Dirichlet series

In mathematics, a Dirichlet series is any series of the form

\sum _{{n=1}}^{{\infty }}{\frac {a_{n}}{n^{s}}},

where s is complex, and a is a complex sequence. It is a special case of general Dirichlet series.

Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.

Combinatorial importance

Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.

Suppose that A is a set with a function w: A → N assigning a weight to each of the elements of A, and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement (A,w) a weighted set.) Suppose additionally that a_n is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:

{\mathfrak {D}}_{w}^{A}(s)=\sum _{{a\in A}}{\frac {1}{w(a)^{s}}}=\sum _{{n=1}}^{{\infty }}{\frac {a_{n}}{n^{s}}}

Note that if A and B are disjoint subsets of some weighted set (U, w), then the Dirichlet series for their (disjoint) union is equal to the sum of their Dirichlet series:

{\mathfrak {D}}_{w}^{{A\uplus B}}(s)={\mathfrak {D}}_{w}^{{A}}(s)+{\mathfrak {D}}_{w}^{{B}}(s).

Moreover, and perhaps a bit more interestingly, if (A, u) and (B, v) are two weighted sets, and we define a weight function w: A × B → N by

w(a,b)=u(a)v(b),

for all a in A and b in B, then we have the following decomposition for the Dirichlet series of the Cartesian product:

{\mathfrak {D}}_{w}^{{A\times B}}(s)={\mathfrak {D}}_{u}^{{A}}(s)\cdot {\mathfrak {D}}_{v}^{{B}}(s).

This follows ultimately from the simple fact that $n^{{-s}}\cdot m^{{-s}}=(nm)^{{-s}}.$

Examples

The most famous of Dirichlet series is

\zeta (s)=\sum _{{n=1}}^{{\infty }}{\frac {1}{n^{s}}},

which is the Riemann zeta function.

Treating these as formal Dirichlet series for the time being in order to be able to ignore matters of convergence, note that we have:

{\begin{aligned}\zeta (s)&={\mathfrak {D}}_{{{\mathrm {id}}}}^{{{\mathbb {N}}}}(s)=\prod _{{p\,{\mathrm {prime}}}}{\mathfrak {D}}_{{{\mathrm {id}}}}^{{\{p^{n}:n\in {\mathbb {N}}\}}}(s)=\prod _{{p\,{\mathrm {prime}}}}\sum _{{n\in {\mathbb {N}}}}{\mathfrak {D}}_{{{\mathrm {id}}}}^{{\{p^{n}\}}}(s)\\&=\prod _{{p\,{\mathrm {prime}}}}\sum _{{n\in {\mathbb {N}}}}{\frac {1}{(p^{n})^{s}}}=\prod _{{p\,{\mathrm {prime}}}}\sum _{{n\in {\mathbb {N}}}}\left({\frac {1}{p^{s}}}\right)^{n}=\prod _{{p\,{\mathrm {prime}}}}{\frac {1}{1-p^{{-s}}}}\end{aligned}},

as each natural number has a unique multiplicative decomposition into powers of primes. It is this bit of combinatorics which inspires the Euler product formula.

Another is:

{\frac {1}{\zeta (s)}}=\sum _{{n=1}}^{{\infty }}{\frac {\mu (n)}{n^{s}}}

where μ(n) is the Möbius function. This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character χ(n) one has

{\frac {1}{L(\chi ,s)}}=\sum _{{n=1}}^{{\infty }}{\frac {\mu (n)\chi (n)}{n^{s}}}

where L(χ, s) is a Dirichlet L-function.

