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A note on the invariants of the $L$ -functions

This paper clarifies the precise meaning of a previous statement regarding invariants of $L$ -functions by demonstrating that every such invariant admits a rational extension, thereby formally establishing that a complex-valued function of the functional equation's data is an invariant if and only if it remains stable under multiplication and factorial formulae.

Original authors: Jerzy Kaczorowski, Alberto Perelli

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jerzy Kaczorowski, Alberto Perelli

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 you are trying to identify a specific person in a crowded room. You have a list of their features: height, eye color, and a unique tattoo. But here's the catch: the person can change their clothes, wear a hat, or even put on a disguise. Sometimes, they might look like a completely different person to a casual observer, but you know it's the same person underneath.

In the world of advanced mathematics (specifically number theory), L-functions are like these people. They are complex mathematical objects that hold deep secrets about prime numbers. To understand an L-function, mathematicians look at its "functional equation," which is like a DNA strand describing how the function behaves.

This equation has a specific set of ingredients (data): a few numbers and a special pattern involving the Gamma function (a fancy version of the factorial, $5! = 5 \times 4 \times 3 \times 2 \times 1$ ).

The Problem: The "Disguise" Dilemma

The authors, Kaczorowski and Perelli, are dealing with a tricky situation. The same L-function can be written down in many different ways. It's like taking a photo of our person, then cropping it, zooming in, or adding a filter. The underlying person is the same, but the list of numbers (the "data") looks different.

Mathematicians want to find invariants. An invariant is a property that never changes, no matter how you dress up the equation. For example, the "degree" of the function is like the person's height; it stays the same whether they wear a hat or not.

In a previous paper, the authors made a bold, somewhat vague claim:

"A property is an invariant if and only if it stays the same when you apply two specific mathematical rules: the Multiplication Rule and the Factorial Rule."

Think of these rules as two magic spells:

The Multiplication Spell: You can multiply the whole equation by a constant.
The Factorial Spell: You can use the rule $s \times \Gamma(s) = \Gamma(s+1)$ to shift numbers around.

The Catch: The "Factorial Spell" is messy. When you cast it, it doesn't just change the numbers; it sometimes attaches a weird, extra fraction (a rational function) to the equation. It's like the person suddenly wearing a giant, clunky backpack.

The problem was: How do you check if a property is an invariant if the equation has this weird backpack attached? The old definition of "invariant" only worked for the clean equation. If the equation got messy, the definition broke down. The authors' previous claim was a bit like saying, "If you can recognize the person even when they wear a hat, you can recognize them even when they wear a backpack," but they hadn't explained how to recognize them with the backpack on.

The Solution: The "Rational Extension"

This new paper is the "How-To" guide for that claim. The authors introduce a brilliant new concept called a Rational Extension.

Here is the analogy:
Imagine you have a special scanner that can identify a person.

The Old Scanner: Only works if the person is wearing their standard outfit (the clean equation). If they put on a backpack (the messy rational function), the scanner breaks.
The New Scanner (Rational Extension): This is a super-sensor. It can look at the person and the backpack. It has a special setting that says, "I know this backpack is just an accessory. I will ignore the extra weight and still identify the person correctly."

The authors prove that:

Every true invariant has a "Rational Extension." You can always build this super-sensor.
If your super-sensor works (is "stable") under both magic spells (Multiplication and Factorial), then the property is a true invariant.

How They Did It (The "Backpack" Logic)

To prove this, they focused on a specific type of invariant called H-invariants (named after a mathematician named H). These are like calculating the "average weight" of the person's features.

They showed that even when the Factorial Spell adds a backpack (a rational function), you can mathematically adjust the "average weight" calculation to cancel out the backpack's effect.

Expanding: If the spell adds a term like $(\lambda s + \mu - 1)$ , the new calculation subtracts the weight of that specific term.
Contracting: If the spell removes a term, the calculation adds it back.

Because the math of these "Bernoulli polynomials" (the fancy formulas used to calculate the weight) is so perfect, the extra weight of the backpack always cancels out exactly. The result is that the final number remains the same, regardless of the backpack.

The Big Takeaway

The paper solves a philosophical and technical headache in mathematics.

Before: "We think this property is an invariant because it survives the magic spells, but we can't prove it when the spells make the equation messy."
After: "We have built a universal translator (the Rational Extension). If a property survives the magic spells even when the equation is messy, we can now rigorously prove it is a true invariant. We have a recipe to build this translator for any property."

In short, the authors took a "heuristic" (a gut feeling) and turned it into a solid, working machine. They showed that the "messy" cases aren't a bug; they are just a different outfit, and with the right tools (Rational Extensions), we can see the true identity of the L-function underneath.

1. Problem Statement

The paper addresses a foundational ambiguity regarding the definition of invariants for $L$ -functions within the extended Selberg class ( $S^\sharp$ ).

