Almost sure bounds for weighted sums of Rademacher random multiplicative functions

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic formula for the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ under this assumption. We study the sum $\displaystyle\sum_{n\leq x}2^{\omega(n)}$ unconditionally too.

Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length o...

The paper considers estimates for the asymptotics of summation functions of bounded multiplicative arithmetic functions. Several assertions on this subject are proved and examples are considered.

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...

We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} \chi(n)$ typically exhibit "better than squareroot cance...

Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.

Let $d_k(n) = \sum_{n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{N} \to \mathbb{C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in \mathbb{N}$. In this paper we improve Mangerel's results which extend the Matom\"aki-Radziwi{\l\l} theorem to divisor bounded multiplicative functions. In particular, we prove that for sufficiently large $X \geq 2$, any $\epsilon>0$ and $h \geq ...

The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic functions. Several assertions on this topic are proved and examples are considered.

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...

Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The Legendre path attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to \infty$, these paths converge...