On a Möbius double sum

1. Problem Statement

The paper investigates the asymptotic behavior and explicit upper bounds of the double sum:
$$S_\varepsilon(X) = \sum_{d,e \le X} \frac{\mu(d)\mu(e)}{[d,e]^{1+\varepsilon}}$$
where:

• $\mu$ is the Möbius function.
• $[d, e]$ denotes the least common multiple (LCM) of $d$ and $e$.
• $\varepsilon \ge 0$ is a real parameter.
• $X$ is a positive real number.

Context and Motivation:

• Analytic Number Theory: This sum arises naturally in the study of zero-density estimates for the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions. Specifically, it represents the "diagonal contribution" in the calculation of mollified second moments of these functions.
• Convergence: The sum converges even when $\varepsilon = 0$. Previous work by Dress, Tenenbaum, and Iwaniec established that $S_0(X)$ converges to a constant as $X \to \infty$. Helfgott later computed this constant to be approximately $0.4408$.
• Gap in Knowledge: While the limit is known, there were no rigorous, uniform, and explicit upper bounds for $S_\varepsilon(X)$ across various ranges of $X$, particularly for small $\varepsilon$ (close to $0^+$) and large $X$. Such bounds are crucial for optimizing mollifiers in zero-density theorems.

2. Methodology

The authors employ a combination of direct computation, analytic estimates of multiplicative functions, and careful error term analysis.

A. Decomposition and Reformulation

The authors transform the double sum into a single sum involving the Euler totient function $\phi$ and the summatory function of the Möbius function.
Using the identity $\sum_{d,e} \dots = \sum_{\ell} \frac{\mu^2(\ell)\phi(\ell)}{\ell^2} \left( \sum_{m \le X/\ell, (m,\ell)=1} \frac{\mu(m)}{m} \right)^2$, the problem reduces to bounding the inner sum $m_q(Y) = \sum_{n \le Y, (n,q)=1} \frac{\mu(n)}{n}$.

B. Estimates for the Summatory Function $m_q(X)$

A core component of the methodology is establishing sharp, explicit bounds for $m_q(X)$ and its perturbed version $m_q(X; 1+\varepsilon)$.

• Lemma 2.6 & 3.1: The authors derive explicit bounds for $|m_q(X)|$ and $|m_q(X; 1+\varepsilon)|$ using previous results (Helfgott, Ramaré) and refining the error terms. They introduce auxiliary multiplicative functions $g_0(q)$ and $g_2(q)$ to handle the dependence on the modulus $q$.
• Perturbation Analysis: For $\varepsilon > 0$, they utilize an expansion:
  $$m_q(X; 1+\varepsilon) = \frac{m_q(X)}{X^\varepsilon} + \frac{q^{1+\varepsilon}}{\phi_{1+\varepsilon}(q)\zeta(1+\varepsilon)} + \varepsilon \Delta_q(X, \varepsilon)$$
  This allows them to relate $S_\varepsilon(X)$ back to $S_0(X)$ plus error terms involving $\Delta_q$.

C. Mean Value Estimates for Multiplicative Functions

To bound the resulting single sums, the authors apply general mean-value theorems for multiplicative functions (Theorems 4.1 and 4.2, based on work by Tenenbaum and others).

• They analyze functions of the form $f(n) = \frac{\phi(n)h(n)}{n^\alpha}$ where $h(n)$ involves the auxiliary functions $g_0, g_2$.
• Theorem 4.2 provides explicit error terms for sums $\sum_{n \le X} \mu^2(n)f(n)$, distinguishing cases based on the exponent $\alpha$ (specifically $\alpha=1$, $\alpha < 1/2$, and $\alpha = 1/2$).
• Infinite Products: Rigorous bounds for infinite Euler products and sums over primes are computed using explicit bounds for the prime counting function $\pi(t)$ and Chebyshev function $\theta(t)$ (Lemma 4.4). The authors set a precision threshold $M = 10^8$ to separate finite computations from tail estimates.

D. Computational Verification

• Direct Computation: For small $X$ (specifically $X < 10^7$), the authors perform direct computations using an optimized algorithm that leverages the recursive structure of $S_0(\ell) - S_0(\ell-1)$.
• Optimization: For larger $X$, they split the sum into a "head" (small $\ell$) and a "tail" (large $\ell$). The tail is bounded using the analytic estimates derived in Sections 2–4, while the head is bounded by optimizing parameters (like the split point $D$) and using the precomputed bounds for $S_0(X)$.

3. Key Contributions

1. Uniform Explicit Bounds: The paper provides the first set of rigorous, uniform upper bounds for $S_\varepsilon(X)$ that are valid for all $X > 0$ and $\varepsilon \in [0, 1/25]$.
2. Refinement of $S_0(X)$: The authors improve the known bounds for the case $\varepsilon = 0$, providing specific upper limits for different ranges of $X$ (e.g., $0.445$ for $422 \le X < 10^7$).
3. Handling the Perturbation $\varepsilon$: They successfully extend the bounds to $\varepsilon > 0$ by controlling the error terms introduced by the perturbation, showing that $S_\varepsilon(X)$ remains close to $S_0(X)$ for small $\varepsilon$.
4. Algorithmic Efficiency: The derivation of identity (12) for the difference $S_0(\ell) - S_0(\ell-1)$ significantly reduces the computational cost of evaluating the sum for large $X$.

4. Main Results

Theorem 1.1 (Case $\varepsilon = 0$):
For $X > 0$, $0 \le S_0(X) \le C(X)$, where $C(X)$ is defined piecewise:

• $0.445$ for $422 \le X < 10^7$
• $0.5976$ for $10^7 \le X < 10^{12}$
• $0.4988$ for $10^{12} \le X < 10^{16}$
• $0.467$ for $10^{16} \le X < 10^{33}$
• $0.4664$ for $X \ge 10^{33}$

Theorem 1.2 (Case $0 \le \varepsilon \le 1/25$):
The authors provide explicit upper bounds for $S_\varepsilon(X)$ depending on the ranges of $X$ and $\varepsilon$.

• Example: For $X \ge 10^{33}$ and $\varepsilon \le 1/100$, $S_\varepsilon(X) \le 0.4794$.
• Example: For $10^{12} \le X < 10^{16}$ and $\varepsilon \le 1/25$, $S_\varepsilon(X) \le 0.593$.
• The bounds are generally slightly larger than those for $\varepsilon=0$, reflecting the perturbation, but remain tight enough for practical applications in analytic number theory.

5. Significance

• Zero-Density Estimates: The primary significance lies in the application to zero-density theorems for $\zeta(s)$. The efficiency of these theorems depends heavily on the quality of the bounds for the mollified second moments. By providing explicit, uniform bounds for $S_\varepsilon(X)$, the authors enable more precise and stronger zero-density results.
• Explicit Constants: In analytic number theory, asymptotic results ($O$-notation) are often insufficient for explicit applications (e.g., verifying the Riemann Hypothesis up to a certain height). This paper replaces asymptotic estimates with concrete numerical constants, making the results immediately usable in computational number theory.
• Methodological Advancement: The paper demonstrates a robust framework for handling double sums involving the Möbius function and LCMs, combining deep analytic tools (mean value theorems, Euler products) with high-precision numerical verification. This approach can be adapted to similar problems involving other arithmetic functions.

In summary, Ramaré and Zuniga-Alterman have bridged the gap between theoretical convergence and practical explicit bounds for a critical sum in analytic number theory, providing essential tools for refining zero-density estimates of the Riemann zeta function.