A Problem on Partitions With a Prime Modulus p ≧3

On a class of partitions with distinct summands

“…Since W~ (r/2)1/2/! {T(r/2)1/2} (see (11.13) in [1]) we have from Theorem 7 Corollary 7.1. As n -* oo «a(») = 7r(2r/(12np + A^I^Tirll)1'2}^ + 0(exp{ -cn1/2})).…”

“…where St{z) > 0, (h,k) = 1,0 = h< k. Theorem 1 of [1] states that n 9n Fa(exp{2nih / k -2nz j k}) = coa(h, k)exp{n(B j z -Az) / 6pk)…”

A Correction of Some Theorems on Partitions

“…This is clearly false (in general) if A = I2pn since (0, fe) = k. Thus, Theorem 4 of [1] has been established only if n > A\ I2p.…”

“…For the argument used to establish the required estimate 0(n1/3fe2/3 + £) for the exponential sums involved does not hold if A = -I2np. Thus, until (if ever) the necessary estimates contained in Theorems 2 and 3 of [1] and Theorems 2 through 5 of [2] can be justified for n = + A/ I2p we must exclude these values of n from consideration.…”

Rademacher-type formulas for restricted partition and overpartition functions

“…These practitioners included Grosswald [24,25], Haberzetle [26], Hagis [27][28][29][30][31][32][33][34][35], Hua [38], Iseki [39][40][41], Lehner [42], Livingood [43], Niven [51], and Subramanyasastri [63].…”