Local square mean in the hyperbolic circle problem and sums of Salié sums

This paper improves the exponent in the local $L^2$-norm estimate for the error term of the hyperbolic circle problem for the modular group, conditionally on a twisted Linnik-Selberg-type conjecture regarding sums of Salié sums.

The Gist

Imagine you are standing in the middle of an infinite, curved room (mathematicians call this the Hyperbolic Plane). The walls of this room are strange: as you move away from the center, the space expands faster than you can walk.

Now, imagine you drop a pebble at a specific spot, $z$. This pebble creates a ripple, but instead of spreading out in a circle like a stone in a pond, it creates a "hyperbolic circle."

The Big Question: The Circle Problem

In this paper, the author, András Biró, is trying to solve a counting puzzle. He asks: "If I draw a hyperbolic circle of a certain size around my pebble, how many 'ghost' copies of that pebble (created by the symmetries of the room) will fall inside that circle?"

Mathematicians have known for a long time that there is a formula to predict this number. However, the formula isn't perfect; it has a little bit of "noise" or error.

• The Old Rule: For a long time, the best anyone could say about this error was that it was roughly the size of the circle's radius raised to the power of $2/3$.
• The Goal: Biró wants to prove that this error is actually much smaller than we thought. He wants to show the error is closer to the radius raised to the power of something smaller than $9/14$ (which is roughly $0.64$, slightly better than $0.66$).

The "Local" Twist

Usually, mathematicians look at the error for the entire room. But Biró is looking at a local view. Imagine taking a magnifying glass to a small, specific corner of the room and measuring the error just there.

• The Analogy: Think of a weather map. The "pointwise" bound is like saying, "It might rain anywhere in the city." The "local square mean" is like saying, "If I stand in this specific park, the average amount of rain I get over a week is much less than the worst-case storm."
• Biró previously proved that in this specific local view, the error is smaller than the general rule. But he wants to prove it's even smaller.

The Secret Weapon: Salié Sums

To get this better result, Biró needs to count the "ghosts" very precisely. The ghosts are arranged in a pattern that looks like a complex dance. To understand the dance, he has to analyze a specific mathematical tool called Salié sums.

• The Metaphor: Imagine the ghosts are dancers. The Salié sum is a way of listening to the music they are dancing to. Sometimes, the music has a rhythm where the dancers cancel each other out (one steps left, one steps right), making the total movement zero.
• The Problem: In the past, mathematicians just assumed the dancers moved randomly, so they estimated the worst-case scenario (everyone stepping in the same direction). This gave a "safe" but loose estimate.
• The Breakthrough: Biró realizes that if the music has a specific, hidden rhythm, the dancers will cancel each other out much more effectively.

The Big Gamble: The Conjecture

Here is the catch. To prove that the dancers cancel out perfectly, Biró needs to assume a specific "conjecture" (a mathematical guess that hasn't been proven yet).

• The "Twisted Linnik-Selberg" Conjecture: This is a fancy name for a rule about how these Salié sums behave. It's like saying, "If the music has a certain beat, the dancers will definitely cancel out."
• The Result: IF this conjecture is true, THEN Biró can prove that the error in his counting problem is significantly smaller than anyone has ever proven before.

How He Did It (The Recipe)

1. Smoothing the Edges: Instead of counting ghosts in a sharp, jagged circle, he uses a "smooth" circle (like a soft blur) to make the math easier to handle.
2. Breaking it Down: He breaks the huge problem into tiny, manageable chunks.
3. The Explicit Formula: He uses a special "recipe" (an explicit formula) he found in a previous paper to describe exactly how the ghosts are arranged.
4. The Cancellation: He applies the "Conjecture" to show that in the most difficult parts of the calculation, the numbers cancel each other out (like positive and negative charges neutralizing).
5. The Final Count: Because of this cancellation, the total "noise" (error) is much lower than the old estimates.

Why Does This Matter?

In the world of number theory, these "error terms" are like the difference between a rough sketch and a high-definition photograph.

• The Old Way: We knew the picture was roughly right, but it was blurry.
• Biró's Way: He is sharpening the lens. Even though he has to assume a guess (the conjecture) to do it, he shows us exactly how much clearer the picture could be.

If this conjecture is eventually proven true (which many mathematicians believe it is), then Biró's result becomes a permanent, unshakeable fact: We can count these hyperbolic ghosts with much higher precision than we ever thought possible.

Summary in One Sentence

András Biró uses a clever mathematical guess about the rhythm of number patterns to prove that the "noise" in counting points on a curved surface is much quieter than we previously believed.

Technical Summary

1. Problem Statement

The paper addresses the Hyperbolic Circle Problem for the modular group $\Gamma = \text{PSL}_2(\mathbb{Z})$.

• Definition: Let $z, w$ be points in the upper half-plane $\mathbb{H}$. The problem concerns estimating the number of elements in the $\Gamma$-orbit of $z$ that lie within a hyperbolic circle of radius $R$ centered at $w$.
• Formulation: Using the variable $X = 2\cosh R$, the counting function is defined as:
  $$N(z, X) := |\{ \gamma \in \Gamma : 4u(\gamma z, z) + 2 \le X \}|$$
  where $u(z, w) = \frac{|z-w|^2}{4\text{Im}(z)\text{Im}(w)}$ is related to the hyperbolic distance.
• Known Bounds: The asymptotic behavior is $N(z, X) \sim 3X$. The classical pointwise error bound, established by Selberg (unpublished), is:
  $$|N(z, X) - 3X| = O_z(X^{2/3})$$
  This $X^{2/3}$ bound has not been improved for any Fuchsian group of finite volume in the pointwise sense.
• Local Square Mean: The paper focuses on the local $L^2$-norm of the error term over a compact set $\Omega \subset \mathbb{H}$. In a previous work [B1], the author proved that the root-mean-square error satisfies:
  $$\left( \int_{\Omega} (N(z, X) - 3X)^2 d\mu_z \right)^{1/2} = O_{\Omega, c}(X^c) \quad \text{for any } c > \frac{9}{14}$$
  The goal of this paper is to improve the exponent $c = 9/14$ conditionally.

