Counting numbers in multiplicative sets: Landau versus Ramanujan

Abstract

A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n ≤ x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant γ = 0.57721566 . . .. This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (and his coauthors).

On the sum of integers from some multiplicative sets and some powers of integers

“…The following may also be observed on sums of integers in N k and at most k − 1 powers of 2, k ≥ 3 or sums of integers in N k and at most k powers of 2, k ≥ 3. By Theorem 1 of [2],…”

“…Let M p be the set of positive integers that are norms of ideals of Z[exp(2πi/p)], p ≥ 3 a prime. This set of integers has the property that for a prime q = p, a power of q, q a divides an integer in M p if and only if q a ≡ 1 mod p. The integers in M p that are relatively prime to p all 1 mod p. By Theorem 1 of [2], |{n ∈ M p |0 < n < x}| ≍ x (log x) 1−1/(p−1) . Lemma 5.…”

Errata and Addenda to Mathematical Constants

“…(numerically unchanged, but π is replaced by π 2 ). Moree [185] expressed such constants somewhat differently:…”

On the error term of the logarithm of the lcm of a quadratic sequence

“…The infinite sum in (2) appears in other mathematical contexts: as it is pointed in [7] this sum is closely related to multiplicative sets whose elements are non-hypotenuse numbers (i.e. integers which could not be written as the hypothenuse of a right triangle with integer sides).…”