Bounds for exponential sums with random multiplicative coefficients

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...

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 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 $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to\infty$, almost surely. The case $\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a ...

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...

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...

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.

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville function this improves on the lower bound $\gg X^{c/\log\log X}$ due to Balog and Perelli (1998). For the M\"{o}bius function this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa (2001). The large discrepancy between these lower bo...

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 $c > 2$, $$V_f(x) = \Omega ((\log \log \log x)^{1/c} )$$ almost surely.

Let $f$ be a real polynomial with irrational leading co-efficient. In this article, we derive distribution of $f(n)$ modulo one for all $n$ with at least three divisors and also we study distribution of $f(n)$ for all square-free $n$ with at least two prime factors. We study exponential sums when weighted by divisor functions and exponential sums over square free numbers. In particular, we are interested in evaluating \begin{align*} \sum_{n\leq N}\tau(n)e\left(f(n)\right) ~\t...

We prove that for the Steinhaus Random Variable $z(n)$ \[\mathbb{E}\left(\left|\sum_{n\in E_{N, m}}z(n)\right|^6\right)\asymp |E_{N, m}|^3 \text{ for } m\ll(\log\log N)^{\frac{1}{3}},\] where \[E_{N, m}:=\{1\leq n:\Omega(n)=m\}\] and $\Omega(n)$ denotes the number of prime factors of $N$.