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Formal Dirichlet series

From Encyclopedia of Mathematics - Reading time: 2 min

A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$ L(a,s)=n=1a(n)ns with addition and multiplication defined by L(a,s)+L(b,s)=n=1(a+b)(n)ns L(a,s)L(b,s)=n=1(ab)(n)ns where (a+b)(n)=a(n)+b(n) is the pointwise sum and (ab)(n)=k|na(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 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)=nanns=p(1+apps+ap2p2s+) and is totally multiplicative if the Euler product is of the form L(a,s)=nanns=p(1apps)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

How to Cite This Entry: Formal Dirichlet series (Encyclopedia of Mathematics) | Licensed under CC BY-SA 3.0. Source: https://encyclopediaofmath.org/wiki/Formal_Dirichlet_series
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