We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\varepsilon>0$, almost surely $|\sum_{n\le x}f(n)|\ll_{\varepsilon,f}\sqrt{x}(\log\log x)^{1/4+\varepsilon}$. Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.
Let $f$ be a Steinhaus or Rademacher random multiplicative function. We use methods from the theory of critical chaos to improve on the best known upper bound for partial sums of random multiplicative functions. In particular, our results imply that for any $\varepsilon>0$, almost surely $$ \Big|\sum_{n\le x}f(n)\Big|...
Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation h...
Let $M_n$ be an $n\times n$ matrix with independent uniform sign entries. We prove that there exist absolute constants $C,c>0$ such that, for all sufficiently large $n$, \[ \mathbb{P}\!\left( \left|\operatorname{Per}(M_n)\right| \ge e^{-Cn}\sqrt{n!} \right) \ge 1-n^{-c}. \] This establishes the exponential scale lower...
We prove that the bound \[ \max \{ |16A|,|16f(A)| \} \gg_m |A|^{\frac{3}{2}+\frac{1}{162}} \] holds for any polynomial $f$ with degree $m \geq 2$ and any finite $A \subset \mathbb R$. This shows that the classical Jarn\'{i}k obstruction to growth beyond exponent $3/2$, which occurs for general strictly convex functions...
We establish an asymptotic formula for squarefree polynomials over a finite field whose coefficients are in $\{0,1\}$. This resolves a function-field analogue of a conjecture of Erd\H{o}s, Mauduit and S\'ark\"ozy. More generally, we estimate the probability of a polynomial $\sum \varepsilon_it^i$ being squarefree, wher...
For each real $b$, exact upper and lower bounds on the probability $\mathsf P(X\ge b)$ over all random variables $X$ with log-concave p.d.f.'s such that $\mathsf E X=0$ and $\mathsf E X^2=1$ are obtained, as well as the best constant factor $C$ in the inequality $\mathsf P(X\ge b)\le C e^{-b}$ for all real $b\ge0$. Exp...
I. Pinelis· 0 citations
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