A correction of some theorems on partitions

Abstract

Theorem 4 in [1] gives a convergent series representation for pa(n), the number of partitions of a positive integer n into positive summands of the form mp ? aj. Here p is an odd prime and aj is an element of a set a consisting of r positive residues of p each of which is less than p/ 2. It is stated that the theorem holds for n > A/12p, where A = rp 26 Y aj(p aj). In the proof of this theorem the estimate 0(n '13k213 +6) is used for certain complicated exponential sums. The proof of this estimate given in Theorem 2 of [1] depends on the fact that (A 12pn, k) = 0(n). This is clearly false (in general) if A = 12pn since (0, k) = k. Thus, Theorem 4 of [1] has been established only if n > Al 12p. Similar remarks apply to Theorem 6 in [2] in which a convergent series is obtained for qa(n), the number of partitions of n into distinct positive summands of the form mp ? aj. Here it is asserted that the theorem holds for n > -Al 12p. However, the proof given is valid only if n > Al 12p. For the argument used to establish the required estimate 0(n'13k213 + C) for the exponential sums involved does not hold if A = 12np. 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/ 12p we must exclude these values of n from consideration. We conclude by giving a simple necessary condition for A =+ 12pn. From the definition of A given above and the fact that either aj or p aj is even we see that if A = + 12pn then 121r.