🔢 mathematics

Sums of Kloosterman sums formed with modular symbols

This paper investigates sums of Kloosterman sums formed with modular symbols by employing Tauberian methods to establish estimates for related Ramanujan sums, constructing an analogous zeta function to prove cancellation results, and providing numerical evidence for their independence from classical Kloosterman sums while formulating an analogue of Linnik's conjecture.

Original authors: Nikolaos Diamantis, Solomon Friedberg, Fredrik Strömberg

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikolaos Diamantis, Solomon Friedberg, Fredrik Strömberg

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 world of numbers as a giant, bustling city where every building has a secret code. Some of these codes are called Kloosterman sums. Think of them as the city's daily traffic report: they count how many cars (numbers) pass through a specific intersection (a modulus $c$ ) while following a very strict, winding route. Usually, these traffic reports are chaotic, but mathematicians have long suspected that if you add up enough of them, the chaos cancels itself out, leaving a perfectly smooth, quiet street. This is the famous "Linnik-Selberg cancellation" idea.

Now, imagine a new kind of traffic report. Instead of just counting cars, this report is "twisted" by a modular symbol. If a standard traffic report is a simple headcount, a modular symbol is like asking every car to sing a specific note as it passes. These notes aren't just random; they come from a special, hidden song (a "cusp form") that the city itself is humming. The authors of this paper, Nikolaos Diamantis, Solomon Friedberg, and Fredrik Strömberg, decided to investigate what happens when you add up these "singing" traffic reports.

The Great Cancellation Mystery

The main goal of the paper is to see if these "singing" reports also cancel out. In the standard world, if you add up the traffic reports for all intersections up to a certain size, the total should be tiny—so tiny it's almost zero (mathematically, it grows slower than any power of the size). The authors wanted to know: Does the "singing" version do the same thing?

To answer this, they built a mathematical machine called a twisted Selberg zeta function. You can think of this machine as a giant radio tuner. If you tune it to the right frequency, it picks up the "noise" of the traffic reports. If the reports cancel out perfectly, the radio should be silent. If there are "ghosts" in the system—mathematicians call them exceptional eigenvalues—the radio would pick up a loud, persistent hum.

What They Found (and What They Didn't)

The authors didn't just guess; they built a rigorous mathematical framework to prove how this machine behaves. They showed that the machine can be tuned to hear frequencies up to a certain point (specifically, where the real part of the frequency is greater than $1/2$ ).

Here is the big reveal: The data suggests that the "singing" reports do indeed cancel out.

The authors explicitly state they cannot prove this in full for every case, so they relied on numerical consistency. When they added up the twisted sums in their simulations, the result was surprisingly small. In fact, their data suggests that the sum grows so slowly that it's essentially negligible, just like the standard traffic reports. This is a huge deal because, in the world of these equations, if the sums cancel out, it implies that those "ghosts" (the exceptional eigenvalues) do not exist.

The paper explicitly does not prove that these sums are uncorrelated with the standard, non-singing sums. Instead, they present data suggesting that the "singing" reports and the "silent" reports are completely uncorrelated. It's as if the cars singing a song have no idea what the other cars are doing; they appear to be two separate, independent universes. Based on this data, the authors formulate a conjecture that they are indeed uncorrelated.

How Sure Are They?

The authors are very careful about their confidence. They didn't just prove this with a single, unbreakable theorem for every possible case (that would be the "holy grail" of the field). Instead, they did two things:

Mathematical Proof: They proved that if these sums didn't cancel out, it would mean there are these "ghost" eigenvalues. They established a precise formula (Theorem 6.7) that shows the sum is made of a main part (which vanishes if there are no ghosts) and a tiny error term.
Numerical Experiments: Since they couldn't prove the "ghosts" don't exist for every city, they ran massive computer simulations. They checked thousands of different "cities" (levels $N$ ) and millions of intersections (up to $c \le 10^7$ in some cases).

The results of these simulations are the key. The data suggests strongly that the sums cancel out perfectly. When they looked at the "noise" in their simulations, it behaved exactly as if the ghosts were absent. They even tried to fit a "ghost" into their data, but the ghost refused to show up.

