Let @math denote the number of monic irreducible polynomials in @math of degree @math . We show that for a fixed degree @math , the sequence @math converges @math -adically to an explicitly determined rational function @math . Furthermore we show that the limit @math is related to the classic necklace polynomial @math by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity
The results in @cite_11 are expressed in terms of expected values of factorization statistics. In this paper we focus on first moments as they lead to a cleaner statement for Theorem . The only difference between expected values and first moments of factorization statistics is whether or not one divides by the total mass'' of the space of polynomials considered. This difference is simply a factor of @math for general univariate polynomials, but is more subtle for squarefree polynomials and multivariate polynomials as it affects the family of characters determining the coefficients. The equivalence between Theorem (2) and [Thm. 2.5] Hyde follows from [Prop. 4.2] Hyde Lagarias . Alternatively, Theorem (2) appears as stated in [Prop. 4.1] CEF .
@cite_11
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We use generating functions to relate the expected values of polynomial factorization statistics over @math to the cohomology of ordered configurations in @math as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.