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A Halász-type asymptotic formula for logarithmic means and its consequences

This paper establishes a sharp Halász-type asymptotic formula for logarithmic means of 1-bounded multiplicative functions, which is then used to significantly improve bounds on the lower limits of such means, confirm a conjecture regarding Rademacher random multiplicative functions, and provide a converse theorem for small absolute values.

Original authors: Oleksiy Klurman, Alexander P. Mangerel

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Oleksiy Klurman, Alexander P. Mangerel

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 predict the weather, but instead of rain and sunshine, you are tracking the behavior of numbers. Specifically, you are looking at a special class of numbers called multiplicative functions.

Think of these functions as "personality traits" assigned to every integer. For example, a number might be "friendly" (+1), "hostile" (-1), or "neutral" (0). The rule is simple: if you multiply two numbers, their combined personality is just the product of their individual personalities.

The mathematicians in this paper, Oleksiy Klurman and Alexander P. Mangerel, are trying to answer a very old, very stubborn question: If you add up the "personalities" of all numbers up to a certain point, does the total balance out, or does it swing wildly to one side?

Here is a breakdown of their breakthrough, explained through everyday analogies.

1. The Problem: The "Tug-of-War" of Numbers

Imagine a giant tug-of-war. On one side, you have all the numbers up to $x$ . Each number pulls the rope either left (negative) or right (positive).

The Old Way (Cesàro Mean): Mathematicians used to look at the total sum of the rope's position. This is like looking at the rope's absolute displacement. The problem? If the rope is being pulled by a "pretender" (a number that acts like a specific pattern, like the Liouville function), the rope can get stuck in a weird spot, making it hard to predict the future.
The New Way (Logarithmic Mean): The authors decided to look at the rope differently. Instead of counting every pull equally, they gave more weight to the smaller numbers and less weight to the huge ones. It's like listening to a choir: you hear the quiet singers (small numbers) just as clearly as the loud ones (large numbers) because you adjust the volume. This is called a Logarithmic Mean.

2. The Big Discovery: The "Magic Constant"

The authors found a precise formula to predict where this weighted rope will end up. But here is the twist: to get the prediction right, they had to look slightly beyond the current number $x$ .

They introduced a mysterious constant, $w_0$ (which is roughly 1.58).

The Analogy: Imagine you are trying to predict the traffic jam at 5:00 PM. Usually, you look at the cars currently on the road. But this new formula says, "To know exactly where the jam will be, you actually need to know about the cars that will be on the road at 5:00 PM plus a little bit of extra time."
Why? Because the "echo" of the numbers slightly larger than $x$ actually influences the average of the numbers up to $x$ . It's a bit like how the sound of a bell you just rang continues to echo and affects the silence you hear a split second later.

3. Application A: The "Bad Luck" Limit

For a long time, mathematicians wondered: How negative can this sum get?

The Old Result: Granville and Soundararajan (20 years ago) proved that the sum could get very negative, but they thought the limit was like a slow, heavy fog: $1/(\log \log x)^{3/5}$ .
The New Result: Klurman and Mangerel proved the fog is actually much thinner. They showed the sum can't get too negative. It's bounded by something like $1/(\log x)^{0.36}$ .
The Metaphor: Imagine a boat in a storm. The old theory said the boat could sink 100 meters deep. The new theory says, "No, the boat might dip, but it will never sink deeper than 10 meters." This is a massive improvement in our understanding of how "bad" these numbers can behave.

4. Application B: The "Random Coin Flip"

What if we assign these +1 and -1 values completely at random, like flipping a coin for every prime number?

The Question: How often will the total sum be negative?
The Intuition: You might think it happens 50% of the time.
The Reality: The authors proved that for these random numbers, the chance of the sum being negative is astronomically small. It's not just "rare"; it's so rare that it's like winning the lottery every day for a million years.
The Metaphor: If you flip a coin a billion times, you expect roughly half heads and half tails. But if you are summing these specific "coin flip numbers" in a logarithmic way, the universe seems to conspire to keep the total positive. The probability of it being negative drops so fast it's practically zero.

5. Application C: The "Imposter" Detection

Finally, they looked at the opposite problem: What if the sum is very small (close to zero)?

