Partitions into Segal–Piatetski–Shapiro sequences

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