2. Methodology

The proof strategy involves a sophisticated combination of analytic number theory, spectral theory, and arithmetic geometry.

A. Smoothing and Reduction

The author employs a smoothing technique using a parameter $d$ and a smooth function $\eta_0$ to define a smoothed counting function $N_{d,J}(z, X)$. The error term is decomposed into contributions from hyperbolic elements. The problem reduces to estimating the square integral of the hyperbolic contribution, denoted as $I_{d,X,J}(\tau)$.

• Previous work [B1] bounded this integral by $O(X^\epsilon d^{5/2} X^{-1/2})$ by using a trivial upper bound for the class numbers of pairs of quadratic forms.
• This paper aims to improve this bound by exploiting cancellation in the summation rather than just using upper bounds.

B. Arithmetic Reduction to Salié Sums

The core of the improvement lies in the arithmetic structure of the error term.

1. Class Numbers: The error term involves sums over the number of $\text{SL}_2(\mathbb{Z})$-equivalence classes of pairs of quadratic forms, denoted $h(d_1, d_2, t)$.
2. Explicit Formula: Instead of using the trivial upper bound for $h$, the author utilizes an explicit formula for these class numbers derived in a previous work [B2]. This formula expresses $h$ in terms of Salié sums and Jacobi symbols.
3. Critical Regions: The summation is split into "critical" and "non-critical" regions. In the critical regions (where the trivial bound is insufficient), the explicit formula reveals that the sum involves terms of the form:
   $$\sum \alpha_G(t_1, t_ f) \left( \frac{t_1^2-4}{D} \right) T(\dots)$$
   where $T$ represents Salié sums.

C. Analytic Techniques

To handle the sums involving Salié sums, the author applies:

• Poisson Summation: Applied to the variable $f$ (related to the discriminant) to transform the sum into a dual sum.
• Fourier Analysis: The zeroth term of the Poisson summation is shown to be small due to the specific structure of the smoothing functions and the alternating signs in the definition of $N_{d,J}$.
• Salié Sum Estimates: The non-zero terms in the Poisson dual sum involve Salié sums of the form $T(m, n; c)$. The magnitude of these sums determines the final error bound.

3. Key Contributions and Assumptions

The Main Conjecture

The improvement of the exponent relies on Conjecture 1, a "Twisted Linnik-Selberg-type conjecture" for sums of Salié sums.

• Statement: It posits that sums of Salié sums $T(m, n; c)$, twisted by smooth functions and multiplicative inverses, exhibit significant cancellation. Specifically, for a sum over moduli $c$ in a range $[C, 2C]$, the sum is bounded by $(mnC)^\epsilon$.
• Distinction: Unlike standard Linnik-Selberg conjectures for Kloosterman sums, this conjecture includes a "twist" involving a multiplicative inverse $L$ and allows parameters to grow slowly with $C$.

Main Result (Theorem 1.2)

Assuming Conjecture 1 is true, the author proves that the exponent in the local square mean estimate can be improved:
$$\left( \int_{\Omega} (N(z, X) - 3X)^2 d\mu_z \right)^{1/2} = O_{\Omega, c}(X^c) \quad \text{for some } c < \frac{9}{14}$$
The paper does not specify the exact value of the new $c$ but demonstrates that the method allows for a strict improvement over $9/14$.

4. Technical Details of the Proof

• Lemma 3.1: Improves a previous lemma regarding the sum of square-free parts of differences of squares, providing better control over the arithmetic weights.
• Lemma 3.12 & 3.13: Decompose the error integral into sums $\Sigma_1$ and $\Sigma_2$ involving Salié sums.
• Lemma 3.19: This is the crucial link. It shows that if the "Statement" (a specific instance of Conjecture 1 regarding sums of Salié sums with specific smooth weights) holds, then the error term $I_{d,X,J}(\tau)$ gains a saving of $X^{-\beta}$, leading to the improved exponent.
• Handling Coprimality: The proof involves intricate manipulations to handle coprimality conditions (using Möbius inversion) and the properties of the Hilbert symbol to ensure the Salié sums are well-defined and can be estimated.

5. Significance and Limitations

• Significance: This is the first result to improve the exponent $9/14$ for the local $L^2$-norm of the hyperbolic circle problem error term. It demonstrates that the "square mean" error is significantly smaller than the pointwise error ($X^{2/3}$), provided deep conjectures about exponential sums hold.
• Conditional Nature: The result is conditional on Conjecture 1. The author notes that partial results towards similar conjectures exist (e.g., for Kloosterman sums in [St]), suggesting that an unconditional proof might be possible via an "averaged" version of the conjecture.
• Scope: The result is currently specific to $\Gamma = \text{PSL}_2(\mathbb{Z})$. The author notes that for general finite index subgroups, the explicit formula for class numbers is not known, preventing a similar improvement. For general finite volume Fuchsian groups, the bound remains $X^{2/3}$.

Conclusion

András Biró's paper represents a significant advancement in the analytic theory of the hyperbolic circle problem. By shifting from upper bounds to explicit formulas and leveraging the cancellation properties of Salié sums (under a plausible conjecture), the author successfully lowers the exponent for the local $L^2$-error from $9/14$ to a strictly smaller value. This work bridges the gap between the classical pointwise bounds and the conjectured optimal bounds, highlighting the deep connection between hyperbolic geometry and the distribution of Salié sums.