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Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol

Petr Kucheriaviy

TL;DR

The paper investigates the positivity of partial sums for a random completely multiplicative ±1 function and related weighted sums, establishing a precise y-threshold above which the partial sums up to x stay nonnegative with probability 1−o(1). It develops a saddle-point and smooth-number framework, leveraging the Dickman function and Halász-type theorems, to bound the likelihood of negative partial sums and to bound sums of f(n)/n via high-moment methods and Rankin’s trick. A Kerr–Klurman-type improvement sharpens bounds on the probability that ∑_{n≤x} f(n)/n is negative, with further refinements possible under a smooth Halász conjecture. The paper also connects these random-model results to Legendre symbols χ_p(n) by relating their positivity probabilities to the random-model analogues, while addressing the impact of potential Siegel zeros on the correlation structure. Overall, it provides rigorous probabilistic control over positivity phenomena for random multiplicative functions and links between random models and character sums.

Abstract

Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of $f(n)$ up to $x$ are nonnegative. We prove that for $y \ge C \frac{(\log x)^2 \log_2 x}{\log_3 x}$ this probability equals $1 - o(1)$. We also study the probability $P_x'$ that $\sum_{n \le x} \frac{f(n)}{n}$ is negative. We prove that $P_x' \ll \exp \left( - \exp \left( \frac{\log x \log_4 x}{(1 + o(1)) \log_3 x} \right) \right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that $P_x' \ll \exp(-x^α)$ for some $α> 0$. Let $χ_p(n) = \left( \frac{n}{p} \right)$ be the Legendre symbol modulo $p$. For a prime $p$ chosen uniformly at random from $(x, 2x]$, we express the probability that all partial sums of $\frac{χ_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative.

Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol

TL;DR

The paper investigates the positivity of partial sums for a random completely multiplicative ±1 function and related weighted sums, establishing a precise y-threshold above which the partial sums up to x stay nonnegative with probability 1−o(1). It develops a saddle-point and smooth-number framework, leveraging the Dickman function and Halász-type theorems, to bound the likelihood of negative partial sums and to bound sums of f(n)/n via high-moment methods and Rankin’s trick. A Kerr–Klurman-type improvement sharpens bounds on the probability that ∑_{n≤x} f(n)/n is negative, with further refinements possible under a smooth Halász conjecture. The paper also connects these random-model results to Legendre symbols χ_p(n) by relating their positivity probabilities to the random-model analogues, while addressing the impact of potential Siegel zeros on the correlation structure. Overall, it provides rigorous probabilistic control over positivity phenomena for random multiplicative functions and links between random models and character sums.

Abstract

Let be a random completely multiplicative function such that with probabilities independently at each prime. We study the conditional probability, given that for all , that all partial sums of up to are nonnegative. We prove that for this probability equals . We also study the probability that is negative. We prove that , which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that for some . Let be the Legendre symbol modulo . For a prime chosen uniformly at random from , we express the probability that all partial sums of are nonnegative in terms of the probability that partial sums of are nonnegative.

Paper Structure

This paper contains 16 sections, 27 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Partial sums of $f(n)$
  3. Partial sums of $\frac{f(n)}{n}$
  4. Proof of Theorem \ref{['thm sum f(n)']}
  5. Proof of Lemma \ref{['keylemma1']}
  6. Proof of Theorem \ref{['Kerr and Klurman improvement']} and Theorem \ref{['cor from conj']}
  7. Some discussions
  8. Lower bound for $\sum_{n \le x} g(n)$
  9. Applying Rankin's trick
  10. Upper bound on $\Psi_f(x, y)$
  11. Corollary from Halász's theorem
  12. The end of the proof of Theorem \ref{['Kerr and Klurman improvement']}
  13. Proof of Theorem \ref{['cor from conj']}
  14. Proof of Proposition \ref{['high moment sum f(n) ineq']}
  15. Proof of Theorem \ref{['chi(n)/n']}
  16. ...and 1 more sections

Key Result

Theorem 1

There exist $C > 0, x_0 > 0$ such that for any $x > x_0$ and any we have $\mathop{\mathrm{\mathbb{P}}}\nolimits\left(f \in \mathcal{L}_x^+ \mid f(p) = 1 \, (p \le y)\right) = 1 - o(1)$ as $x$ tends to infinity.

Theorems & Definitions (51)

  • Theorem 1
  • Conjecture 1
  • Corollary 1
  • Theorem 2
  • Conjecture 2
  • Theorem 3
  • Theorem 4
  • Corollary 2
  • Lemma 2.1
  • proof : Deduction of Theorem \ref{['thm sum f(n)']} from Lemma \ref{['keylemma1']}
  • ...and 41 more