Reflections around Euler's theorem on the sum of reciprocals of primes

Abstract

We revisit the influential and historical method of Euler, based on Euler products, that proves the divergence of the sum of reciprocals of primes. Our key innovation rests on the adaptation of Euler products with the help of suitable truncations, and thereby not introducing too much of an error along the way. We then give a totally discrete derivation of the Euler product identity over any multiplicative set that avoids the justification of infinite processes such as Cauchy products and rearrangement of absolutely convergent series. Armed with this approach, we study various aspects of the interaction between a prime set P and its multiplicative offspring N P . Euler's theorem is extended to all prime sets P, where N P satisfies a mild growth condition on its size. Then using normalized values of the sum of divisors function with arguments coming from N P , we show how Diophantine approximation type results are available for convergent harmonic series over N P . Here, in principle, P can be taken as the set of all primes with a twin or more generally as the set of all primes arising from the Hardy-Littlewood prime tuples conjecture. Exploiting the multiplicative structure of the sum of divisors function, we offer an analog of Gronwall's inequality for prime sets possessing relative natural density and the Mertens-Meissel constant. Lastly, we discuss how lower bounds on the size of N P in certain ranges imply the existence of a considerable amount of primes in P together with limitations and converses of some of our deductions. As a bonus, we discover a particular case of this interaction that stays undecided, and also show a dichotomy in the behavior of N P and N Q for any two complementary sets of primes P and Q.