Correlations of multiplicative functions with their partial sums

Technical Summary: Correlations of Multiplicative Functions with Their Partial Sums

Problem Statement
This paper investigates the correlation between multiplicative functions and their corresponding summatory (partial sum) functions, specifically focusing on the Möbius function $\mu(n)$ and its summatory function $M(n)$ (the Mertens function), as well as the Liouville function $\lambda(n)$ and its summatory function $L(n)$. The author defines a specific logarithmic average correlation measure:
$$\langle a(n)A(n-1) \rangle(T) = \frac{1}{\zeta(1+\delta(T))} \sum_{n \le T^{1-c}} \frac{a(n)A(n-1)}{n^{1+\delta(T)}}$$
where $0 < c < 1$ and $\delta(T)$ is a suitably chosen parameter decaying as $T \to \infty$. The central problem is to determine the asymptotic behavior of these correlations under the assumption of the Riemann Hypothesis (RH) and the Simplicity of Zeros Conjecture (SZC). The study aims to clarify whether these functions exhibit anticorrelation with their partial sums and to derive implications for the growth rate of $1/\zeta'(\rho)$, where $\rho$ denotes the nontrivial zeros of the Riemann zeta function.

Methodology
The proofs rely on analytic number theory techniques, specifically the explicit formulas linking summatory functions to the zeros of $\zeta(s)$. The methodology proceeds as follows:

1. Explicit Formulas: The author utilizes explicit formulas for $M(n-1)$ and $L(n-1)$ derived from Perron's formula and the calculus of residues. Under RH and SZC, these are expressed as sums over the nontrivial zeros $\rho = 1/2 + i\gamma$:
   • $M(n-1) \approx \sum_{|\gamma|<T} \frac{n^\rho}{\rho \zeta'(\rho)} + \text{error terms}$.
   • $L(n-1) \approx \frac{n^{1/2}}{\zeta(1/2)} + \sum_{|\gamma|<T} \frac{\zeta(2\rho)n^\rho}{\rho \zeta'(\rho)} + \text{error terms}$.
2. Finite Sum Approximations: The paper establishes finite versions of Dirichlet series identities for $\mu(n)$, $\mu^2(n)$, $\lambda(n)$, and $\lambda^2(n)$ using partial summation. These approximations include explicit error bounds dependent on $N$ (the truncation limit) and $\epsilon$.
3. Laurent Expansion and Parameter Selection: The core of the proof involves evaluating the weighted sum $\sum \frac{a(n)A(n-1)}{n^{1+\delta}}$. The author expands $\zeta(1+\delta)$ and $\zeta(1-\rho+\delta)$ using Laurent series around $\delta=0$. A critical step is the construction of a specific function $\delta(T)$ (dependent on $T$ and the zeros $\rho$) that ensures the convergence of error terms while isolating the dominant contributions from the zero sums.
4. Limit Analysis: By setting the truncation limit $N = T^{1-c}$ and letting $T \to \infty$, the author isolates the constant terms and the sums over zeros, demonstrating that the error terms vanish or are bounded by $O(1/N)$.

Key Contributions and Results
The paper presents two main theorems describing the asymptotic limits of the correlations:

• Theorem 1 (Möbius Correlation): Under RH and SZC, the correlation between $\mu(n)$ and $M(n-1)$ is given by:
  $$\langle \mu(n)M(n-1) \rangle(T) = -\frac{3}{\pi^2} \left(1 - T^{(c-1)\delta(T)}\right) + \sum_{0<\gamma<T} \frac{1}{|\rho \zeta'(\rho)|^2}$$
  The result suggests a negative bias (anticorrelation) dominated by the term $-3/\pi^2$.

• Theorem 2 (Liouville Correlation): Under RH and SZC, the correlation between $\lambda(n)$ and $L(n-1)$ is given by:
  $$\langle \lambda(n)L(n-1) \rangle(T) = \frac{1}{2} \left( \frac{1}{\zeta^2(1/2)} - 1 + T^{(c-1)\delta(T)} \right) + \sum_{0<\gamma<T} \left| \frac{\zeta(2\rho)}{\rho \zeta'(\rho)} \right|^2$$
  The leading term $\frac{1}{2}(\frac{1}{\zeta^2(1/2)} - 1)$ is approximately $-0.2655$, also indicating a negative correlation.

• Numerical Evidence: The paper provides numerical plots for $N \le 10^7$ showing that the computed logarithmic averages for both $\mu(n)M(n-1)$ and $\lambda(n)L(n-1)$ are robustly negative, aligning with the theoretical predictions.

Significance and Implications
The primary significance of these results lies in their corollaries regarding the magnitude of the derivative of the zeta function at its zeros, $1/\zeta'(\rho)$.

• Corollary 1: If the right-hand side of Theorem 1 is negative (as suggested by the numerical data), it implies an effective upper bound:
  $$\frac{1}{\zeta'(\rho)} = o(|\rho|) \quad \text{as } |\gamma| \to \infty$$
• Corollary 2: Similarly, if the right-hand side of Theorem 2 is negative, it implies:
  $$\frac{1}{\zeta'(\rho)} = o(|\rho| \log \log |\gamma|) \quad \text{as } |\gamma| \to \infty$$

The paper notes that finding effective upper bounds for $1/|\zeta'(\rho)|$ is a longstanding open problem. While lower bounds are well-studied, effective upper bounds are less understood. The author cautions that these numerical results must be interpreted carefully, citing the historical example of Mertens' conjecture (which was disproven despite strong numerical evidence) to emphasize that numerical trends do not constitute proof. However, the combination of the derived asymptotic formulas and the observed negative correlations suggests that effective upper bounds on $|1/\zeta'(\rho)|$ may exist, providing a new perspective on the distribution of the zeta function's zeros.