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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

The celebrated seventy year old, innocent looking problem of D.H. Lehmer [5] asking for composite numbers, if any, satisfying the relation φ(n)|(n − 1), where φ(n) is the Euler totient, is still unsolved. This is easily seen to be equivalent to asking the

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 Q k ( n ) {Q_k}(n) be the number of k-free integers ⩽ n \leqslant n and d ( Q k ) d({Q_k}) the Schnirelmann density of the k-free integers. If k ⩾ 5 k \geqslant 5 , it is shown that Q k ( n ) / n = d ( Q k ) {Q_k}(n)/n = d({Q_k}) for some n satisfying 6 k / 2 ⩽ n > 6 k {6^k}/2 \leqslant n > {6^k} and certain other properties, and that \[ d ( Q k ) ⩾ 1 − 2 − k − 3 − k − 5 − k + ( 3 − k + 2 ⋅ 5 − k ) ( 6 k − 3 k + 1 ) − 1 . d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}. \] d ( Q k ) d({Q_k}) and the n for which Q k ( n ) / n = d ( Q k ) {Q_k}(n)/n = d({Q_k}) are found for 7 ⩽ k ⩽ 12 7 \leqslant k \leqslant 12 .

In 1929, in the course of proving certain results stated by Ramanujan concerning his continued fraction, G. N. Watson proved an identity involving five infinite products and an infinite series. In 1938, Watson proved another identity which again involved five products. Finally in 1961, one more quintuple product identity was established, this time by Basil Gordon. We show here that all these identities are equivalent. Also, with the help of the quintuple product identity and Jacobi’s triple product identity, we establish two new identities involving only series.

This paper generalizes and extends the recent results of George Andrews on Euler pairs. If S 1 {S_1} and S 2 {S_2} are nonempty sets of natural numbers, we define ( S 1 , S 2 ) ({S_1},{S_2}) to be an Euler pair of order r r whenever q r ( S 1 ; n ) = p ( S 2 ; n ) {q_r}({S_1};n) = p({S_2};n) for all natural numbers n n , where q r ( S 1 ; n ) {q_r}({S_1};n) denotes the number of partitions of n n into parts taken from S 1 {S_1} , no part repeated more than r − 1 r - 1 times ( r > 1 ) (r > 1) , and p ( S 2 ; n ) p({S_2};n) the number of partitions of n n into parts taken from S 2 {S_2} . Using a method different from Andrews’, we characterize all such pairs, and consider various applications as well as an extension to vector partitions.

We characterize all multiplicative arithmetical functions f k ( r ) {f_k}(r) such that an identity of the form \[ ∑ r = 1 ∞ f k ( r ) c k ( n , r ) = q k ( n ) g ( k ) , g ( k ) ≠ 0 , \sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,} \] holds for all n, where q k ( n ) {q_k}(n) is the characteristic function of the set of k-free integers and c k ( n , r ) {c_k}(n,r) is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking f k ( r ) = μ ( r ) / J k ( r ) {f_k}(r) = \mu (r)/{J_k}(r) and g ( k ) = ζ ( k ) g(k) = \zeta (k) .

Let n be a positive integer and let T be any nonempty set of positive integers. By (n, T) = 1 we men n is relatively prime to each element of T. Hence n can be written as n = n1n2 where n1 is the largest divisor of n such that (n1, T)= 1. Let P be a property associated with positive integers. We shall say that a positive integer n is a P-number if it satisfies the property P. If in the above representation of n, the integer n1 is a P-number. then we shall say that n is a quasi P-number relative to T, or, simply, a quasi P-number. In particular, for k a positive integer >1, n is quasi k-free for given set T) if n1 is k-free.

Let k and r be fixed integers such that 1 r k . Any positive integer n of the form n = a k b , where b is r -free, is called a ( k, r )-integer. In this paper we prove that if Q k , r ( x ) denotes the number of ( k , r )-integers ≤ x , then Q k,r (x) = xζ(k) ζ(r) + Δ k,r (x) , where Δ k,r (x) = O(x 1 r exp [−B log 3 5 x ( log log x) − 1 5 ]) , B being a positive constant depending on r and the O -estimate is uniform in k . On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ k , r ( x ) and prove that 1 x ∫ 1 α δ k,r (t)dt=0(x 1 k ω(x)) or 0(x 3/(4r+1) ω(x)) , according as k ≤ (4r + 1) 3 or k > (4r + 1) 3 , where ω ( x ) = exp [ B log x (log log x ) −1 ].

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.

Two theorems are given about partitions in which the multiplicity of the parts satisfies certain conditions. One of these theorems generalizes a recent result of Andrews concerning partitions in which a part with an odd multiplicity occurs at least 2 r + 1 2r + 1 times.