Related papers: On a Tur\'an conjecture and random multiplicative …

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…

We prove that if $f$ is a random completely multiplicative function, conditional $f(p)=1$ for each prime $p \le (\log x)^{2-\epsilon}$, the probability that $\sum_{1\le n \le N}f(n)\ge 0$ for all $N\le x$ is $o(1)$ as $x \rightarrow…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

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…

Let $\varepsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x \to +\infty$, $$ \sum_{n \leqslant x} f(n) \ll \sqrt{x} (\log_2 x)^{\frac{3}{4}+ \varepsilon}. $$

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\…

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…

Inspired by the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, $\sum_{n\leq…

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…

For completely multiplicative functions f(n) taking values 1 and -1, under certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log log x)sqrt(log x)) times as n runs through the integers <= x.

Let $f$ be a completely multiplicative function that assumes values inside the unit disc. We show that if $\sum_{n<x} f(n) \ll x/(\log x)^A$, $x>2$, for some $A>2$, then either $f(p)$ is small on average or $f$ pretends to be $\mu(n)n^{it}$…

As early as the 1930s, P\'al Erd\H{o}s conjectured that: {\em for any multiplicative function $f:\mathbb{N}\to\{-1,1\}$, the partial sums $\sum_{n\leq x}f(n)$ are unbounded.} Considering this conjecture, in this paper we consider…

Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum_{n \leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. We show that for any constant $c > 2$, $$V_f(x) =…

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N…

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are…

We show that for a random polynomial \[ F(X) = \sum_{n=1}^{N} f(n) X^{n-1}, \] where $f(n)$ is a random completely multiplicative function taking values in $\{\pm 1\}$, one has \[ \limsup_{N \to \infty} \mathbb{P}\big[F(X) \text{ is…

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated…

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum…

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^*$. We prove that the partial sums…

In this paper, we study the probability that some weighted partial sums of a random multiplicative function $f$ are positive. Applying the characteristic decomposition, we obtain that if $S$ is a non-empty subset of the multiplicative…