However, the paper admits a small caveat: it's theoretically possible that the ghosts exist but are so quiet (their "amplitude" is zero) that the math can't hear them. But the authors argue this is highly unlikely, like a ghost that is so shy it never speaks.

The "Ghost" Problem

The "exceptional eigenvalues" are the paper's main villain. In the world of modular forms, there's a famous conjecture (the Selberg eigenvalue conjecture) that says these ghosts shouldn't exist at all. If they did, they would break the perfect cancellation of the sums.

The authors' work provides a new way to test this conjecture. By showing that the "singing" sums cancel out in their simulations, they are essentially saying, "We looked for the ghosts, and we didn't find them." They formulate a conjecture (Conjecture 7.3) stating that for any "ghost" that might exist, there is a specific "song" (a modular symbol) that would make it scream loud enough to be heard. Since their simulations show no screaming, the ghosts are likely non-existent.

The Bottom Line

In simple terms, this paper is a detective story. The detectives (the authors) investigated a new type of number pattern (twisted Kloosterman sums) to see if it behaves like the old, well-known patterns. They built a special radio (the twisted zeta function) to listen for "ghosts."

The Finding: The radio is silent in the data. The sums appear to cancel out beautifully.
The Implication: This silence strongly suggests that the "ghosts" (exceptional eigenvalues) don't exist, supporting a major mathematical conjecture.
The Twist: The "singing" numbers appear to be completely unrelated to the "silent" numbers, though the authors present this as a conjecture based on data rather than a proven fact.

The authors haven't solved the entire mystery with a single, unshakeable proof for every possible case, but their combination of rigorous math and massive computer simulations provides the strongest evidence yet that the "ghosts" are just that—ghosts. The city is quiet, the traffic is canceling out, and the song is pure.

Technical Summary: Sums of Kloosterman Sums Formed with Modular Symbols

Problem Statement
The paper investigates the analytic properties and asymptotic behavior of sums involving "twisted" Kloosterman sums, denoted $S^*(m, n; c)$ . These sums are constructed by weighting classical Kloosterman sums with modular symbols $\langle f, \gamma \rangle$ associated with a holomorphic cusp form $f$ of weight 2 on a congruence subgroup $\Gamma_0(N)$ . The classical case, $S(m, n; c)$ , is central to the eigenvalue $1/4$ conjecture (Selberg's conjecture), where cancellation in the sums $\sum S(m, n; c)/c$ implies the non-existence of exceptional eigenvalues for the Laplacian. The authors aim to determine whether analogous cancellation occurs for the twisted sums $S^*(m, n; c)$ and to understand the relationship between the twisted Selberg zeta function $Z^*_{m,n}(s)$ and the exceptional spectrum.

Methodology
The authors employ a combination of analytic number theory techniques, spectral theory, and numerical computation:

Twisted Eisenstein Series: The study is anchored in the non-holomorphic Eisenstein series $E^*_a(z, s)$ twisted by modular symbols. The Fourier coefficients of this series involve the twisted Kloosterman sums. The authors analyze the meromorphic continuation and growth of these coefficients.
Tauberian Methods: To estimate sums of twisted Ramanujan sums ( $S^*(n, 0; c)$ ), the authors utilize Tauberian theorems. A significant technical challenge addressed is the presence of infinitely many poles of $E^*_a(z, s)$ on the critical line $\text{Re}(s) = 1/2$ , in addition to potential exceptional poles. The authors carefully shift the line of integration, managing these poles one by one using precise bounds on the Eisenstein series and its derivatives.
Sobolev Spaces and Perturbation Theory: For the case where $m, n \neq 0$ , the authors analyze the twisted Selberg zeta function $Z^*_{m,n}(s)$ . They adapt the perturbation theory approach of Phillips-Sarnak and Petridis, utilizing pseudo-Laplacians and Sobolev inequalities. By introducing a family of unitary characters $\chi_\epsilon$ derived from the modular symbols, they express $Z^*_{m,n}(s)$ as a derivative of a standard twisted zeta function with respect to the perturbation parameter $\epsilon$ at $\epsilon=0$ . This allows them to establish meromorphic continuation and growth bounds for $Z^*_{m,n}(s)$ in the region $\text{Re}(s) > 1/2$ .
Numerical Investigations: The authors implement an algorithm based on the Eichler integral and Fast Fourier Transforms (FFT) to compute modular symbols and twisted Kloosterman sums for large moduli ( $c$ up to $10^7$ ). They perform statistical analyses on the partial sums and the correlation between classical and twisted sums.

