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Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Πp|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers.

The asymptotic expansion of ]]> ]]> ]]> \sum\limits_{n\le x}e^*(n)$ is given, where $e^*(n)$ is defined by $\sum{\frac{{e^*(n)}}{{n^s}}}\eg \prod\limits_{n=2}^\infty (1-1/n^s)$.

We give the asymptotics of the sum \(\sum\nolimits_{x \leqslant n \leqslant x + h} {f(n)}\), where \(h \gg x^{7/12 + \varepsilon }\), for the multiplicative functions \(f(n) = z^{\omega (n)} , z^{\omega (n)} |\mu (n)|, 1/d_k (n)\), and \(nd(n)\).

(2002). A Modified Problem of Pillai and Some Related Questions. The American Mathematical Monthly: Vol. 109, No. 6, pp. 554-559.

the following assertion is proved. If f is a completely multiplicative function taking values of modulus 1, such that the set e of limit points of \(f(n + 1)\bar f(n)\) has k ≨ 3 distinct values, then f(n) = nirF(n) with some r e R, and Fk(n) = 1 for every n.

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 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 .

Let k be a fixed integer ⩾ 1 \geqslant 1 and define τ k ( n ) = Σ d k / n 1 {\tau _k}(n) = {\Sigma _{{d^k}/n}}1 . Thus τ 1 ( n ) {\tau _{1}}(n) is the ordinary divisor function and τ k ( n ) {\tau _k}(n) is the number of kth powers dividing n. We derive the asymptotic behaviour as n → ∞ n \to \infty of P k ( n ) {P_k}(n) defined by \[ ∑ n = 0 ∞ P k ( n ) x n = ∏ n = 1 ∞ ( 1 − x n ) − τ k ( n ) . \sum \limits _{n = 0}^\infty {{P_k}(n){x^n} = \prod \limits _{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .} \] Thus P k ( n ) {P_k}(n) is the number of partitions of n where we recognize τ k ( m ) {\tau _k}(m) different colours of the integer m when it occurs as a summand in a partition. The case k = 2 k = 2 is of special interest since the number f ( n ) f(n) of semisimple rings with n elements when n = q 1 l 1 q 2 l 2 … n = q_1^{{l_1}}q_2^{{l_2}} \ldots is given by f ( n ) = P 2 ( l 1 ) P 2 ( l 2 ) … f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots .

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.

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.

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 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.

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.

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) .

(1965). The Brauer-Rademacher Identity. The American Mathematical Monthly: Vol. 72, No. 2, pp. 135-138.

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.

P(a,b,c) = 2 r~a 2 k~b Σ ι~cr=\ k=\ l=\ Then P(a, b, c) + P(a, c, b) + P(b, c, a) + P(b, a, c)