Horizontal moments of Kloosterman sums and a density version of sieves

Denote by $\mathrm{K}\mathrm{l}(a,n)$ the normalized Kloosterman sum modulo $n$. In this paper, we study the moments of $\mathrm{\mid }\mathrm{K}\mathrm{l}(1,n)\mathrm{\mid }$ on average over $n⩽X$, and show that \begin{align*} \sum_{n\leqslant X}|$\mathrm{K}\mathrm{l}$(1,n)|\asymp X(\log X)^{\frac{8}{3π}-1}. \end{align*} The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato–Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato–Tate distribution.