Almost sure bounds for weighted sums of Rademacher random multiplicative functions

If $f: \mathbb{N} \rightarrow \{\pm1\}$ is a sample of the random completely multiplicative function, we show that almost surely $\sum_{n \le x} \frac{f(n)}{\sqrt{n}}$ changes signs infinitely many times, answering a question of Aymone.

We show that if $f$ is the random completely multiplicative function, the probability that $\sum_{n\le x}\frac{f(n)}{n}$ is positive for every $x$ is at least $1-10^{-45}$, while also strictly smaller than $1$. For large $x$, we prove an asymptotic upper bound of $O(\exp(-\exp( \frac{\log x}{C\log \log x })))$ on the exceptional probability that a particular truncation is negative, where $C$ is some positive constant.

We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We...

We consider a sequence $\{f(p)\}_{p\ {\rm prime}}$ of independent random variables taking values $\pm 1$ with probability $1/2$, and extend $f$ to a multiplicative arithmetic function defined on the squarefree integers. We investigate upper bounds for $\Psi_f(x,y)$, the summatory function of $f$ on $y$-friable integers $\leq x$. We obtain estimations of the type $\Psi_f(x,y) \ll \Psi(x,y)^{1/2+\epsilon}$, more precise formulas being given in suitable regions for $x,y$. In the...

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random variables with $\mathbb{E} f(p)<0$, $\forall p\in \mathcal{P}$. The function $f$ is called strongly biased (towards classical M\"obius function), if $\sum_{p\in\mathcal{P}}\frac{f(p)}{p}=-\infty$ a.s., and it is weakly biased if $\sum_{p\in\mathcal...

Tur\'an observed that logarithmic partial sums $\sum_{n\le x}\frac{f(n)}{n}$ of completely multiplicative functions (in the particular case of the Liouville function $f(n)=\lambda(n)$) tend to be positive. We develop a general approach to prove two results aiming to explain this phenomena. Firstly, we show that for every $\varepsilon>0$ there exists some $x_0\ge 1,$ such that for any completely multiplicative function $f$ satisfying $-1\le f(n)\le 1$, we have $$\sum_{n\le x}\...

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...

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 ${...

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\varepsilon>0$ we show that, with high probability, $$ \exp( (\log N)^{1/2-\varepsilon} ) \ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp( (\log N)^{1/2+\varepsilon}). $$

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums associated to their products. More precisely, we let $f_j$, $j=1,2$ be such that $|f_j(n)|\leq \tau(n)^\kappa$ for some $\kappa\in\mathbb{N}$, and assume their partial sums satisfy $\left|S(x_j,f_j)\right|\geq \eta x_j (\log x_j)^{2^\kappa-1}$ for s...