View source for 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 operation|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 $L(\delta,s)=1$) 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 arithmetic function|multiplicative]] if and only if there is a formal [[Euler identity]] beween the Dirichlet series $L(a,s)$ 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 function|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}} {{MR|0108510}}
* 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}}

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