Sums over primitive sets with a fixed number of prime factors
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A primitive set is one in which no element of the set divides another. Erdős conjectured that the sum f(A) :=∑n∈A 1/nlogn taken over any primitive set A would be greatest when A is the set of primes. More recently, Banks and Martin have generalized this conjecture to claim that, if we let Nk represent the set of integers with precisely k prime factors(counted with multiplicity), then we have f(N1) > f(N2) > f(N3) > ···. The first of these inequalities was established by Zhang; we establish the second. Our methods involve explicit bounds on the density of integers with precisely k prime factors. In particular, we establish an explicit version of the Hardy-Ramanujan theorem on the density of integers with k prime factors.
Bayless, J., Kinlaw, P., & Klyve, D. (2019). Sums over primitive sets with a fixed number of prime factors. Mathematics of Computation, 88(320), 3063–3077. https://doi.org/10.1090/mcom/3416
Mathematics of Computation
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