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On transcendence of non-periodic continued fractions associated with modular forms and arithmetic functions

This paper establishes sufficient conditions for the non-periodicity of sequences of various arithmetic functions modulo $m$ and utilizes these results to prove the transcendence of specific continued fractions constructed from these functions, including the Ramanujan tau function and Euler's totient function.

Original authors: Tapas Chatterjee, Sagar Mandal

Published 2026-02-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tapas Chatterjee, Sagar Mandal

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 standing in front of an endless, magical tapestry. This tapestry is woven with numbers, and the pattern it creates is determined by special mathematical rules called arithmetic functions. Some of these rules are famous, like the Ramanujan tau function (a superstar of number theory), while others are more humble, like counting how many ways you can group numbers together.

The paper you are asking about is like a detective story. The authors, Tapas Chatterjee and Sagar Mandal, are trying to solve two big mysteries about these number patterns:

The Mystery of the Broken Loop: Do these number patterns ever repeat themselves in a perfect, predictable cycle?
The Mystery of the Magic Number: If we take these patterns and weave them into a specific type of infinite fraction (called a continued fraction), do we create a number that is so unique and complex that it can never be the solution to a simple algebraic equation? (Such numbers are called transcendental, like $\pi$ or $e$ ).

Here is the story of their discovery, explained simply.

Part 1: The Broken Loop (Non-Periodicity)

Imagine you are listening to a song. If the song is periodic, it's like a catchy chorus that repeats every 30 seconds: La-la-la, boom-boom, La-la-la, boom-boom. You can predict exactly what comes next because the pattern is locked in a loop.

The authors wanted to know: Do these number sequences have a "catchy chorus"?

They looked at sequences created by taking famous numbers (like the Ramanujan tau function or Euler's totient function) and dividing them by a specific number $m$ to see what the remainder is. For example, if the sequence is $1, 5, 12, 7...$ and we divide by 5, the remainders are $1, 0, 2, 2...$

The Discovery:
The authors proved that for many of these famous sequences, there is no catchy chorus. The pattern never settles into a repeating loop. It keeps changing in a way that is unpredictable and non-repeating forever.

How did they prove it?
They used a clever trick involving prime numbers (numbers like 2, 3, 5, 7, 11 that can't be divided by anything else).

Think of the sequence as a machine that takes a prime number and spits out a remainder.
The authors showed that if you feed this machine different prime numbers, it spits out remainders that are "too different" to ever fit into a repeating cycle.
They used a mathematical "magnifying glass" (Dirichlet's Theorem) to find specific primes that force the pattern to break any potential loop.

Why does this matter?
It tells us that these number patterns are inherently chaotic and complex. They aren't just simple, repeating rhythms; they are wild, free-flowing streams of data.

Part 2: The Magic Number (Transcendence)

Now, imagine you take that non-repeating sequence of remainders and use it to build a Continued Fraction.

Think of a continued fraction like a Russian nesting doll, but instead of dolls, you have fractions inside fractions.
$0 + \frac{1}{d_1 + \frac{1}{d_2 + \frac{1}{d_3 + \dots}}}$
The numbers $d_1, d_2, d_3$ are the remainders from our sequences.

The authors asked: If we build a number using these non-repeating remainders, is the resulting number "special"?

In math, there are two types of numbers:

Algebraic Numbers: These are numbers that are solutions to simple equations (like $x^2 - 2 = 0$ , where $x = \sqrt{2}$ ). They are "tame."
Transcendental Numbers: These are numbers that are so wild they cannot be the solution to any simple equation with whole numbers. They are "untamable." Famous examples are $\pi$ (the ratio of a circle's circumference to its diameter) and $e$ (the base of natural logarithms).

The Discovery:
The authors proved that if you build a number using the non-repeating sequences they studied (like the Ramanujan tau function or the totient function), the result is always a Transcendental Number.

The Analogy:
Imagine you are building a tower out of bricks.

If the bricks follow a simple, repeating pattern (like Red, Blue, Red, Blue), the tower might be stable and predictable (Algebraic).
But the authors showed that if you use bricks that follow a wild, non-repeating pattern (like the sequences they studied), the resulting tower is so structurally unique and complex that it defies all standard rules of geometry. It becomes a "Transcendental" tower.

