Petrow–Young Subconvexity Bound

• The paper establishes a Weyl-type subconvexity bound for Dirichlet L-functions with an exponent of 1/6, surpassing the classical convexity barrier.
• It utilizes innovative analytic and spectral methods, including the approximate functional equation, delta method, and Kuznetsov formula.
• The result leads to improved distribution exponents for divisor functions like d3(n) and d4(n), advancing equidistribution in arithmetic progressions.

The Petrow–Young subconvexity bound is a breakthrough in analytic number theory that establishes a Weyl-type subconvex estimate for Dirichlet $L$-functions of primitive characters, with profound implications for the distribution of divisor functions in arithmetic progressions. The method enables significant improvements on classical “square root barrier” results, enabling new exponents of distribution for $d_3(n)$ and $d_4(n)$ and advancing the study of mean values of arithmetic functions in residue classes.

1. Formal Statement of the Petrow–Young Subconvexity Bound

Let $\chi$ be a primitive Dirichlet character modulo a cube-free integer $q$, and let $L(s,\chi) = \sum_{n \ge 1} \chi(n)n^{-s}$ for $s = \tfrac12 + it$. Petrow and Young prove the following Weyl-type subconvexity bound: $$L\!\Bigl(\tfrac12+it,\chi\Bigr)\;\ll_{\varepsilon}\;(q(1+|t|))^{\tfrac16+\varepsilon}$$ for every $\varepsilon>0$ (Parry, 2024, Aydemir et al., 18 Jan 2026).

Equivalently, the "exponent" $\theta = 1/6$ may be taken in the general form

$$L\!\left(\tfrac12+it,\chi\right) \ll_\varepsilon (q(1+|t|))^{\theta+\varepsilon}.$$

This subconvex estimate sits strictly below the classical convexity bound with exponent $1/4$, and permits a power-saving improvement crucial to arithmetic equidistribution problems.

2. Outline of the Proof and Main Techniques

The proof proceeds by several innovative analytic and spectral techniques:

• Approximate Functional Equation: The $L$-function is expressed in terms of two dual sums, each of length about $(qT)^{1/2}$.
• Circle/Delta Method: Detection of $n = m$ via a $\delta$-symbol or spectral trace formula on $\mathrm{PGL}_2$.
• Reciprocity and Kuznetsov Formula: The sum is analyzed using Kloosterman sums, whose structure is accessed via the Kuznetsov trace formula on $\Gamma_0(q)$.
• Spectral Reciprocity: Introduction of a duality between $\mathrm{GL}_1$-twists and sums over $\mathrm{GL}_2$ Fourier coefficients.
• Amplification and Stationary Phase Analysis: An amplification method (inspired by Duke–Friedlander–Iwaniec) is used to harvest the central values, and oscillatory integrals are bounded by the stationary phase method.

The consequence is a “hybrid” subconvex bound, yielding $\theta=1/6$ in both $q$-aspect and $t$-aspect, and thus establishing the stated Weyl-type estimate (Aydemir et al., 18 Jan 2026).

3. Applications to Exponent of Distribution Beyond the Square-Root Barrier

The Petrow–Young bound serves as the analytic foundation for achieving exponents of distribution for divisor functions $d_k(n)$ beyond the $1/2$ barrier in arithmetic progressions:

• Four-fold Divisor Function ($d_4(n)$): Parry demonstrates that $d_4(n)$ possesses exponent of distribution $4/7$ on average over residue classes, breaking the $q \asymp x^{1/2}$ limit for the first time. The main steps involve smoothing, application of an Ivić multidimensional Voronoi formula, and separation into diagonal/off-diagonal terms. The off-diagonal regime is controlled by employing the Petrow–Young subconvexity in bounding bilinear forms involving $L$-functions, culminating in the second-moment estimate

$$\sum_{a\bmod q}\bigl|A(a/q)\bigr|^2 \ll_\varepsilon x^{3/2+\varepsilon}q^{7/8}$$

for prime $q \le x^{4/7 - \varepsilon}$. This leads to individual error terms $E(x;q,a)$ of size $\ll x^{1-\delta}$, confirming equidistribution of $d_4(n)$ up to $q\le x^{4/7}$ (Parry, 2024).

