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Sums of Apostol's Möbius functions of order $k$

This paper provides complete affirmative solutions to the conditional version and partial solutions to the unconditional version of A. Bege's 2001 conjecture regarding sums of Apostol's Möbius functions of order $k$ over integers coprime to $q$ , while also establishing a mean square estimate for the associated error term.

Original authors: Reo Terada

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Reo Terada

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 count a very specific type of "mathematical grain" scattered across a vast, infinite field. This grain is called the Apostol-Möbius function. It's a special rule that assigns a number (1, -1, or 0) to every whole number based on how its "prime building blocks" are arranged.

For a long time, mathematicians knew that if you count these grains up to a certain point $x$ , the total isn't just a random mess. It follows a predictable pattern: a main straight line (the "average" amount) plus a little bit of "wiggle" or "noise" (the error term).

This paper, written by Reo Terada, is about measuring that "noise" more precisely, especially when we add a new rule to the game: we only count the grains that don't share any factors with a specific number $q$ (like only counting grains that are "coprime" to $q$ ).

Here is a breakdown of what the paper achieves, using simple analogies:

1. The Main Goal: Taming the Noise

Think of the total count of these grains as a car driving down a highway.

The Main Road ($Akx$): This is the smooth, predictable path the car is supposed to follow.
The Bumpy Road ( $E_{k,q}(x)$ ): This is the "error term" or the noise. The car wobbles a bit off the straight line.

Previous mathematicians (like Apostol, Bege, and others) had maps for how bumpy this road was.

The Old Map: Said the bumps were roughly the size of $x^{1/k}$ .
Bege's Conjecture (The Dream Map): Suggested the bumps were actually much smaller and smoother, shrinking exponentially as you drove further.

What Terada did: He proved that Bege's "Dream Map" is mostly correct. He showed that the bumps are indeed much smaller than previously thought, but with a slight caveat: the size of the bumps depends on the specific rules of the road (the number $k$ and the number $q$ ).

2. The Two New Maps (Theorems)

The paper provides two different maps for this bumpy road, depending on whether we assume a famous, unproven mathematical theory called the Riemann Hypothesis is true.

Map A: The "Real World" Map (Unconditional Result)

The Scenario: We don't assume the Riemann Hypothesis is true. We are working with what we know for sure.
The Result: Terada proves that the bumps (the error) are very small. They shrink faster than anyone expected, following a formula that looks like a steep slide down a hill.
The Catch: The size of the bumps still depends heavily on the "order" $k$ (how complex the grain rule is).
Analogy: Even without a perfect map of the universe, Terada proved that the car's wobble is tiny, provided you know exactly how bumpy the specific road segment is.

Map B: The "Perfect World" Map (Conditional on Riemann Hypothesis)

The Scenario: We assume the Riemann Hypothesis is true (a giant, unproven rule about how prime numbers are distributed).
The Result: Under this assumption, the road becomes incredibly smooth. The bumps shrink even further, making the car's path almost perfectly straight.
Significance: This completely solves a specific guess (conjecture) made by a mathematician named Bege in 2001. It confirms that if the universe follows the Riemann rules, the "noise" in this counting problem is negligible.

3. The "Mean Square" Estimate (The Average Wobble)

The paper also looks at the Mean Square of the error.

Analogy: Imagine you take a photo of the car's wobble every second and measure how far it is from the center line. Then, you square those distances (to make them all positive) and take the average.
The Result: Terada calculated exactly how much "energy" is in these wobbles over a long period. He found that while the car wobbles, the average amount of wobble is predictable and follows a specific growth rate. This helps mathematicians understand the "typical" behavior of the error, not just the worst-case scenario.

4. The "Coprime" Twist

A key part of this paper is that it doesn't just count all grains; it counts grains that are coprime to $q$ .

Analogy: Imagine you are counting grains, but you have a sieve. If a grain shares a secret code (a factor) with the number $q$ , you throw it away.
The Discovery: Terada showed that even with this sieve, the same rules for the "bumps" apply. The size of the bumps is directly related to how many "square-free" divisors the number $q$ has (a fancy way of counting how many ways you can break $q$ down without repeating prime factors).

Summary

In plain English, this paper is a precision engineering report for a specific mathematical counting problem.

It confirms that a long-standing guess (Bege's Conjecture) about how small the counting errors are is correct.
It provides a new, tighter formula for these errors that works even without assuming the Riemann Hypothesis is true.
It gives a complete solution for the errors if we do assume the Riemann Hypothesis is true.
It extends all these results to a more complex scenario where we filter out numbers that share factors with a specific number $q$ .

The paper doesn't talk about building bridges or curing diseases; it is purely about cleaning up the math to see the underlying patterns of numbers more clearly.

Technical Summary: Sums of Apostol's Möbius Functions of Order $k$

Problem Statement
The paper addresses the asymptotic behavior of the summatory function of the generalized Möbius function $\mu_k(n)$ of order $k$ , specifically under the condition that the summation index $n$ is coprime to a fixed integer $q$ . The classical Möbius function $\mu(n)$ corresponds to the case $k=1$ . For $k \ge 2$ , T. M. Apostol (1970) established that $\sum_{n \le x} \mu_k(n) = A_k x + E_k(x)$ , where the error term $E_k(x)$ is $O(x^{1/k} \log x)$ .

