Dirichlet series with integer coefficients as a UFD

Question

I recall the following question from Ulam's book "Unsolved math problems": show that the ring of Dirichlet series with integer coefficients is a factorial ring. I believe that soon after Ulam wrote his book, this problem was solved; essentially, this ring is isomorphic to the ring of formal power series in infinitely many variables (one for each prime) with integer coefficients, and once it is reformulated this way it looks much less mysterious. However, my actual question is not about the proof, - I was always curious if a result like that (in its original formulation) can be really applied or interpreted from the Dirichlet series point of view. Has anyone heard/thought of any reasonable number-theoretic interpretation of this fact, or does the interpretation as a power series ring kill any hope of that sort?

Answer 1

First, I want to nail down a reference to the solution to the problem alluded to above: the Dirichlet ring of functions f: Z⁺ -> Z with pointwise addition and convolution product is a UFD. This was proved by L. Durst in his 1961 master's thesis at Rice University and independently by Cashwell and Everett. References:

Deckard, Don; Durst, L. K. Unique factorization in power series rings and semigroups. Pacific J. Math. 16 1966 239--242.

Cashwell, E. D.; Everett, C. J. Formal power series. Pacific J. Math. 13 1963 45--64.

Note that the latter is the second paper of Cashwell and Everett on the subject of factorization in Dirichlet rings. They also had a 1959 paper:

Cashwell, E. D.; Everett, C. J. The ring of number-theoretic functions. Pacific J. Math. 9 1959 975--985.

Ulam makes reference to the 1959 paper in his statement of the problem in the first (1960) edition of his book. In the 1964 paperback edition of his book, he announces the problem as solved by 1961 papers of Buchsbaum and Samuel.

I want to make the remark that since the original 1959 paper of Cashwell and Everett certainly makes the connection to formal power series rings, there is more to the solution of the problem than this. (Indeed, the 1961 result of Samuel, that if R is a regular UFD, then R[[t]] is again a UFD, seems to be the key.)

There is a 2001 arxiv preprint of Durst which claims a more elementary proof of the result:

http://arxiv.org/PS_cache/math/pdf/0105/0105219v1.pdf

As to whether the result has number-theoretic significance: not as far as I know. I should say that I first learned of the Cashwell-Everett theorem by reading an analytic number theory text -- Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, p. 26 -- but the author seems to mention it just for culture.

Finally, remember that this is a result about factorization in the ring of formal Dirichlet series, an inherently algebraic beast. So perhaps the following result is more relevant:

Bayart, Frédéric; Mouze, Augustin Factorialité de l'anneau des séries de Dirichlet analytiques. (French) [The ring of analytic Dirichlet series is factorial] C. R. Math. Acad. Sci. Paris 336 (2003), no. 3, 213--218

Answer 2

I believe so. Dirichlet series are a special case of a construction called the reduced incidence algebra of a poset, and the poset of integers under division is precisely a product of chains, one for each prime. Finally, the reduced incidence algebra of a chain is just the ring of power series in one variable.

Answer 3

Factoring arithmetic functions in the ring $\text{Arith}_{\mathbb{Z}}$ of all arithmetic functions over $\mathbb{Z}$ into irreducibles is unlikely to have much use as far as their analytic properties are concerned. For example, any integer-valued multiplicative function is a unit in $\text{Arith}_{\mathbb{Z}}$, and one can say very little analytically about all multiplicative functions, no less all units $f$ in $\text{Arith}_{\mathbb{Z}}$, which need only satisfy $f(1) = \pm 1$. Moreover, any integer-valued function $f$ satisfying $f(1) = 0$ and $f(p) = 1$ for at least one prime $p$ (e.g., the prime counting function $f(n) = \pi(n)$) is irreducible in $\text{Arith}_{\mathbb{Z}}$, and, again, there is little one can say analytically about such a large class functions. (For example, since there are uncountably many irreducibles in $\text{Arith}_{\mathbb{Z}}$, there is no obvious analogue of the prime number theorem that might hold.) Irreducible factorizations of arithmetic functions in $\text{Arith}_{\mathbb{Z}}$ are just too coarse of a decomposition to have any analytic applications. So, any application of the theorem of unique factorization in $\text{Arith}_{\mathbb{Z}}$ would likely have to be algebraic, rather than analytic, in nature. Part of the reason that the result has found little application algebraically is that classifying the irreducibles of $\mathbb{Z}[[X]]$, no less $\mathbb{Z}[[X_2,X_3,X_5,\ldots]] \cong \text{Arith}_{\mathbb{Z}}$, is not easy. On the other hand, unique factorizations of multiplicative functions (which are all units in the ring of arithmetic functions) as infinite Euler products is quite useful, both algebraically and analytically. These are interpreted as infinite sums in the lambda ring $\text{Mult}_{\mathbb{Z}}$ of all multiplicative functions over $\mathbb{Z}$, which is isomorphic to $\prod_p \Lambda(\mathbb{Z})$, where $\Lambda$ is the universal lambda ring functor.