Partitions into Segal–Piatetski–Shapiro sequences

Let κ be any positive real number and m∈ℕ∪{∞} be given. Let p_κ , m(n) denote the number of partitions of n into the parts from the Segal–Piatestki–Shapiro sequence (⌊ℓ ^κ⌋ )_ℓ∈ℕ with at most m possible repetitions. In this paper, we establish some asymptotic formulas of Hardy–Ramanujan type for p_κ , m(n) . As a necessary step in the proof, we prove that the Dirichlet series ζ _κ (s)=∑ _n≥ 1⌊ n^κ⌋ ^-s can be continued analytically beyond the imaginary axis except for simple poles at s=1/κ -j, (0≤ j< 1/κ , j∈ℤ) .