Other identities include

{\frac {\zeta (s-1)}{\zeta (s)}}=\sum _{{n=1}}^{{\infty }}{\frac {\varphi (n)}{n^{s}}}

where $\varphi$ (n) is the totient function,

{\frac {\zeta (s-k)}{\zeta (s)}}=\sum _{{n=1}}^{{\infty }}{\frac {J_{k}(n)}{n^{s}}}

where J_k is the Jordan function, and

\zeta (s)\zeta (s-a)=\sum _{{n=1}}^{{\infty }}{\frac {\sigma _{{a}}(n)}{n^{s}}}

{\frac {\zeta (s)\zeta (s-a)\zeta (s-2a)}{\zeta (2s-2a)}}=\sum _{{n=1}}^{\infty }{\frac {\sigma _{a}(n^{2})}{n^{s}}}

{\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}}=\sum _{{n=1}}^{{\infty }}{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}

where σ_a(n) is the divisor function. By specialisation to the divisor function d=σ₀ we have

\zeta ^{2}(s)=\sum _{{n=1}}^{{\infty }}{\frac {d(n)}{n^{s}}}

{\frac {\zeta ^{3}(s)}{\zeta (2s)}}=\sum _{{n=1}}^{{\infty }}{\frac {d(n^{2})}{n^{s}}}

{\frac {\zeta ^{4}(s)}{\zeta (2s)}}=\sum _{{n=1}}^{{\infty }}{\frac {d(n)^{2}}{n^{s}}}.

The logarithm of the zeta function is given by

\log \zeta (s)=\sum _{{n=2}}^{\infty }{\frac {\Lambda (n)}{\log(n)}}\,{\frac {1}{n^{s}}}

for Re(s) > 1. Here, Λ(n) is the von Mangoldt function. The logarithmic derivative is then

{\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-\sum _{{n=1}}^{\infty }{\frac {\Lambda (n)}{n^{s}}}.

These last two are special cases of a more general relationship for derivatives of Dirichlet series, given below.

Given the Liouville function λ(n), one has

{\frac {\zeta (2s)}{\zeta (s)}}=\sum _{{n=1}}^{\infty }{\frac {\lambda (n)}{n^{s}}}.

Yet another example involves Ramanujan's sum:

{\frac {\sigma _{{1-s}}(m)}{\zeta (s)}}=\sum _{{n=1}}^{\infty }{\frac {c_{n}(m)}{n^{s}}}.

Another example involves the Möbius function:

{\frac {\zeta (s)}{\zeta (2s)}}=\sum _{{n=1}}^{\infty }{\frac {|\mu (n)|}{n^{s}}}\equiv \sum _{{n=1}}^{\infty }{\frac {\mu ^{2}(n)}{n^{s}}}.

Analytic properties of Dirichlet series

Given a sequence {a_n}_{n ∈ N} of complex numbers we try to consider the value of

f(s)=\sum _{{n=1}}^{\infty }{\frac {a_{n}}{n^{s}}}

as a function of the complex variable s. In order for this to make sense, we need to consider the convergence properties of the above infinite series:

If {a_n}_{n ∈ N} is a bounded sequence of complex numbers, then the corresponding Dirichlet series f converges absolutely on the open half-plane of s such that Re(s) > 1. In general, if a_n = O(n^k), the series converges absolutely in the half plane Re(s) > k + 1.

If the set of sums a_n + a_{n + 1} + ... + a_{n + k} is bounded for n and k ≥ 0, then the above infinite series converges on the open half-plane of s such that Re(s) > 0.

In both cases f is an analytic function on the corresponding open half plane.

In general the abscissa of convergence of a Dirichlet series is the intercept on the real axis of the vertical line in the complex plane such that there is convergence to the right of it, and divergence to the left. This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes.

In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.

Abscissa of convergence

Assume that $\textstyle \sum _{n=1}^{\infty }a_{n}n^{-s_{0}}$ converges for some $\textstyle s_{0}\in \mathbb {C} ,Re(s_{0})>0$ .

Then $\textstyle A(N)=\sum _{n=1}^{N}a_{n}=o(N^{s_{0}})$ . Proof : note that $\textstyle (n+1)^{s}-n^{s}=\int _{n}^{n+1}sx^{s-1}dx={\mathcal {O}}(n^{s-1})$ . Let $\textstyle B(N)=\sum _{n=1}^{N}{\frac {a_{n}}{n^{s_{0}}}}=L+o(1)$ where $L=\sum _{n=1}^{\infty }a_{n}n^{-s_{0}}$ , by summation by parts we have $A(N)=\textstyle \sum _{n=1}^{N}{\frac {a_{n}}{n^{s_{0}}}}n^{s_{0}}=B(N)N^{s_{0}}+\sum _{n=1}^{N-1}B(n)(n^{s_{0}}-(n+1)^{s_{0}})$ $=o(N^{s_{0}})+\sum _{n=1}^{N-1}{\mathcal {O}}(n^{s_{0}-1})=o(n^{s_{0}})$ .