A. Bege (2001) conjectured significantly sharper bounds for the error term $E_{k,q}(x)$ in the coprime case $\sum_{n \le x, (n,q)=1} \mu_k(n) = A_{k,q} x + E_{k,q}(x)$ . Bege's conjecture proposed two versions:

Unconditional: $E_{k,q}(x) \ll \theta(q) x^{1/k} \exp\left(-D \frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\right)$ .
Conditional (assuming the Riemann Hypothesis): $E_{k,q}(x) \ll \theta(q) x^{2/(2k+1)} \exp\left(A \frac{\log x}{\log \log x}\right)$ .

While recent works by Banerjee et al. (2023) and Martin and Yip (2025) resolved these conjectures for the case $q=1$ , the general case for arbitrary $q \ge 1$ remained open. This paper aims to generalize these recent results to arbitrary $q$ and provide a mean square estimate for the error term.

Methodology
The author employs analytic number theory techniques, primarily relying on Dirichlet series, Perron's formula, and contour integration. The core methodological steps include:

Decomposition of $\mu_k$ : The function $\mu_k$ is decomposed via Dirichlet convolution as $\mu_k = f_k * c_k$ . The function $f_k$ is defined such that its Dirichlet series relates to $\zeta^{-2}(s)$ , while $c_k$ is an auxiliary function with controlled growth properties.
Splitting the Sum: The summatory function is split into ranges based on a parameter (e.g., $d \le \rho z$ vs. $d > \rho z$ ) to isolate the main term from the error term. This allows the application of different estimates for different parts of the sum.
Dirichlet Series Analysis: The paper utilizes the Euler product expansions of the relevant Dirichlet series. Key lemmas establish the absolute convergence of these series in specific half-planes and provide bounds for the coefficients of the auxiliary functions (e.g., $b_k(n)$ and $d_k(n)$ ).
Contour Integration: Perron's formula is used to express the partial sums as complex integrals. The integration contours are shifted to the left (towards the critical line) to extract main terms (residues) and bound the remaining integrals.
- For the unconditional proof (Theorem 1), the contour is shifted to a region utilizing zero-free regions of the Riemann zeta-function, leading to the exponential decay factor involving $(\log x)^{3/5}$ .
- For the conditional proof (Theorem 2), assuming the Riemann Hypothesis (RH), the contour is shifted to $\text{Re}(s) = 1/2$ . Bounds on $\zeta(s)$ and $\zeta^{-1}(s)$ under RH (using the $\omega$ -function notation) are applied to estimate the integrals.
Mean Square Estimation: For Theorem 3, the author analyzes the mean square of the discrepancy $\Xi_{k,q}(x)$ (the error term adjusted for a secondary main term). This involves Parseval's identity applied to the Dirichlet series, converting the integral of the square of the function into an integral of the squared modulus of the associated Dirichlet series along a vertical line.

Key Contributions and Results

Theorem 1 (Unconditional Estimate): The paper proves that for any $x \ge 3$ , $k \ge 2$ , and $q \ge 1$ :
$E_{k,q}(x) \ll_k \theta(q) x^{1/k} \exp\left(-D_0 k^{-8/5} \frac{(\log x)^{3/5}}{(\log \log x)^{1/5}}\right)$
where $D_0$ is an absolute positive constant. This provides an affirmative solution to the unconditional version of Bege's conjecture, allowing the implied constant and the constant $D$ to depend on $k$ . It also slightly improves the $q=1$ result of Banerjee et al.
Theorem 2 (Conditional Estimate): Assuming the Riemann Hypothesis, the paper proves:
$E_{k,q}(x) \ll \theta(q) x^{1/(k+1)} \exp\left(C_0 \frac{\log x}{\log \log x}\right)$
where $C_0$ is an absolute positive constant. This completely solves the conditional part of Bege's conjecture for general $q$ . It refines the $q=1$ result of Martin and Yip by providing a more precise error term structure.
Theorem 3 (Mean Square Estimate): The paper establishes a bound for the mean square of the adjusted error term $\Xi_{k,q}(x) = E_{k,q}(x) - B_{k,q}x^{1/(k+1)}$ :
$\int_1^T \Xi_{k,q}^2(x) \, dx \ll C_{k,q}^2 T^{1+1/k} \exp\left(G_0 (\log T)^{G_1 k / (G_1 k + 1)}\right)$
where $C_{k,q}$ is a constant depending on $q$ and $k$ , and $G_0, G_1$ are absolute constants. This quantifies the average size of the error term's oscillation.

Significance
The paper claims to provide "affirmative solutions" to the conditional version of Bege's conjecture completely and the unconditional version partially (in the sense that the constants depend on $k$ ). By generalizing the recent breakthroughs for $q=1$ to arbitrary $q$ , the work extends the understanding of the distribution of higher-order Möbius functions in arithmetic progressions. The results confirm that the error terms behave as Bege predicted, with the magnitude of the error controlled by the number of square-free divisors of $q$ ( $\theta(q)$ ) and the order $k$ . The mean square estimate further characterizes the fluctuation of these sums, offering a deeper insight into the error term's behavior beyond simple upper bounds.