Large fluctuations of extended Rademacher random multiplicative functions

Introduction

Main result

A Rademacher random completely multiplicative function $f : \mathbb{N} \to\{-1,1\}$ is obtained by taking $(f(p))_p$ to be independent Rademacher random variables and extending $f$ completely multiplicatively. Throughout the paper, we refer to this model as the extended Rademacher RMF. The more familiar Rademacher RMF is supported only on squarefree integers, and it goes back to Wintner [32], who introduced it as a probabilistic model for the Möbius function. Although it is clear nowadays that the Rademacher RMF does not capture much behavior of Möbius function, we believe that the extended Rademacher RMF is a good model to predict the typical behavior of Dirichlet quadratic characters. Write $f(p)=\epsilon_p$ and

In this paper, our purpose is to prove a quantitative two-sided fluctuation result for $S_f(x)$. One can distinguish this model from the Rademacher RMF $g(n)$ supported on squarefree integers, by noting that $g(n)=\mu^2(n)f(n)$, where $\mu$ is the Möbius function, so that $g(n)=0$ unless $n$ is squarefree. The distinction is substantial. The variables $g(n)$, indexed by squarefree $n$, has perfect orthogonality, whereas $\mathbb{E}f(n)$ equals $1$ when $n$ is a square and equals $0$ otherwise; in particular, $\mathbb{E}S_f(x)=\lfloor\sqrt{x}\rfloor$. The two Dirichlet series are related by

Thus the completely multiplicative property introduces a zeta factor with a pole at $s=1/2$. This pole is the analytic expression of the contribution from squares and can be a principal obstruction to transferring some results for $g$ directly to $f$.

The study of the large fluctuations of the partial sums of RMFs has been very active. In the Rademacher case, Wintner [32] and Erdős [12] obtained some of the earliest results in

2020 Mathematics Subject Classification. Primary 11K65; Secondary 11N37, 60F15.

Key words and phrases: extended Rademacher random multiplicative functions, partial sums, multiplicative chaos, large fluctuations. this direction, and Halász [15] later obtained an important almost sure $\Omega$-result. Harper [17] subsequently improved Halász’s result by means of Gaussian comparison. His later work proved that, for every deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which

He also proved the corresponding result for the Steinhaus RMF; see [20]. This was the first almost sure lower bound for the partial sums which exceeds $\sqrt{x}$ by a factor tending to infinity. The proof conditions on the values at the smaller primes and applies a multivariate Gaussian approximation to the contribution from the remaining large primes. Its main difficulty is to show that sufficiently many of the resulting conditional covariances are small. Although the theorem is stated in the absolute value form (1.2), in the Rademacher case the conditional symmetry of the large prime vector gives the corresponding large fluctuations of either sign.

Harper further suggested that the exponent $1/4$ in (1.2) should be optimal. More precisely, the conjectured upper bound is that, for every $\varepsilon> 0$, almost surely

One heuristic for this exponent comes from the low moments of the partial sums. The results of Harper [19] show that their typical size is smaller than $\sqrt{x}$ by a factor of order $(\log\log x)^{1/4}$. Combining this with the usual $\sqrt{\log\log x}$ correction suggested by the law of the iterated logarithm leads precisely to the scale $\sqrt{x}(\log\log x)^{1/4}$. The relevant low and high moment estimates for the underlying random Euler products are developed in [18, 19].

We give a quick summary of the substantial progress on the upper bounds: see, for example, [7, 9, 15, 26, 27] and the references therein. Very recently, Durkan and Pearce-Crump [10] obtained the matching upper bound

almost surely.

They proved the same result for the Steinhaus RMF. Together with Harper’s lower bound, this determines the optimal iterated logarithmic exponent in both classical RMFs and settles Harper’s conjecture. In an independent work, Verreault [31] obtained the sharper estimate

almost surely,

for both the Rademacher and Steinhaus RMFs, where $\log_k$ denotes the $k$-fold iterated logarithm. We also note that in the weighted case, Atherfold [2] proved that almost surely

and obtained the sharper exponent $1/4+\varepsilon$ under the restriction $P^{+}(n)>\sqrt{x}$. He conjectured that this exponent is optimal. He also showed that almost surely

for extended Rademacher RMFs. See also the work by Hardy on the Steinhaus case [16].