• Ternary Divisor Function ($d_3(n)$): The techniques transfer to $d_3(n)$, and by similar analysis the exponent of distribution is improved from $2/3$ to $8/11$ after averaging over residue classes modulo a prime $q$. In this context, the Petrow–Young bound is invoked at several points:
  • Bounding bilinear sums involving $d_3(n)$ via Dirichlet character orthogonality,
  • Employing Perron's formula to express sums over $d_3(n)\chi(n)$ in terms of $L(s,\chi)^3$,
  • Applying hybrid and averaged Weyl-type bounds for $L$-functions within $t$ and $q$-aspects.
  • The resulting mean square error is

$$\sum_{a=1}^q |E_x(q,a)|^2 \ll_\varepsilon x^{3/2+\varepsilon} q^{11/16}$$

for $q \le x^{8/11}$, thus confirming substantial progress over prior results (Aydemir et al., 18 Jan 2026).

4. Technical Preconditions and Functional Ranges

The subconvexity result and its applications are subject to several precise restrictions:

• The character $\chi$ must be primitive of conductor $q$, with $q$ required to be cube-free in Petrow–Young (prime $q$ is allowed).
• All bounds are uniform in $t \in \mathbb{R}$ and for all $\varepsilon > 0$.
• In distribution problems for $d_k(n)$, the modulus $q$ is required to satisfy $q \le x^{\theta}$, with $\theta=4/7$ for $d_4(n)$, $\theta=8/11$ for $d_3(n)$ under averaging, as set by the detailed optimization of second moment bounds.

5. Consequences for Divisor Functions and Related Problems

The principal consequences include:

• $d_4(n)$ Equidistribution: Establishes equidistribution in prime modulus arithmetic progressions beyond the conventional range, achieving power-saving error terms for $q\le x^{4/7}$ (Parry, 2024).
• $d_3(n)$ Improvements: Improves the exponent for equidistribution of $d_3(n)$ up to modulus $q\le x^{8/11}$ on average (Aydemir et al., 18 Jan 2026).
• Framework for Higher Divisor Functions: The analytic structure provided by Petrow–Young suggests that, for $k\geq5$, any suitable subconvex bound for the relevant $L$-functions could yield nontrivial distribution exponents for $d_k(n)$. This insight shows the generality and reach of the underlying analytic approach.

A table summarizing key exponents:

Function	Classical Exponent	With P-Y Bound	Modulus Range
$d_3(n)$	$2/3$	$8/11$	$q \le x^{8/11}$
$d_4(n)$	$1/2$	$4/7$	$q \le x^{4/7}$

6. Key Formulas and Estimate Summary

The following central formulas encapsulate the main analytic advances:

• Petrow–Young Weyl Bound:

$$L(\tfrac12 + it, \chi) \ll_\varepsilon (q(1+|t|))^{1/6 + \varepsilon}$$

• Fourth-Moment Bound (for $d_4(n)$):

$$\int_{-T}^{T} |L(\tfrac12 + it, \chi)|^4\,dt \ll qT (qT)^{1/3+\varepsilon}$$

• Second-Moment for Error Terms:

$$\sum_{a\bmod q} |E(x;q,a)|^2 \ll_\varepsilon x^{3/2+\varepsilon} q^{7/8}$$

for $d_4(n)$, and

$$\sum_{a=1}^q |E_x(q,a)|^2 \ll_\varepsilon x^{3/2+\varepsilon} q^{11/16}$$

for $d_3(n)$.

These expressions delineate the pathway from the subconvexity bound to explicit equidistribution results in arithmetic progressions.

7. Impact and Future Directions

The Petrow–Young subconvexity bound provides an essential analytic tool for advancing beyond natural convexity-barrier limits in the study of distribution of arithmetic functions. Its hybrid and uniformly sharp nature is indispensable for mean value theorems and divisor distribution problems. A plausible implication is that further enhancements in subconvex bounds for Dirichlet or related automorphic $L$-functions would directly enable stronger exponents for various arithmetic function distribution results, particularly for higher $d_k(n)$ and analogous settings. The method’s paradigm—melding spectral reciprocity, advanced trace formulae, and optimized moment estimates—serves as a blueprint for subsequent progress in analytic number theory and automorphic forms.