Expansion, divisibility and parity: an explanation

After seeing how questions on the finer distribution of prime factorization -- considered inaccessible until recently -- reduce to bounding the norm of an operator defined on a graph describing factorization, we will show how to bound that norm. In essence, the graph is a strong local expander, with all eigenvalues bounded by a constant factor times the theoretical minimum (i.e., the eigenvalue bound corresponding to Ramanujan graphs). The proof will take us on a walk from graph theory to linear algebra and the geometry of numbers, and back to graph theory, aided, along the way, by a generalized sieve. This is an expository paper; the full proof has appeared as a joint preprint with M. Radziwi\{l}\{l}.

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