Partitions into Segal–Piatetski–Shapiro sequences

Abstract

Let \(\kappa \) be any positive real number and \(m\in \mathbb {N}\cup \{\infty \}\) be given. Let \(p_{\kappa , m}(n)\) denote the number of partitions of n into the parts from the Segal–Piatestki–Shapiro sequence \((\lfloor \ell ^{\kappa }\rfloor )_{\ell \in \mathbb {N}}\) with at most m possible repetitions. In this paper, we establish some asymptotic formulas of Hardy–Ramanujan type for \(p_{\kappa , m}(n)\). As a necessary step in the proof, we prove that the Dirichlet series \(\zeta _\kappa (s)=\sum _{n\ge 1}\lfloor n^{\kappa }\rfloor ^{-s}\) can be continued analytically beyond the imaginary axis except for simple poles at \(s=1/\kappa -j, ~(0\le j< 1/\kappa , j\in \mathbb {Z})\).