# Difference between revisions of "Formal Dirichlet series"

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

− | + | $$ | |

− | + | (a+b)(n) = a(n)+b(n) | |

− | + | $$ | |

is the [[pointwise operation|pointwise]] sum and | 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 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 $ | ||

The ring of formal Dirichlet series over $\mathbb{C}$ is isomorphic to a ring of formal power series in countably many variables. | 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 $ | + | 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 ) | L(a,s) = \sum_n a_n n^{-s} = \prod_p (1+a_p p^{-s} + a_{p^2} p^{-2s} + \cdots ) | ||

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* 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}} | * 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}} | * 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}} | ||

+ | |||

+ | {{TEX|done}} |

## Latest revision as of 20:24, 2 March 2018

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 $L(\delta,s)=1$) 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 $$ L(a,s) = \sum_n a_n n^{-s} = \prod_p (1+a_p p^{-s} + a_{p^2} p^{-2s} + \cdots ) $$ 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

**How to Cite This Entry:**

Formal Dirichlet series.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Formal_Dirichlet_series&oldid=36940