The factorial function n! has a Dirichlet series representation through the Riemann zeta function. Extending this to number fields requires replacing integers with ideals, n! with a generalized factorial (the product of norms of ideals up to a bound), and the Riemann zeta function with the Dedekind zeta function.
The generalized Legendre factorial of a number field K at a bound x is the product of N(a)^floor(x/N(a)) over all nonzero ideals a of the ring of integers O_K, where N(a) is the ideal norm. This reduces to the ordinary factorial when K = Q.
The authors establish the Dirichlet series for these generalized factorials, with asymptotic error terms of order O(n · exp(-c · sqrt(log n))). The error term matches the best known bounds on the prime number theorem in number fields — the factorial's asymptotics are controlled by the same zero-free region of the Dedekind zeta function that controls the prime ideal theorem.
The connection is not accidental. The logarithm of the generalized factorial is a sum over prime ideals, weighted by the Legendre function (floor(x/p^k) summed over k). This sum is a smoothed version of the prime-counting function, and its asymptotics inherit the analytic structure of the Dedekind zeta function.
The factorial as a Dirichlet series. The error as a zero-free region. Classical number theory — the interplay between multiplicative structure and analytic continuation — extended to its natural generality.