Multivariate mutual information

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In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of

n=1aneλns,

where an, s are complex numbers and {λn} is a strictly increasing sequence of positive numbers that tends to infinity.

A simple observation shows that an 'ordinary' Dirichlet series

n=1anns,

is obtained by substituting λn=logn while a power series

n=1an(es)n,

is obtained when λn=n.

Fundamental theorems

If a Dirichlet series is convergent at s0=σ0+t0i, then it is uniformly convergent in the domain

|arg(ss0)|θ<π2,

and convergent for any s=σ+ti where σ>σ0.

There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of s. In the latter case, there exist a σc such that the series is convergent for σ>σc and divergent for σ<σc. By convention, σc= if the series converges nowhere and σc= if the series converges everywhere on the complex plane.

Abscissa of convergence

The abscissa of convergence of a Dirichlet series can be defined as σc above. Another equivalent definition is

σc=inf{σ:n=1aneλns converges for any s where Re(s)>σ}.

The line σ=σc is called the line of convergence. The half-plane of convergence is defined as

σc={s:Re(s)>σc}.

The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series.

On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical line, then this line must be the line of convergence. The proof is implicit in the definition of abscissa of convergence. An example would be the series

n=11nens,

which converges at s=πi (alternating harmonic series) and diverges at s=0 (harmonic series). Thus, σ=0 is the line of convergence.

Suppose that a Dirichlet series does not converge at s=0, then it is clear that σc0 and an diverges. On the other hand, if a Dirichlet series converges at s=0, then σc0 and an converges. Thus, there are two formulas to compute σc, depending on the convergence of an which can be determined by various convergence tests. These formulas are similar to the Cauchy-Hadamard theorem for the radius of convergence of a power series.

If ak is divergent, i.e. σc0, then σc is given by

σc=lim supnlog|a1+a2++an|λn.

If ak is convergent, i.e. σc0, then σc is given by

σc=lim supnlog|an+1+an+2+|λn.

Abscissa of absolute convergence

A Dirichlet series is absolutely convergent if the series

n=1|aneλns|,

is convergent. As usual, an absolutely convergent Dirichlet series is convergent, but the converse is not always true.

If a Dirichlet series is absolutely convergent at s0, then it is absolutely convergent for all s where Re(s)>Re(s0). A Dirichlet series may converge absolutely for all, for no or for some values of s. In the latter case, there exist a σa such that the series converges absolutely for σ>σa and converges non-absolutely for σ<σa.

The abscissa of absolute convergence can be defined as σa above, or equivalently as

σc=inf{σ:n=1aneλns converges absolutely for any s where Re(s)>σ}.

The line and half-plane of absolute convergence can be defined similarly. There are also two formulas to compute σa.

If |ak| is divergent, then σa is given by

σa=lim supnlog(|a1|+|a2|++|an|)λn.

If |ak| is convergent, then σa is given by

σa=lim supnlog(|an+1|+|an+2|+)λn.

In general, the abscissa of convergence does not coincide with abscissa of absolute convergence. Thus, there might be a strip between the line of convergence and absolute convergence where a Dirichlet series is conditionally convergent. The width of this strip is given by

0σaσcL:=lim supnlognλn.

In the case where L= 0, then

σc=σa=lim supnlog|an|λn.

All the formulas provided so far still hold true for 'ordinary' Dirichlet series by substituting λn=logn.

Analytic functions

A function represented by a Dirichlet series

f(s)=n=1aneλns,

is analytic on the half-plane of convergence. Moreover, for k=1,2,3,...

f(k)(s)=(1)kn=1anλnkeλns.

Further generalizations

A Dirichlet series can be further generalized to the multi-variable case where λnk, k = 2, 3, 4,..., or complex variable case where λnm, m = 1, 2, 3,...

References

  • G. H. Hardy, and M. Riesz, The general theory of Dirichlet's series, Cambridge University Press, first edition, 1915.
  • E. C. Titchmarsh, The theory of functions, Oxford University Press, second edition, 1939.
  • Tom Apostol, Modular functions and Dirichlet series in number theory, Springer, second edition, 1990.
  • A.F. Leont'ev, Entire functions and series of exponentials (in Russian), Nauka, first edition, 1982.
  • A.I. Markushevich, Theory of functions of a complex variables (translated from Russian), Chelsea Publishing Company, second edition, 1977.
  • J.-P. Serre, A Course in Arithmetic, Springer-Verlag, fifth edition, 1973.

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