On a conjecture of Goldmakher

Technical Summary: On a Conjecture of Goldmakher

Problem Statement
The paper addresses a conjecture proposed by Goldmakher in 2009 regarding the behavior of logarithmically averaged partial sums of 1-bounded, completely multiplicative functions. Let $\mathcal{F}$ denote the collection of such functions $f: \mathbb{N} \to \mathbb{C}$ with $|f(n)| \le 1$. For $f \in \mathcal{F}$, the logarithmic partial sum is defined as $L_f(x) = \sum_{n \le x} f(n)/n$.

Goldmakher's conjecture posits that the magnitude of $L_f(x)$ is controlled by the "pretentious distance" of $f$ from the constant function $1$. Specifically, it asserts that for any $1 \le y \le x$:
$$\sum_{n \le x, p|n \implies p \le y} \frac{f(n)}{n} \ll 1 + (\log y) e^{-D(f, 1; y)^2},$$
where $D(f, g; z) = \left( \sum_{p \le z} \frac{1 - \text{Re}(f(p)\overline{g(p)})}{p} \right)^{1/2}$.

Morally, this conjecture suggests that $L_f(x)$ can only be large (or unbounded) if $f$ "pretends" to be the constant function $1$(i.e.,$D(f, 1; x)^2$ grows slowly). This stands in contrast to ordinary partial sums, where large values are known to occur when $f$ pretends to be $n^{it}$ for some $t \in \mathbb{R}$. While previous work by Granville and Soundararajan established a weaker bound with an exponent of $1/2$ in the exponential term, and later work by Granville and the author showed an optimal exponent of $\lambda \approx 0.8221$ for certain constructions, the specific form of Goldmakher's conjecture (with exponent $1$) remained unrefuted.

Methodology
The paper constructs a counterexample by analyzing the relationship between the logarithmic sums $L_f(x)$ and the Dirichlet series $L(s, f) = \sum f(n)n^{-s}$. The core strategy involves the following steps:

1. Perron's Formula and Peak Analysis: The author relates the partial sums to the behavior of the function $L(\sigma + it, f)\zeta(\sigma + it)$ near the line $\sigma = 1 + 1/\log x$. Goldmakher's conjecture implicitly assumes that the dominant contribution to the integral comes from the peak at $t=0$ (corresponding to $f$ pretending to be $1$).
2. Construction of a "Competing" Peak: The author constructs a specific function $f$ designed to create a dominant peak at a non-zero frequency $t = \xi$ (where $\xi \to 0$ as $x \to \infty$). This is achieved by defining $f(p)$ on primes such that the real part of the logarithmic derivative is optimized at $\sigma + i\xi$ rather than $\sigma$.
3. Defining the Counterexample Function: For a large scale $x$, the author selects a parameter $\xi = (\log x)^{-\alpha}$ and defines $f(p) = R(p) + iI(p)$ based on the values of $\cos(\xi \log p)$ and $\sin(\xi \log p)$. The real part $R(p)$ is chosen to be slightly less than $\cos(\xi \log p)$ in specific intervals to ensure that the sum $\sum_{p \le x} \text{Re}((1-f(p))(1-p^{-i\xi}))/p$ is negative, thereby making $|L(\sigma + i\xi, f)|$ significantly larger than $|L(\sigma, f)|$.
4. Inequality Derivation: The construction ensures that:
   $$|L(\sigma + i\xi, f)\zeta(\sigma + i\xi)| \gg (\sigma - 1)^{-c} |L(\sigma, f)\zeta(\sigma)|$$
   for some constant $c > 0$. This additional factor $(\sigma - 1)^{-c}$ is crucial, as it allows the peak at $\xi$ to dominate the integral representation of the partial sums, violating the bound predicted by the conjecture.
5. Inductive Patching: To prove the existence of a single function $f$ that violates the bound infinitely often (rather than just at a single scale), the author uses an inductive argument. They construct a sequence of functions $f_k$ and scales $x_k^*$, where each $f_{k+1}$ agrees with $f_k$ up to a certain point but is modified for larger primes to create a new violation at a larger scale.

Key Results
The paper establishes two main theorems:

• Theorem 1.3 (Finite Scale Counterexample): For any constant $C > 0$, there exists a threshold $x_0(C)$ such that for every $x \ge x_0(C)$, there exists a function $f \in \mathcal{F}$ satisfying:
  $$L_f(x) \ge C \left( 1 + (\log x) e^{-D(f, 1; x)^2} \right).$$
  This demonstrates that the bound in Goldmakher's conjecture fails for any fixed constant $C$ at sufficiently large scales.

• Theorem 1.2 (Global Counterexample): There exists a single function $f \in \mathcal{F}$ such that:
  $$\limsup_{x \to \infty} \frac{|L_f(x)|}{1 + (\log x) e^{-D(f, 1; x)^2}} = \infty.$$
  This definitively disproves Goldmakher's conjecture.

Significance
The paper resolves a specific open problem in the theory of multiplicative functions by showing that the size of logarithmic partial sums cannot be bounded solely by the pretentious distance to the constant function $1$ with an exponent of $1$.

The author notes that while previous results (such as those by Granville and the author) suggested the conjecture might be false due to the optimality of the exponent $\lambda \approx 0.8221$, those specific counterexamples satisfied $L_f(x) \ll 1$ and thus did not directly disprove the conjecture. The significance of this work lies in constructing a function where the logarithmic sums are genuinely unbounded relative to the predicted bound, demonstrating that "pretending" to be $n^{i\xi}$ for small $\xi$ can indeed drive the logarithmic sums to be arbitrarily large, even when the distance to $1$ is large.

The work relies on standard tools in analytic number theory, including Mertens' theorem, Perron's formula, and the Halberstam-Richert bounds, but applies them in a novel construction to separate the behavior of logarithmic sums from the standard pretentious distance heuristic.