18 problems

Helson's conjecture.

Harper's conjecture. For every $\varepsilon>0$, almost surely

Let $f$ be a random completely multiplicative function taking values in $\\{1,-1\\}$, with the values $f(p)$ chosen independently and uniformly at random over the primes. Let…

Let $f\colon\mathbb N\to\mathbb C$ be a Steinhaus random multiplicative function, meaning that the values $f(p)$ at primes are independent and uniformly distributed on the complex…

Chatterjee's conjecture. The normalized partial sum should not converge in distribution to $\mathcal{CN}(0,1)$ in either the Rademacher or Steinhaus case.

Uniform smooth-number cancellation conjecture. The paper conjectures that its main estimate should hold uniformly for every $2\leqslant y\leqslant x$:

Let $P(n)$ denote the largest prime factor of $n$, let $f$ be a Steinhaus random multiplicative function, and fix $A>0$ and a sufficiently small $\varepsilon>0$. For $T>0$, exclude…

Slow-divergence conjecture. The quantities ${\mathcal B}_{2k,n}$ diverge slowly enough in both $k$ and $n$ that Assumption holds whenever $a_n=o(b_n)$. The conjecture would provide…

Let ${\mathcal P}$ be the set of prime numbers, let $(X_p)_{p\in{\mathcal P}}$ be independent random variables with ${\mathbb P}(X_p=\pm1)=1/2$, and define … when $j$ is square fre…

Helson's conjecture. The expected absolute value of the partial sum should exhibit better-than-square-root cancellation:

Let $\alpha(n)$ be the random completely multiplicative function appearing in the paper, let $\mu$ be the Möbius function, and let $m_{x,\infty}$ be the random measure defined in t…

Let $f\in\mathcal{M}$. Suppose that … for some $\theta\in(0,1)$, and that there exists $c>0$ such that $|f(p^k)|^2=O(2^{k(1-c)})$ and $|f(p)|<1/c$ for all $k\geq 2$ and primes $p$.…

Najnudel's conjecture. For $P(x)=x(x+1)$, and more generally for the displayed polynomial family, this normalized sum should converge in distribution to $\mathcal{CN}(0,1)$.

Let $S_{N,m}:=\sum_{n\in E_{N,m}}z(n)$, where $E_{N,m}:=\{1\le n:\Omega(n)=m\}$ and $z$ is the completely multiplicative Steinhaus random function. Write…

In the setting of Steinhaus chaos, let $c(k)$ denote the constant in the conjectured comparison $\|S_{N,m}\|_{2k}\asymp\|S_{N,m}\|_2$ for $m<\beta\log\log N$ and $\beta<c(k)$. Sixt…

Let $(z(p))_{p\ \text{prime}}$ be Steinhaus random variables, extended completely multiplicatively to $\boldsymbol{N}$. For $E_{N,m}:=\{1\le n:\Omega(n)=m\}$, set…

Fractional-moment conjecture. For $0\leq k<1$ and $0\leq\sigma<1/2$,

Random-multiplicative-function moment conjecture. As $N\rightarrow\infty$,