Almost sure upper bound for random multiplicative functions

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong...

We consider the random functions $S_N(z):=\sum_{n=1}^N z(n) $, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that ${\Bbb E} |S_N|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $({\Bbb E} |S_N|^q)^{1/q}\gg_{q} \sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(\theta) \big|^{2k}\, d\theta $$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not pe...

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x, x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x/(\log x)^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x, x+H]$ wit...

The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self-contained, and p...

We provide a simple proof that the partial sums $\sum_{n\leq x}f(n)$ of a Rademacher random multiplicative function $f$ change sign infinitely often as $x\to\infty$, almost surely.

We investigate when the better than square-root cancellation phenomenon exists for $\sum_{n\le N}a(n)f(n)$, where $a(n)\in \mathbb{C}$ and $f(n)$ is a random multiplicative function. We focus on the case where $a(n)$ is the indicator function of $R$ rough numbers. We prove that $\log \log R \asymp (\log \log x)^{\frac{1}{2}}$ is the threshold for the better than square-root cancellation phenomenon to disappear.

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in {\mathcal A}} f(n)$, where ${...

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $\alpha f$. Our result applies to $d_z$, the $z$-th divisor function, a...

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by $\sum_{x<n \leqslant x+y} f(n)$, exhibits the phenomenon of \textit{better than square-root cancellation}. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio $\log\big(\frac{x}{y}\big)$ is around $\sqrt{\log\log x}$. By modeling characters by random multiplicative functions, we give a sharp bound ...