Description
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Abstract: The Bateman-Horn Conjecture predicts how often an
irreducible polynomial assumes prime values. We will discuss how
with sufficient averaging in the coefficients of the polynomial
(exponential in the size of the inputs), one can not only prove
Bateman-Horn results on average but also pin down precise
information about the distribution of prime values at finite but
growing scales. We will prove that 100% of polynomials satisfy the
appropriate analogue of the Poisson Tail Conjecture, in the sense
that the distribution of the gaps between consecutive prime values
around the average spacing is Poisson.
We will also study the frequencies of sign patterns of the Liouville
function evaluated at the consecutive outputs of f; viewing f as a
random variable, we establish the limiting distribution for every
sign pattern.
A key input behind all of our arguments is Leng's recent
quantitative work on the higher-order Fourier uniformity of the von
Mangoldt and Mobius functions (in turn relying on Leng, Sah, and
Sawhney's quantitative inverse theorem for the Gowers norms).
This talk is based on joint work with Noah Kravitz and Max Xu.
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Andras Biro is inviting you to a scheduled Zoom meeting.
Topic: szeminarium
Time: Nov 17, 2026 02:00 PM Budapest
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