potential second zero $\beta_1$ from 1 (specifically $1-\beta_1$ ).

The strategy proceeds by contradiction:

Assume the existence of a real zero $\beta_1$ close to 1.
Construct a function $G_k(z)$ related to the $k$ -th derivative of the logarithmic derivative of $F(z)$ .
Use Proposition 2.1 to show that the presence of an initial "exceptional" zero $\beta_\epsilon$ forces $|G_k(1+\epsilon)|$ to be large.
Use Proposition 2.2 (via zero repulsion) to show that if $\beta_1$ exists, $|G_k(1+\epsilon)|$ must be small (bounded by a multiple of $1-\beta_1$ ).
By comparing these estimates and utilizing bounds on sums over prime ideals (Proposition 5.1), the author derives a contradiction unless the zero-free region is sufficiently wide.

Key Contributions and Results
The paper establishes the following unconditional results, where constants $c$ are ineffective (dependent on $n, F, \epsilon$ , and potentially fixed representations):

Standard $L$ -functions (Theorem 1.1): For all $\epsilon > 0$ , there exists an ineffective constant $c_4 > 0$ such that if $\sigma \geq 1 - c_4 C_\pi^{-\epsilon}$ , then $|L(\sigma, \pi)| \geq c_4 C_\pi^{-\epsilon}$ . This holds for all $\pi \in \mathcal{F}_n$ , including self-dual cases where previous unconditional results were weaker.
Rankin–Selberg $L$ -functions (Theorem 1.2): Fixing $\pi'$ , for all $\epsilon > 0$ , there exists an ineffective constant $c_9 > 0$ such that if $\sigma \geq 1 - c_9 C_\pi^{-\epsilon}$ , then $|L(\sigma, \pi \times \pi')| \geq c_9 C_\pi^{-\epsilon}$ . This result is uniform in $\pi$ and subsumes previous results on character twists.
Prime Number Theorems (Theorems 3.1 & 3.2): The zero-free regions imply improved error terms in the prime number theorems for these $L$ -functions. Specifically, for self-dual cases, the error term $E(x; \pi)$ satisfies $E(x; \pi) \leq c_{14} e^{-\sqrt{\log x}}$ for $\log x \geq c_{13} C_\pi^\epsilon$ , matching the behavior of real Dirichlet characters.
Brauer–Siegel Type Theorems (Theorem 3.3): The results generalize the Brauer–Siegel theorem, showing that $\lim_{j \to \infty} \frac{\log |L(1, \pi_j)|}{\log C_{\pi_j}} = 0$ for sequences of representations with growing conductors.

Significance and Claims
The paper claims to provide the first unconditional, uniform Siegel-type lower bounds for standard and Rankin–Selberg $L$ -functions without recourse to unproven hypotheses on the modularity of Rankin–Selberg $L$ -functions (such as the existence of automorphic tensor products).

The author emphasizes that the method is genuinely new. Previous approaches based on the non-negativity of Dirichlet coefficients of auxiliary series (as used by Hoffstein, Ramakrishnan, and others) fail to produce these results for arbitrary self-dual $\pi$ or general Rankin–Selberg products without assuming modularity. The paper notes that while Goldfeld's proof of Siegel's theorem for Dirichlet $L$ -functions was a direct generalization of earlier work, the current problem for $\text{GL}(n)$ requires the specific combination of the two power sum inequalities (Propositions 2.1 and 2.2) to overcome the lack of modularity.

The results are described as "ineffective" due to the nature of the zero-repulsion arguments, but they represent a significant improvement in the dependence on the analytic conductor $C_\pi$ , moving from polynomial decay ( $C_\pi^{-\delta}$ ) to arbitrarily small polynomial decay ( $C_\pi^{-\epsilon}$ ). The paper explicitly states that the approach is necessary for the general case and that proofs of the limitations of previous methods will appear in forthcoming work by Nawapan Wattanawanichkul.

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