The Cast of Characters

The paper investigates a whole "family" of these number patterns:

The Ramanujan Tau Function ( $\tau(n)$ ): The rockstar of the group. It comes from the study of "Modular Forms" (shapes that look the same when you stretch or twist them in specific ways).
Eisenstein Series ( $E_k$ ): Another group of modular forms that act like a chorus of numbers.
Totient Functions ( $\phi, \Phi, J_k$ ): These are like "counting machines." They count how many numbers are "friendly" (coprime) to a specific number.
Divisor Sums ( $\sigma$ ): These add up all the factors of a number.

The authors showed that all of these, when looked at through the lens of remainders, refuse to repeat. And because they refuse to repeat, they can be used to build those "untamable" transcendental numbers.

The Big Picture

Why should a regular person care?

Understanding Chaos: It helps us understand that even in the rigid world of whole numbers, there is infinite complexity and unpredictability.
New Magic Numbers: The paper gives us a recipe to cook up infinitely many new transcendental numbers. Before this, we mostly knew about $\pi$ and $e$ . Now, we have a whole kitchen full of new, strange numbers derived from these ancient number patterns.
The Power of Patterns: It shows that by studying how numbers behave when we divide them (remainders), we can unlock deep secrets about the nature of reality and mathematics itself.

In short, the authors took a bunch of famous number sequences, proved they never get bored and repeat themselves, and then used that endless variety to construct new, magical numbers that exist beyond the reach of simple equations.

1. Problem Statement

The paper addresses two interconnected problems in number theory and transcendental number theory:

Non-Periodicity of Arithmetic Sequences: Determining sufficient conditions under which the sequence $\{f(n) \pmod m\}_{n \ge 1}$ is non-periodic (i.e., not eventually periodic) for various arithmetic functions $f(n)$ and moduli $m$ .
Construction of Transcendental Numbers: Utilizing the non-periodicity of these sequences to construct specific continued fractions that represent transcendental numbers.

The authors focus on sequences derived from:

Modular Forms: The Ramanujan tau function $\tau(n)$ and Fourier coefficients of normalized Eisenstein series $E_k(z)$ .
Arithmetic Functions: Nathanson's totient function $\Phi(n)$ , Euler's totient function $\phi(n)$ , the sum of divisors function $\sigma(n)$ , their Dirichlet convolution $\sigma * \phi(n)$ , Jordan's totient function $J_k(n)$ , and the unitary totient function $\phi^*(n)$ .

2. Methodology

The authors employ a combination of analytic number theory, modular form theory, and Diophantine approximation techniques.

A. Criteria for Non-Periodicity

The core of the methodology relies on proving non-periodicity by contradiction using the Dirichlet Prime Number Theorem. The authors establish two general theorems (Theorems 1.9 and 1.10) that serve as the foundation for their specific results:

Theorem 1.9: If an arithmetic function $f$ satisfies $f(p) \mid f(n)$ whenever $p \mid n$ , and if $m \nmid f(p)$ for infinitely many primes while $m \mid f(p)$ for primes $p \equiv 1 \pmod{K(m)}$ , then $\{f(n) \pmod m\}$ is non-periodic.
Theorem 1.10: If $f$ is multiplicative for square-free integers, and there exist distinct residues $r_1, r_2$ in $\mathbb{Z}/m\mathbb{Z}$ such that $f(p) \equiv r_1 \pmod m$ for $p \equiv 1 \pmod m$ and $f(p) \equiv r_2 \pmod m$ for infinitely many other primes, then the sequence is non-periodic.

These general theorems are applied to specific functions by leveraging known congruence properties:

Ramanujan $\tau(n)$ : Utilizing Ramanujan's congruences (e.g., $\tau(n) \equiv n\sigma_1(n) \pmod 5$ ) and the multiplicativity of $\tau(n)$ .
Eisenstein Series: Using the explicit Fourier expansions of $E_k(z)$ which relate coefficients to divisor sums $\sigma_{k-1}(n)$ .
Nathanson's Functions: Utilizing the relation $g(n+1) - g(n) = \frac{1}{2}\Phi(n+1)$ and properties of $\Phi(n)$ established in prior work (Lemma 2.3).