The Insight: If the sum is tiny, the numbers aren't random. They are "pretending" to be a specific, famous pattern called the Liouville function (which is a very specific, chaotic pattern of +1s and -1s).
The Metaphor: If you walk into a room and the temperature is exactly 70°F, you might think it's just a normal day. But if the temperature is exactly 70.000000°F every single day, you'd suspect someone is controlling the thermostat. The authors proved that if the sum is this "perfectly balanced," the numbers must be mimicking that specific "thermostat" (the Liouville function).

Summary

This paper is a masterclass in refining our view.

New Lens: They stopped looking at the "raw" sum and started looking at the "weighted" (logarithmic) sum.
New Insight: They realized the future (numbers slightly larger than $x$ ) influences the present average.
New Limits: They proved that these numbers can't be as "bad" (negative) as we feared, and if they are "too good" (balanced), they are definitely faking it.

In short, they took a chaotic, unpredictable crowd of numbers and found a hidden rhythm, proving that even in the wild world of prime numbers, there are strict rules governing how they balance out.

1. Problem Statement and Motivation

The paper addresses a central problem in analytic number theory: understanding the asymptotic behavior of partial sums of multiplicative functions. Specifically, it focuses on logarithmic partial sums defined as:
$L_f(x) := \sum_{n \le x} \frac{f(n)}{n}$
where $f: \mathbb{N} \to \mathbb{C}$ is a multiplicative function bounded by 1 (i.e., $|f(n)| \le 1$ ).

Context and Limitations of Previous Work:

Halász's Theorem: The classical result by Halász provides upper bounds for the standard (Cesàro) partial sums $M_f(x) = \sum_{n \le x} f(n)$ . These bounds depend on the "pretentious distance" $D(f, n^{it}; x)$ , which measures how much $f$ resembles a character $n^{it}$ .
Logarithmic vs. Cesàro: While $L_f(x)$ can be bounded using $M_f(x)$ via partial summation, this approach is often not sharp. The behavior of $L_f(x)$ is distinct because it is less sensitive to $f$ "pretending" to be $n^{it}$ for large $|t|$ .
The Gap: Existing asymptotic formulas for $L_f(x)$ either require rigid distributional assumptions on $f$ at primes or yield error terms that are not sharp when $|L_f(x)| = O(1)$ .
Specific Open Problems:
1. Granville-Soundararajan Conjecture: Determining the optimal lower bound for $L_f(x)$ when $f$ is completely multiplicative and takes values in $[-1, 1]$ . Previous bounds were of the form $-(\log \log x)^{-3/5}$ , while examples existed showing values as low as $-C/\log x$ .
2. Random Turán Problem: Estimating the probability that $L_f(x) < 0$ for a Rademacher random multiplicative function.
3. Inverse Problems: Classifying functions where $|L_f(x)|$ is exceptionally small.

2. Methodology and Key Innovations

The authors develop a new, sharp asymptotic formula for $L_f(x)$ that improves upon previous estimates by relating it to the mean value of a related non-negative function $g = 1 * f$ (Dirichlet convolution with the constant function 1).

Core Innovation: The Constant $w_0$
The most significant technical novelty is the introduction of the constant $w_0 = e^{1-\gamma}$ (where $\gamma$ is the Euler-Mascheroni constant). The main term in their asymptotic formula is not $\tilde{M}_g(x)$ (the mean of $g$ up to $x$ ), but rather $\tilde{M}_g(w_0 x)$ .

Derivation: The authors use Perron's formula to express $L_f(x)$ and $\tilde{M}_g(w_0 x)$ as contour integrals involving the Dirichlet series $L(s, f)$ and $L(s, g) = L(s, f)\zeta(s)$ .
Cancellation: By subtracting the two integrals, they isolate the difference:
$L_f(x) - \tilde{M}_g(w_0 x) \propto \int L(s, f) \left( \frac{s w_0^{1-s}}{s-1} - \zeta(s) \right) \dots ds$
Using the Laurent expansion of $\zeta(s)$ near $s=1$ , they show that the term in the parentheses vanishes to first order at $s=1$ specifically when the shift is $w_0$ . This cancellation is crucial for obtaining a sharp error term.

