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Difference between revisions of "Dirichlet series"

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In [[mathematics]], a '''Dirichlet series''' is an infinite series whose terms involve successive [[positive integer]]s raised to powers of a variable, typically with integer, real or complex coefficients.  If the series converges, its value determines a function of the variable involved.   
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In [[mathematics]], a '''Dirichlet series''' is an infinite [[series (mathematics)|series]] whose terms involve successive [[positive integer]]s raised to powers of a variable, typically with integer, real or complex coefficients.  If the series converges, its value determines a function of the variable involved.   
  
 
Formally, let ''s'' be a variable and <math>a_n</math> be a sequence of real or complex coefficients.  The associated Dirichlet series is
 
Formally, let ''s'' be a variable and <math>a_n</math> be a sequence of real or complex coefficients.  The associated Dirichlet series is
  
:<math>\sum_{n=1}^\infty a_n n^{-s} . \,</math>.
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:<math>\sum_{n=1}^\infty a_n n^{-s} . \,</math>
  
 
Over the complex numbers the series will have an [[abscissa of convergence]] ''S'', a real number with the property that the series converges for all complex numbers ''s'' with real part <math>\Re s > S</math> and that ''S'' is the "smallest" number with this property ([[infimum]] of all numbers with this property).  If the series converges for all complex numbers ''s'', we formally say that the abscissa of convergence is infinite.
 
Over the complex numbers the series will have an [[abscissa of convergence]] ''S'', a real number with the property that the series converges for all complex numbers ''s'' with real part <math>\Re s > S</math> and that ''S'' is the "smallest" number with this property ([[infimum]] of all numbers with this property).  If the series converges for all complex numbers ''s'', we formally say that the abscissa of convergence is infinite.
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:<math> \left(\sum a_n n^{-s}\right) \cdot \left(\sum b_n n^{-s} \right) = \sum_{n=1}^\infty \left(\sum_{d\vert n}^n a_d b_{n/d}\right) n^{-s} . \, </math>
 
:<math> \left(\sum a_n n^{-s}\right) \cdot \left(\sum b_n n^{-s} \right) = \sum_{n=1}^\infty \left(\sum_{d\vert n}^n a_d b_{n/d}\right) n^{-s} . \, </math>
  
and these purely algebraic definitions are consistent with the values achieved within the region of convergence: the multiplication formula is known as ''[[Dirichlet convolution]]''.
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and these purely algebraic definitions are consistent with the values achieved within the region of convergence: the multiplication formula is known as ''Dirichlet convolution''.
  
 
==Formal Dirichlet series==
 
==Formal Dirichlet series==
 
Let ''R'' be any [[ring (mathematics)|ring]]: an important special case is the ring of integers.  A '''formal power series''' over ''R'', with variable ''S'' is a formal sum <math>\sum a_n n^{-S}</math> with coefficients <math>a_n \in R</math>.  Addition and multiplication are now defined purely formally, with no questions of convergence, by the formulae above for [[pointwise operation|pointwise]] addition and Dirichlet convolution.  The formal Dirichlet series form a ring, which is an ''R''-[[algebra over a ring|algebra]].
 
Let ''R'' be any [[ring (mathematics)|ring]]: an important special case is the ring of integers.  A '''formal power series''' over ''R'', with variable ''S'' is a formal sum <math>\sum a_n n^{-S}</math> with coefficients <math>a_n \in R</math>.  Addition and multiplication are now defined purely formally, with no questions of convergence, by the formulae above for [[pointwise operation|pointwise]] addition and Dirichlet convolution.  The formal Dirichlet series form a ring, which is an ''R''-[[algebra over a ring|algebra]].

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In mathematics, a Dirichlet series is an infinite series whose terms involve successive positive integers raised to powers of a variable, typically with integer, real or complex coefficients. If the series converges, its value determines a function of the variable involved.

Formally, let s be a variable and be a sequence of real or complex coefficients. The associated Dirichlet series is

Over the complex numbers the series will have an abscissa of convergence S, a real number with the property that the series converges for all complex numbers s with real part and that S is the "smallest" number with this property (infimum of all numbers with this property). If the series converges for all complex numbers s, we formally say that the abscissa of convergence is infinite.

For example

converges for all , but diverges for and so has abscissa of convergence 1.

In the half-plane to the right of the abscissa of convergence, a Dirichlet series determines an analytic function of s.

Dirichlet series may be added and multiplied. If and are Dirichlet series, we may define their sum

and product

and these purely algebraic definitions are consistent with the values achieved within the region of convergence: the multiplication formula is known as Dirichlet convolution.

Formal Dirichlet series

Let R be any ring: an important special case is the ring of integers. A formal power series over R, with variable S is a formal sum with coefficients . Addition and multiplication are now defined purely formally, with no questions of convergence, by the formulae above for pointwise addition and Dirichlet convolution. The formal Dirichlet series form a ring, which is an R-algebra.