B. Construction of Transcendental Numbers

To prove transcendence, the authors utilize a theorem by Adamczewski, Bugeaud, and Davison (Theorem 2.5).

Theorem 2.5: States that if $\theta$ is an irrational number in $(0,1)$ , and a sequence of integers $d_n = 1 + ([n\theta] \pmod k)$ is formed, then the continued fraction $[0; d_1, d_2, \dots]$ represents a transcendental number.
Application: The authors construct numbers $\alpha_{k,m}$ where the partial quotients $d_{n,k,m}$ are derived from the partial sums of the arithmetic functions modulo $m$ (specifically, $d_{n,k,m} = 1 + ([\sum_{r=1}^n f(r) \pmod m / 10^r] \pmod k)$ ).
Logic: Since the sequences $\{f(n) \pmod m\}$ are proven to be non-periodic, the resulting real numbers defined by these series are irrational. The specific structure of the continued fraction partial quotients then guarantees transcendence via Theorem 2.5.

3. Key Contributions and Results

A. Non-Periodicity Results

The paper establishes non-periodicity for a wide range of sequences modulo specific integers:

Ramanujan Tau Function: $\{\tau(n) \pmod m\}$ is non-periodic for $m \in \{5, 7, 8, 9, 691\}$ .
Eisenstein Series Coefficients: The sequences of coefficients for $E_4, E_6, E_8, E_{10}, E_{14}$ are non-periodic modulo $m$ under specific coprimality conditions (e.g., for $E_4$ , $m \ge 3$ and $\gcd(m, 240)=1$ ).
Generalized Totient Functions:
- $\{g(n) \pmod m\}$ is non-periodic for all odd $m \ge 5$ .
- $\{\Phi(n) \pmod m\}$ is non-periodic for all odd $m \ge 5$ .
- $\{\phi(n) \pmod m\}$ is non-periodic for all $m \ge 3$ .
- $\{\sigma(n) \pmod m\}$ and $\{\sigma * \phi(n) \pmod m\}$ are non-periodic for all $m \ge 7$ .
- $\{J_k(n) \pmod m\}$ is non-periodic for $m \ge 3$ and odd $k \ge 3$ .
- $\{\phi^*(n) \pmod m\}$ is non-periodic for all $m \ge 3$ .

B. Transcendence Results

The authors construct infinitely many transcendental numbers $\alpha_{k,m}$ defined by continued fractions $[0; d_1, d_2, \dots]$ . The partial quotients $d_n$ are constructed using the arithmetic functions listed above.

Specific Cases: The paper provides explicit formulas for $d_{n,k,m}$ involving $\tau(n)$ , $E_k$ coefficients, $\Phi(n)$ , $g(n)$ , $\phi(n)$ , $\sigma(n)$ , $\sigma*\phi(n)$ , $J_k(n)$ , and $\phi^*(n)$ .
Conditions: These numbers are proven transcendental for specific ranges of $m$ and $k$ (e.g., $m=5,7,8,9$ for $\tau(n)$ ; $m \ge 5$ odd for $\Phi(n)$ ).

4. Significance

Extension of Previous Work: The paper extends the work of Ayad and Kihel (who proved non-periodicity for $g(n)$ for prime $m \neq 3$ ) to a much broader class of moduli (all odd $m \ge 5$ ) and a wider variety of arithmetic functions.
Bridging Fields: It creates a novel link between the theory of modular forms (specifically the distribution of coefficients modulo $m$ ) and the construction of transcendental numbers.
New Transcendental Numbers: It provides a systematic method to generate new families of transcendental numbers based on deep arithmetic properties of classical functions, rather than relying on ad-hoc constructions.
General Criteria: The establishment of Theorems 1.9 and 1.10 provides a reusable toolkit for researchers to determine the non-periodicity of other arithmetic sequences modulo $m$ .

In summary, the paper rigorously proves that the modular reductions of several fundamental arithmetic sequences are non-periodic and leverages this property to demonstrate the transcendence of associated continued fractions, thereby enriching the landscape of transcendental number theory with examples derived from modular forms and multiplicative number theory.