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Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

On the set F of complex-valued arithmetic functions we construct an infinite family of convolutions, that is, binary operations ψ of the formso that (F, +, ψ) is a commutative ring, for which the unity is unbounded. Here + denotes pointwise addition.

We obtain an asymptotic formula for the number of (k, r)-free integers that do not exceed x. By definition, a (k, r)-free integer is one in whose canonical representation no prime power is in the interval [r, k−1] where 1 < r < k are fixed integers. These include as special cases the r-free integers, the semi r-free integers and the k-full integers. We obtain an asymptotic formula for the number of representations of an integer as the sum of a prime and a (k, r)-free integer, and use the result to prove that every sufficiently large integer can be represented as the sum of a prime and m = abk where a and b are both square free, (a, b) = 1, b > 1 and k is any fixed integer, k ≥ 3.

We give three new partition theorems of the classical Rogers-Ramanujan type which are very much in the style of MacMahon. These are a continuation of four theorems of the same kind given recently by the second author. One of these new theorems, very similar to one of the original Rogers-Ramanuj an - MacMahon type theorems is as follows: Let C(n) denote the number of partitions of n into parts congruent to ±2, ± 3, ±4, ± 5, ±6, ±7 (mod 20). Let D(n) denote the number of partitions of n of the form n = b1 + b2 + … + bt, where bt ≧ 2, bt ≧ bi + 1 and if 1 ≦ i ≦ [(t - 2)/2], bi - bi + 1 ≧ 2. Then C(n) = D(n).

Let d*(n) denote the number of unitary divisors of the positive integer n. For x > 1, let B(x) denote the number of integers n for which nd*(n) ≦ x. Balasubramanian and Ramachandra proved that there exists a positive constant β such that . In this note we give an explicit expression for β as an infinite product, namely where the product is over all primes p.

Let l and k be positive integers. Then for each integer n ≥ 1, define d(n; l, k) to be the number of (positive) divisors of n which lie in the arithmetic progression I mod k. Note that d(n;1,1) = d(n), the ordinary divisor function.

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also letWe recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.

Let σ*(N) denote the sum of the unitary divisors of N, that is,It is easily seen that σ*(N) is multiplicative. In fact σ*(l) = 1 if N > 1 has the prime decomposition . Let us define a positive integer to be unitary perfect whenever σ*(N) = 2N. The first four such numbers are 6, 60, 90 and 87, 360. In a recent abstract [ l ] published by one of us, the last of these numbers was overlooked. No other unitary perfect numbers are known to the authors.

This note proves (in the theorem below) a conjecture made by the author last year through the pages of the Departmental Problem Book. This arose in connection with some other investigations of arithmetic functions.

In a recent paper, Murray Eden [5] generalized the simple identity for the Eulerian product,1.1and obtained the following infinite family of identities:For A= 1,2, 3,…, let1.2where we assume throughout that |x| < 1, empty products equal unity and empty sums equal zero; then1.3As Eden noted, Fh(b;x) is the generating function of ph(m, n) which denotes the number of partitions of n into m parts, in which the largest part appears exactly h times and all other parts are distinct

There is considerable literature concerning the century old result that for arbitrary positive integers a and m, 1.1 where μ(m) is the usual Mobius function. For earlier work on this we refer to L.E. Dickson [4, pp. 84–86] and L. Carlitz [1,2]. Another reference not noted by the above authors is R. Vaidyanathaswamy [6], who noted that the left member of (1.1) represents the number of special fixed points of the m th power of a rational transformation of the n th degree.

A combinatorial proof is given for Uchimura’s identityAs a corollary to this proof we derive a formula for the sum of the nth powers of the divisors of m in terms of partitions of m.

In the first part of this note we give a very simple and elegant proof of the theorem on the order of the error function of the (k, r)-integers, which the authors proved earlier using elaborate calculations. We also obtain an improvement of an earlier result on the order of the same error function on the basis of the Riemann hypothesis.

Arithmetic sums of the formwhere f is an arithmetic function and [ ] is the greatest integer function frequently occur in various situations in the theory of numbers and have much of interest in their own right. Two instances appear in the well-known results.

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

Given a sequence of r distinct real numbers such that the number of terms of every decreasing subsequence is at most m, then there exists an increasing subsequence of more than n terms, where n is the largest integer less than r/m.An extremely simple and elegant proof of the theorem was given by A. Seidenberg [2]. This note is intended to point out that a result analogous to the above holds under a more general setting.