Explicit zero-free regions for automorphic $L$-functions

Abstract: Let $L(s,f)$ be the $L$-function associated with a newform $f$ of even weight $k$, squarefree level $N$ and trivial nebentypus. In this paper, we establish a new explicit zero-free region for $L(s,f)$. More precisely, we prove that $L(s,f)$ does not vanish in the region $\Re(s)\geq 1-\frac{1}{C\log(kN\max(1,|\Im(s)|))}$ with $C=16.7053$ if $|\Im(s)|\geq 1$ or $|\Im(s)|\leq \frac{0.30992}{\log(kN)}$ and $C=16.9309$ if $\frac{0.30992}{\log(kN)}<|\Im(s)|\leq 1$. This improves a result of Hoey et al. where $445.994$ was shown to be an admissible value for $C$.

• The paper establishes explicit zero-free regions for automorphic L-functions by adapting Stečkin’s differencing method and optimizing trigonometric polynomials.
• It provides sharp numerical bounds for the constant C, reducing the previous bound from 445.994 to approximately 16.7053–16.9309, which improves error estimates in related theorems.
• The work offers a clear analytic framework, including pseudocode for verification, that supports practical applications and potential extensions to higher rank L-functions.

Explicit Zero-Free Regions for Automorphic $L$-Functions

Introduction and Context

This paper establishes new explicit zero-free regions for automorphic %%%%1%%%%-functions associated to Hecke newforms of even weight $k$, squarefree level $N$, and trivial nebentypus. The main result is a significant improvement in the explicit constant $C$ governing the zero-free region for $L(s,f)$, where $f$ is a newform. Specifically, the authors prove that $L(s,f)$ does not vanish in the region

$$\operatorname{Re}(s) \geq 1 - \frac{1}{C \log(kN \max(1, | \operatorname{Im}(s) |))}$$

with $C = 16.7053$ for $|\operatorname{Im}(s)| \geq 1$ or $|\operatorname{Im}(s)| \leq 0.30992 / \log(kN)$, and $C = 16.9309$ for $0.30992 / \log(kN) < |\operatorname{Im}(s)| \leq 1$. This sharpens the previous best-known constant $C < 445.994$ due to Hoey et al.

The explicit determination of zero-free regions for $L$-functions is central to analytic number theory, with direct implications for effective error terms in prime number theorems, Sato-Tate distributions, and bounds for sums involving Hecke eigenvalues. The improvement in the constant $C$ enhances the precision of such applications.

Analytic Framework and Techniques

The proof leverages and adapts the classical Stečkin differencing technique, previously used for the Riemann zeta function and Dirichlet $L$-functions, to the setting of $\mathrm{GL}(2)$ automorphic $L$-functions. The approach involves:

• Trigonometric Polynomial Positivity: Construction of a quartic trigonometric polynomial $P_4(\theta; n)$ whose nonnegativity yields inequalities involving Stečkin differences of logarithmic derivatives of auxiliary $L$-functions.
• Symmetric Power $L$-Functions: Detailed analysis of $L(s, \mathrm{Sym}^m f)$ for $m = 0, 1, 2, 3, 4$, including their functional equations and explicit formulae, is required to control the auxiliary terms in the positivity argument.
• Explicit Bounds: Careful bounding of the Stečkin differences for the logarithmic derivatives of the relevant $L$-functions, including the use of explicit digamma function estimates and control of ramified Euler factors.

The authors optimize the coefficients of the trigonometric polynomial to maximize the width of the zero-free region, resulting in the improved constants.

Main Theorem and Numerical Results

The main theorem states:

Let $N$ be squarefree, $k$ even, and $f$ a newform of weight $k$ and level $N$. If $\beta + it$ is a zero of $L(s,f)$ with $\beta > 1/2$, then

$$\beta \leq \begin{cases} 1 - \frac{1}{16.7053 \log(kN|t|)} & \text{if } |t| \geq 1 \ 1 - \frac{1}{16.9309 \log(kN)} & \text{if } 0.30992/\log(kN) < |t| < 1 \ 1 - \frac{1}{16.7053 \log(kN)} & \text{if } |t| \leq 0.30992/\log(kN) \end{cases}$$

This result is unconditional and applies to all newforms of the specified type. The improvement in the constant $C$ is substantial compared to previous work, and the bounds are uniform in $k$ and $N$.

Implementation and Application

Algorithmic Steps

To apply these results in computational or analytic contexts, the following steps are recommended:

1. Parameter Computation: For a given newform $f$ of weight $k$ and level $N$, compute $kN$ and determine the relevant $|t|$ regime.
2. Zero-Free Region Verification: For any $s = \sigma + it$ with $$\operatorname{Re}(s) \geq 1 - 1/(C \log(kN \max(1, |t|)))$$, assert $L(s,f) \neq 0$.
3. Error Term Estimation: In applications such as explicit prime number theorems or Sato-Tate error terms, use the improved constant $C$ to sharpen error bounds.

Example: Effective Sato-Tate Error Terms

Suppose one wishes to bound the error in the Sato-Tate distribution for Hecke eigenvalues $(p)$ of a fixed non-CM newform $f$. The improved zero-free region allows for tighter control of the error term in sums of the form

$\sum_{p \leq x} (p^m)$

by ensuring that the relevant $L$-functions $L(s, \mathrm{Sym}^m f)$ are zero-free in a wider region, thus reducing the contribution from zeros near $s=1$.

Pseudocode for Zero-Free Region Check

def is_in_zero_free_region(sigma, t, k, N): log_kN = math.log(k * N) abs_t = abs(t) if abs_t >= 1 or abs_t <= 0.30992 / log_kN: C = 16.7053 else: C = 16.9309 bound = 1 - 1 / (C * math.log(k * N * max(1, abs_t))) return sigma >= bound

Computational Considerations

• The bounds are explicit and require only elementary computations (logarithms, basic arithmetic).
• For large $k$ and $N$, the region remains effective due to the logarithmic dependence.
• The method is robust for both large and small $|t|$, with careful treatment of the transition region.

Theoretical Implications

The improvement in explicit zero-free regions for automorphic $L$-functions has several theoretical consequences:

• Sharper Explicit Results: Directly improves explicit versions of the prime number theorem for arithmetic progressions and automorphic forms.
• Nonexistence of Siegel Zeros: The result is unconditional and does not rely on the possible existence of exceptional zeros, which are known to be absent for these $L$-functions.
• Framework Extension: The adaptation of Stečkin's method to $\mathrm{GL}(2)$ automorphic $L$-functions opens the possibility for further improvements and generalizations to higher rank $L$-functions.

Future Directions

Potential avenues for further research include:

• Extension to Non-Squarefree Levels: The current method relies on the squarefree condition for $N$; extending to arbitrary levels would require new techniques for controlling the conductor and ramified Euler factors.
• Higher Symmetric Powers: The framework may be adapted to $L(s, \mathrm{Sym}^m f)$ for $m > 4$, with implications for moments and distributions of Hecke eigenvalues.
• Automorphic $L$-Functions on $\mathrm{GL}(n)$: Generalization to higher rank groups could yield explicit zero-free regions for a broader class of $L$-functions.

Conclusion

This work provides a substantial improvement in explicit zero-free regions for automorphic $L$-functions associated to newforms of even weight and squarefree level. The adaptation of Stečkin's differencing technique, combined with careful analytic estimates and optimization of trigonometric polynomial coefficients, yields explicit constants that are significantly sharper than previous results. These advances have direct impact on effective results in analytic number theory and open new directions for further refinement and generalization.