7 problems

Let $k\geq 2$ be an integer, let $\chi$ be a real non-principal Dirichlet character modulo $q$, and let $f=\mu^{(k)}\chi$ or $f=\mu^{(k)}g_\chi$. Then the large-values limsup conje…

Let $\nu_k([V,\infty))$ denote the upper tail of the limiting distribution associated with the summatory function considered in the paper. Meng's Large Deviation Conjecture. The ri…

Let $k\geq 2$ be an integer, let $\mu^{(k)}$ denote the indicator function of the $k$-free integers, and let $h:\mathbb{N}\to\{-1,1\}$ be multiplicative. Aymone's conjecture. For e…

Let $k\geq 2$ be an integer, let $\chi$ be a real Dirichlet character, let $g_\chi:\mathbb{N}\to\{-1,1\}$ be the associated modified completely multiplicative character, and let…

The short-interval k-free integer conjecture. Uniformly for $x^{\epsilon}\leq H\leq x$,

The k-free integer error-term conjecture. For every fixed $\epsilon>0$,

Conjecture on $k$-free character sums. For every $\epsilon>0$, these partial sums satisfy