Formal Dirichlet series

A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$

\[ L(a,s) = \sum_{n=1}^\infty a(n) n^{-s} \ \]

with addition and multiplication defined by

\[ L(a,s) + L(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s} \ \] \[ L(a,s) \cdot L(b,s) = \sum_{n=1}^\infty (a*b)(n) n^{-s} \ \]

where

\[ (a+b)(n) = a(n)+b(n) \ \]

is the pointwise sum and

\[ (a*b)(n) = \sum_{k|n} a(k)b(n/k) \ \]

is the Dirichlet convolution of a and b.

The formal Dirichlet series form a ring $\Omega$, indeed an $R$-algebra, with the zero function as additive zero element and the function $\delta$ defined by $\delta(1)=1$, $\delta(n)=0$ for $n>1$ (so that $D(\delta,s)=1$) as multiplicative identity. An element of this ring is invertible if $a(1)$ is invertible in $R$. If $R$ is commutative, so is $\Omega$; if $R$ is an integral domain, so is $\Omega$. The non-zero multiplicative functions form a subgroup of the group of units of $\Omega$.

The ring of formal Dirichlet series over $\mathbb{C}$ is isomorphic to a ring of formal power series in countably many variables.

The function $a$ is multiplicative if and only if there is a formal Euler identity beween the Dirichlet series $D(a,s)$ and a formal Euler product over primes $$ L(a,s) = \sum_n a_n n^{-s} = \prod_p (1+a_p p^{-s} + a_{p^2} p^{-2s} + \cdots ) $$ and is totally multiplicative if the Euler product is of the form $$ L(a,s) = \sum_n a_n n^{-s} = \prod_p (1 - a_p p^{-s})^{-1} \ . $$

References

• E.D. Cashwell, C.J. Everett,. "The ring of number-theoretic functions", Pacific J. Math. 9 (1959) 975-985 Zbl 0092.04602 MR0108510
• Henri Cohen, "Number Theory: Volume II: Analytic and Modern Tools", Graduate Texts in Mathematics 240, Springer (2008) ISBN 0-387-49894-X Zbl 1119.11002
• Gérald Tenenbaum, "Introduction to Analytic and Probabilistic Number Theory", Cambridge Studies in Advanced Mathematics 46, Cambridge University Press (1995) ISBN 0-521-41261-7 Zbl 0831.11001