Let the Laurent expansion of the Riemann zeta function ξ(s) about s=1 be written in the formIt has been discovered independently by many authors that, in terms of this notation, the coefficient
Let 6,[X] denote a polynomial ring in an indeterminate X over a finite field IF, . Exact formulae are derived for (i) the number of polynomials of degree n in IF,[X] with a specified number k of zeros in F, , and (ii) the average number of zeros and corresponding variance for a polynomial of degree n in F, [X]. The main emphasis is on the case when mulriplicity of zeros is counted.
The object of this note is to state certain theorems, whose proofs together with related results will appear elsewhere. The theorems are mainly concerned with asymptotic enumeration of the isomorphism classes of finite rings or finite-dimensional algebras lying in various naturally-defined categories. Two results concern enumeration of subalgebras or subrings in a given algebra or ring. All the rings and algebras are associative, but need not necessarily have units.
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