Key Contributions and Results

Estimates for Twisted Ramanujan Sums (Theorem 5.1): The authors derive an asymptotic formula for the weighted sum of twisted Ramanujan sums $S^*(n, 0; c)$ . The main term involves the Fourier coefficients of the cusp form $f$ and contributions from exceptional eigenvalues (if they exist). The error term is shown to be $O(X^{1/2+\epsilon})$ , provided the sum is weighted appropriately.
Meromorphic Continuation of Twisted Zeta Function (Theorem 6.6): The paper establishes that the twisted Selberg zeta function $Z^*_{m,n}(s)$ (for $m, n \neq 0$ ) admits a meromorphic continuation to $\text{Re}(s) > 1/2$ . The only potential poles in the strip $(1/2, 1)$ correspond to exceptional eigenvalues of the Laplacian. Furthermore, a growth bound is established: $Z^*_{m,n}(s) \ll |t|^{7/2+\epsilon}(\sigma - 1/2)^{-3}$ .
Cancellation in Twisted Kloosterman Sums (Theorem 6.7): Using the analytic properties of $Z^*_{m,n}(s)$ , the authors prove a cancellation result for the partial sums of twisted Kloosterman sums. Specifically, $\sum_{c<x} S^*(m, n; c)/c$ is shown to be a sum of terms related to exceptional eigenvalues plus an error term of size $O(x^{7\beta/9+\epsilon})$ , where $\beta$ is the exponent of growth of the sums.
Numerical Evidence and Conjectures:
- Absence of Exceptional Eigenvalues: Numerical data for levels $N \le 50$ is consistent with the absence of exceptional eigenvalues. The partial sums of twisted Ramanujan sums exhibit cancellation consistent with the $O(x^{1/2+\epsilon})$ bound, and no main terms corresponding to exceptional eigenvalues are detected.
- Lack of Correlation (Conjecture 7.2): The authors present numerical evidence suggesting that the classical Kloosterman sums $S(m, n; c)$ and the twisted sums $S^*(m, n; c)$ are uncorrelated. The signs of the sums appear independent, and the Pearson correlation coefficient tends to zero.
- Linnik-type Conjecture (Conjecture 7.1): The authors formulate an analogue of Linnik's conjecture for the twisted sums, asserting that $\sum_{c \le x} S^*(m, n; c)/c \ll x^\epsilon$ .

Significance
The paper extends the classical theory of Kloosterman sums and the Selberg zeta function to the setting of modular symbols. It highlights that while the twisted sums share structural similarities with the classical case (e.g., relation to the exceptional spectrum), they introduce new complexities, such as the lack of unitarity in the weights (modular symbols are normally distributed, not of modulus 1) and the presence of infinitely many poles on the critical line.

The authors demonstrate that the cancellation in twisted sums is a viable pathway to investigating the eigenvalue $1/4$ conjecture, though a full proof remains elusive due to the complexity of the residues associated with potential exceptional eigenvalues. The numerical results provide strong empirical support for the non-existence of exceptional eigenvalues and suggest a fundamental independence between classical and twisted arithmetic objects. The work bridges analytic methods (Tauberian theorems, Sobolev spaces) with explicit numerical verification, offering a refined perspective on the spectral theory of automorphic forms.

The Great Cancellation Mystery

What They Found (and What They Didn't)

How Sure Are They?

The "Ghost" Problem

The Bottom Line

More like this