Title: Secondary terms in the counting function of quartic fields
Abstract: A classical question in number theory is: How many degree-n fields exist with discriminant less than X? Asymptotics for this count N_n(X) are known for n=2, n=3 (by work of Davenport--Heilbronn) and for n=4,5 (by work of Bhargava). For n=3, it has been proven (independently by Bhargava--Shankar--Tsimerman and Taniguchi--Thorne) that N_3(X) also has a second main term, and so it is possible to write N_3(X)=c_3X+c_3'X^{5/6} +o(X^{5/6}).
In this talk, I will discuss joint work with Jacob Tsimerman, in which we prove the existence of secondary terms for smoothed counts of quartic fields.
Alexander Smith
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