Formal Dirichlet series

A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$ $$L(a,s)=\sum _{n=1}^{\mathrm{\infty }}a(n)n^{-s}$$ with addition and multiplication defined by $$L(a,s)+L(b,s)=\sum _{n=1}^{\mathrm{\infty }}(a+b)(n)n^{-s}$$ $$L(a,s)\cdot L(b,s)=\sum _{n=1}^{\mathrm{\infty }}(a\ast b)(n)n^{-s}$$ where $$(a+b)(n)=a(n)+b(n)$$ is the pointwise sum and $$(a\ast b)(n)=\sum _{k|n}a(k)b(n/k)$$ is the Dirichlet convolution of $a$ and $b$.

The formal Dirichlet series form a ring $\mathrm{\Omega }$, 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 $\mathrm{\Omega }$; if $R$ is an integral domain, so is $\mathrm{\Omega }$. The non-zero multiplicative functions form a subgroup of the group of units of $\mathrm{\Omega }$.

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 $L(a,s)$ 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 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[edit]

• 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