3 problems

Let $d_k(n)$ be the number of ways to write $n$ as a product of $k$ factors, and let … Define $\beta_k$ as the infimum of the numbers $b_k$ for which … The generalized divisor prob…

Let $d_k(n)$ be the number of ways to write $n$ as a product of $k$ factors, and let … where $P_{k-1}$ is a polynomial of degree $k-1$. Define $\alpha_k$ and $\beta_k$ as the infim…

Let $R_0(x):=\sum_{n\leq x}\frac{\varphi(n)}{n}-\frac{x}{\zeta(2)}$ and, for an interval of size $H$, let … Let $H=\Theta(x^\delta)$ for some fixed $0<\delta\leq 1$. Variance conje…