Namespaces
Variants
Actions

Difference between revisions of "Formal Dirichlet series"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article)
 
(TeX done)
 
Line 1: Line 1:
 
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 $R$
 
+
$$
:<math> L(a,s) = \sum_{n=1}^\infty a(n) n^{-s} \  </math>
+
L(a,s) = \sum_{n=1}^\infty a(n) n^{-s}  
 
+
$$
 
with addition and multiplication defined by
 
with addition and multiplication defined by
 
+
$$
:<math> L(a,s) + L(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s} \  </math>
+
L(a,s) + L(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s}  
:<math> L(a,s) \cdot L(b,s) = \sum_{n=1}^\infty (a*b)(n) n^{-s} \  </math>
+
$$
 
+
$$
 +
L(a,s) \cdot L(b,s) = \sum_{n=1}^\infty (a*b)(n) n^{-s}  
 +
$$
 
where  
 
where  
 
+
$$
:<math> (a+b)(n) = a(n)+b(n) \ </math>
+
(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 $b$.
  
:<math> (a*b)(n) = \sum_{k|n} a(k)b(n/k) \ </math>
+
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 $\Omega$.
 
 
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 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 $D(a,s)$ and a formal [[Euler product]] over primes
+
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 )
Line 35: Line 37:
 
* 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