Context: Every $F \in S^\sharp$ satisfies a functional equation $\gamma(s)F(s) = \omega\gamma(1-s)F(1-s)$ , where the $\gamma$ -factor is defined as $\gamma(s) = Q^s \prod_{j=1}^r \Gamma(\lambda_j s + \mu_j)$ . The parameters $(\omega, Q, \lambda, \mu)$ constitute the "data" of the functional equation.
The Issue: A single $L$ -function $F$ can possess multiple representations of its $\gamma$ -factor due to the properties of the Euler Gamma function, specifically the factorial formula ( $s\Gamma(s) = \Gamma(s+1)$ ) and the multiplication formula (Legendre-Gauss).
The Heuristic: In previous work [3], the authors asserted that a function $I(\omega, Q, \lambda, \mu)$ is an invariant (i.e., its value depends only on $F$ and not the specific representation of $\gamma$ ) if and only if it is "stable" under these transformation formulae.
The Gap: While stability under the multiplication formula is well-defined, stability under the factorial formula is problematic. Applying the factorial formula to a $\Gamma$ -factor transforms the $\gamma$ -factor into a product $R(s)\gamma(s)$ , where $R(s)$ is a non-trivial rational function. This new form $R(s)\gamma(s)$ no longer fits the standard definition of a $\gamma$ -factor (1.1). Consequently, the original invariant function $I$ , which is defined only on the data of the standard form, is technically undefined on the transformed object, rendering the concept of "stability" formally ill-posed.

2. Methodology

To resolve this ambiguity, the authors introduce a rigorous framework based on the concept of a rational extension.

Rational Extension ( $I^*$ ): The authors define a generalized function $I^*(R, \omega, Q, \lambda, \mu)$ $I^{*} (R, ω, Q, λ, μ)$ that depends not only on the standard data but also on a rational function $R(s)$ $R (s)$ . This extension is defined such that when $R(s)$ $R (s)$ is a constant (specifically 1), $I^*$ $I^{*}$ reduces to the original invariant $I$ $I$ .
- Formally: $I^*(1, \omega, Q, \lambda, \mu) = I(\omega, Q, \lambda, \mu)$ .
Stability Definition: The authors define stability for this extended function $I^*$ $I^{*}$ under the transformation of the product $R(s)\gamma(s)$ $R (s) γ (s)$ :
- Multiplication Formula: Transforms $R(s)\gamma(s)$ to $R(s)c\gamma'(s)$ .
- Factorial Formula: Transforms $R(s)\gamma(s)$ to $R'(s)\gamma'(s)$ , where $R'(s)$ is a new rational function derived from the expansion or contraction of the $\Gamma$ -factors.
- Condition: $I^*$ is stable if its value remains unchanged under these transformations of the tuple $(R, \omega, Q, \lambda, \mu)$ .
Construction of the Extension: The paper provides an explicit algorithm to construct such an extension for specific invariants, notably the H-invariants ( $H_F(n)$ ), which involve Bernoulli polynomials.

3. Key Contributions

The paper's primary novelty is the explicit construction of a rational extension that validates the heuristic assertion made in previous literature.

Formalization of Stability: The authors provide the precise mathematical definition of "stability" for invariants under the factorial formula by extending the domain of the invariant function to include rational factors $R(s)$ .
Explicit Construction for H-Invariants:
- They define an extended H-invariant $H^*(n; R\gamma)$ for a product $R(s)\gamma(s)$ .
- If $R(s) = \frac{\prod(s-\beta_j)}{\prod(s-\alpha_j)}$ , the extension is:
  $H^*(n) = 2\sum_{j=1}^r \frac{B_n(\mu_j)}{\lambda_j^{n-1}} + (-1)^n 2^n \left( \sum_{j=1}^m \beta_j^{n-1} - \sum_{j=1}^l \alpha_j^{n-1} \right)$
  where $B_n(x)$ are Bernoulli polynomials.
- They prove that this $H^*(n)$ is stable under both the multiplication and factorial formulae. The proof relies on the functional equation of Bernoulli polynomials: $B_n(x+1) = B_n(x) + nx^{n-1}$ .
General Theorem: The paper establishes a necessary and sufficient condition for a function to be an invariant.

4. Main Results

The central result is stated as a Theorem:

Theorem: A function $I(\omega, Q, \lambda, \mu)$ of the data is an invariant if and only if it possesses a rational extension $I^*(R, \omega, Q, \lambda, \mu)$ that is stable under the multiplication and factorial formulae.

Proof Logic:

( $\Rightarrow$ ): If $I$ is an invariant, it can be expressed as a function of the fundamental invariants of the class ( $\omega_F$ , conductor $q_F$ , and the sequence of H-invariants). Since these fundamental invariants (and their rational extensions) are known to be stable, one can construct $I^*$ by composing the original function with these stable components.
( $\Leftarrow$ ): If $I$ has a stable rational extension $I^*$ , then for any two $\gamma$ -factors of the same $F$ (related by a sequence of transformations), the value of $I^*$ remains constant. Since $I$ is the restriction of $I^*$ to the case where $R=1$ , $I$ must yield the same value for all representations of $F$ , making it an invariant.

5. Significance

Rigorous Foundation: The paper resolves a long-standing technical gap in the theory of the Selberg class. It moves the concept of invariance from a "heuristic" or "intuitive" level to a rigorously defined mathematical property.
Validation of H-Invariants: It confirms the validity of the H-invariants ( $H_F(n)$ ) defined in previous work [4], which are crucial for understanding the arithmetic properties of $L$ -functions, by proving their stability even when the factorial formula introduces rational factors.
Algorithmic Utility: The construction provided offers a concrete method to verify whether a proposed function of the functional equation data is truly an invariant, which is essential for classifying $L$ -functions and studying their properties (such as the degree, conductor, and root number).
General Applicability: While the detailed construction focuses on H-invariants, the theorem applies to any function of the data, providing a universal criterion for invariance in the context of the Selberg class.