Technical Strategy:

Reduction to Completely Multiplicative: They reduce the general multiplicative case to the completely multiplicative case using a convolution argument, noting that for real-valued $f \in [-1, 1]$ , the function $g = 1 * f$ is non-negative.
"Averages of Averages": They employ a strategy similar to Halász's original proof, analyzing the difference $\Delta(x) = L_f(x) - \tilde{M}_g(w_0 x)$ by averaging over scales.
Plancherel and Majorant Principles: The error terms are bounded using $L^2$ estimates of Dirichlet series on vertical lines, utilizing Plancherel's theorem and the Montgomery-Wirsing majorant principle. The error terms are expressed in terms of exponential sums involving $\cos(t \log p)$ .

3. Main Results

Theorem 1.1 (The Asymptotic Formula)

For a multiplicative function $f: \mathbb{N} \to [-1, 1]$ and $g = 1 * f$ , the authors establish:
$L_f(x) = \left(1 + O\left(\frac{1}{\log x}\right)\right) \tilde{M}_g(w_0 x) + \text{Error Term}$
The error term is explicitly bounded by integrals involving functions $H_1(y)$ and $H_2(y; T)$ , which measure the deviation of $f$ from 1 and its correlation with characters $n^{it}$ .

Significance: This formula is sharp in many cases. Crucially, if $f$ is real-valued and completely multiplicative, $g \ge 0$ , making $\tilde{M}_g(w_0 x)$ a non-negative main term.

Corollary 1.3 (Improved Lower Bound)

For any completely multiplicative $f: \mathbb{N} \to [-1, 1]$ :
$L_f(x) \ge -\frac{c}{(\log x)^{1 - 2/\pi}}$
This significantly improves the 20-year-old bound of Granville and Soundararajan, which was $O((\log \log x)^{-3/5})$ . The exponent $1 - 2/\pi$ is shown to be essentially best possible (Theorem 1.4).

Theorem 2.3 (Random Turán Problem)

For a Rademacher random completely multiplicative function $f$ , the probability that $L_f(x) < 0$ decays exponentially:
$\mathbb{P}(L_f(x) < 0) \ll \exp(-x^c)$
for some $c \in (0, 1)$ . This confirms a previously conjectured bound, improving upon previous results that only gave sub-exponential decay.

Theorem 1.4 (Optimality)

The authors construct examples showing that the error term in their asymptotic formula cannot be improved beyond the order of $(\log x)^{-(1-2/\pi)}$ , confirming the sharpness of their lower bound.

Converse Theorems (Corollaries 2.4 & 2.5)

They provide a classification for functions with small $|L_f(x)|$ . If $|L_f(x)|$ is very small, $f$ must "pretend" to be the Liouville function $\lambda(n)$ (i.e., the pretentious distance $D(f, \lambda; x)$ is small). They construct examples showing this bound is also essentially sharp.

4. Applications and Significance

Resolution of Granville-Soundararajan Problem: The paper settles the question of the lower bound for logarithmic sums of bounded multiplicative functions, moving the exponent from a power of $\log \log x$ to a power of $\log x$ .
Random Multiplicative Functions: The result on the random Turán problem provides a much stronger understanding of the sign changes of random multiplicative functions, showing that negative values are exponentially rare.
Methodological Advancement: The introduction of the shift $w_0 = e^{1-\gamma}$ in the mean value term is a novel technique. It demonstrates that logarithmic means are intrinsically linked to the behavior of the function slightly "beyond" $x$ , a phenomenon not captured by standard Cesàro mean analysis.
Sharpness: Unlike many asymptotic results in this field which are "up to constants," this paper provides precise exponents and constructs examples to prove they are optimal.

5. Conclusion

Klurman and Mangerel have established a powerful new asymptotic framework for logarithmic means of multiplicative functions. By refining the connection between $L_f(x)$ and the mean of its Dirichlet convolution $g$ , and by exploiting the specific constant $w_0$ , they have achieved sharp bounds that resolve long-standing open problems regarding the spectrum of these sums, their behavior in the random setting, and the structure of functions with unusually small sums. The work represents a significant step forward in the "pretentious" approach to multiplicative number theory.