It is in general the case that the three standard RMFs may exhibit different levels of difficulty and the Steinhaus case is usually the least technical among the three. So far, we in general know less in the Rademacher case, while the extended Rademacher RMF is arguably even more challenging. This is indeed the case in the fluctuation problem considered here. As we described before, large fluctuations in both the Rademacher and Steinhaus RMFs have been known since the work of Harper [20], whereas no corresponding result was previously available for the extended Rademacher RMF. In fact, even the weaker original question of Erdős remained open¹. The formulation in the 1999 problem collection Some of Paul’s favorite problems asks whether $\limsup_{x\to\infty} |S_f(x)|/\sqrt{x}=+\infty$ almost surely, while the current online formulation, recorded as Erdős Problem #1144, asks the stronger one-sided question whether $\limsup_{x\to\infty} S_f(x)/\sqrt{x}=+\infty$ almost surely [8, 30]. Our main theorem gives a much stronger quantitative two-sided strengthening of both formulations.

Theorem 1.1. Let $f$ be an extended Rademacher RMF. Then for any fixed deterministic function $V(x)\to\infty$, we have, almost surely

Here $A(x)=\Omega_{\pm}(B(x))$ means that $\limsup_{x\to\infty} A(x)/B(x)>0$ and $\liminf_{x\to\infty} A(x)/B(x)<0$ for an eventually positive function $B(x)$.

In particular, the positive limsup assertion implies $\limsup_{x\to\infty} S_f(x)/\sqrt{x}=+\infty$, answering Erdős Problem #1144 affirmatively in the stronger form.

Remark 1.2. It is worth pointing out that our argument may work for a more general model than the extended Rademacher RMF considered here. What is essentially used is the Euler product factorisation

where $F_X(s)=\prod_{p\leq X}(1+\varepsilon_p p^{-s})$ is the squarefree Rademacher Euler product, $Z_X(s)=\prod_{p\leq X}(1-p^{-s})^{-1}$ and $R_X$ is a “well-behaved” factor (in our case $R_X(s)\equiv1$). Under some suitable moment and regularity assumptions on $R_X$, the same argument applies more generally. For instance, the $m$-free extended Rademacher RMF has factorisation

For $m\geq3$ the last product is regular on $\Re s=1/2$, whereas for $m=2$ it cancels $Z_X(2s)$ completely, giving the usual Rademacher RMF. Actually, by combining Harper [20]’s original argument with our conditional symmetry observation used in Section 8, one can obtain the Rademacher RMF analogues of Theorems 1.1 and 1.4.

Another related family is obtained by taking $f(p)$ uniformly among the $k$-th roots of unity and extending completely multiplicatively. Its Euler product decomposition contains the deterministic factor $Z_X(ks)$. Thus $k=2$ is the critical case in which the associated zeta singularity occurs at $s=1/2$, while for $k\geq3$ it lies to the left of the critical line; as $k\to\infty$, the uniform distribution on the $k$-th roots of unity converges weakly to the uniform distribution on the unit circle, corresponding to the Steinhaus RMF. We do not pursue the details of these extensions here.

Remark 1.3. We also remark that our bound should be sharp up to $(\log\log x)^{o(1)}$. One might be able to prove the corresponding upper bound by combining some techniques here and arguments of Caich [9] with the recent developments [10, 31].

It is worth noting that, very recently, Atherfold, Gerspach and Hamdan [3] announced their results on low moments of extended Rademacher RMFs. They show that the typical size can behave very differently with the additional completely multiplicative property, compared with the usual Rademacher case. While our result shows that its influence on large fluctuations is more limited.

Following the general strategy of Harper [20], we first prove a conditional fluctuation estimate for $X^{8/7} \le x \le X^{4/3}$. For $X \ge3$ let

Thus $\mathcal{F}_X$ records the small prime variables, while conditional on $\mathcal{F}_X$ the variables $(f(p))_{p>X}$ remain independent Rademacher variables. Using the conditional Borel–Cantelli lemma (see Lemma 2.2 below), Theorem 1.1 follows from the following result.

Theorem 1.4. Let $f$ be an extended Rademacher RMF. There is an absolute constant $W_0 > 0$ such that for all sufficiently large $X$ and all $W_0 \le W \le\frac{1}{30}\log\log\log X$, we have