Analogues of the binary Goldbach problem for integers with an odd number of prime factors

Abstract: The binary Goldbach problem, which asserts that every large enough even integer is a sum of two primes, is known to be intractable via sieve methods alone, due to the ``parity problem''. A few years ago, Shusterman asked the following relaxation of binary Goldbach that nevertheless suffers from the same parity obstruction: can every large enough even integer be expressible as a sum of two integers, both with an odd number of prime factors?

In this talk I will discuss this and other related problems. In particular, I will explain my recent solution to Shusterman's problem, conditional on the GRH for Dirichlet L-functions (or indeed, something much weaker). The proof involves a careful study of symmetries of exponential sums involving the Liouville function that encode combinatorial data, linked together, somewhat surprisingly, by an algorithm of Pierce towards expressing rational numbers as an alternating series of a given type.

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html