If $\sum _{n=1}^{\infty }a_{n}$ diverges, then the number $\sigma =\lim \sup _{N\to \infty }{\frac {\ln |A(N)|}{\ln N}}=\inf \left\{\sigma ,A(N)={\mathcal {O}}(N^{\sigma })\right\}$ is called the abscissa of convergence of the Dirichlet series : $\sum _{n=1}^{\infty }a_{n}n^{-s}$ converges for $Re(s)>\sigma$ and diverges for $Re(s)<\sigma$ From the definition $\forall \epsilon >0,A(N)={\mathcal {O}}(N^{\sigma +\epsilon })$ , so that $\textstyle \sum _{n=1}^{N}a_{n}n^{-s}=A(N)N^{-s}+\sum _{n=1}^{N-1}A(n)(n^{-s}-(n+1)^{-s})={\mathcal {O}}(N^{\sigma +\epsilon -s})+\sum _{n=1}^{N-1}{\mathcal {O}}(n^{\sigma +\epsilon -s-1})$ which converges as $\textstyle N\to \infty$ whenever $\textstyle Re(s)>\sigma$ . Hence, for every $s$ such that $\sum _{n=1}^{\infty }a_{n}n^{-s}$ diverges, we have $\sigma \geq Re(s)$ , and this finishes the proof. Hardy (1914). "the general theory of dirichlet series" (PDF).

Formal Dirichlet series

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

D(a,s)=\sum _{{n=1}}^{\infty }a(n)n^{{-s}}\

with addition and multiplication defined by

D(a,s)+D(b,s)=\sum _{{n=1}}^{\infty }(a+b)(n)n^{{-s}}\

D(a,s)\cdot D(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 Ω, indeed an R-algebra, with the zero function as additive zero element and the function δ defined by δ(1)=1, δ(n)=0 for n>1 as multiplicative identity. An element of this ring is invertible if a(1) is invertible in R. If R is commutative, so is Ω; if R is an integral domain, so is Ω. The non-zero multiplicative functions form a subgroup of the group of units of Ω.

The ring of formal Dirichlet series over C is isomorphic to a ring of formal power series in countably many variables.^[1]

Derivatives

Given

F(s)=\sum _{{n=1}}^{\infty }{\frac {f(n)}{n^{s}}}

it is possible to show that

F'(s)=-\sum _{{n=1}}^{\infty }{\frac {f(n)\log(n)}{n^{s}}}

assuming the right hand side converges. For a completely multiplicative function ƒ(n), and assuming the series converges for Re(s) > σ₀, then one has that

{\frac {F^{\prime }(s)}{F(s)}}=-\sum _{{n=1}}^{\infty }{\frac {f(n)\Lambda (n)}{n^{s}}}

converges for Re(s) > σ₀. Here, Λ(n) is the von Mangoldt function.

Products

Suppose

F(s)=\sum _{{n=1}}^{{\infty }}f(n)n^{{-s}}

and

G(s)=\sum _{{n=1}}^{{\infty }}g(n)n^{{-s}}.

If both F(s) and G(s) are absolutely convergent for s > a and s > b then we have

{\frac {1}{2T}}\int _{{-T}}^{{T}}\,F(a+it)G(b-it)\,dt=\sum _{{n=1}}^{{\infty }}f(n)g(n)n^{{-a-b}}{\text{ as }}T\sim \infty .

If a = b and ƒ(n) = g(n) we have

{\frac {1}{2T}}\int _{{-T}}^{{T}}|F(a+it)|^{{2}}dt=\sum _{{n=1}}^{{\infty }}[f(n)]^{{2}}n^{{-2a}}{\text{ as }}T\sim \infty .

Integral transforms

The Mellin transform of a Dirichlet series is given by Perron's formula.

Relation to power series

The sequence a_n generated by a Dirichlet series generating function corresponding to:

\zeta (s)^{m}=\sum _{{n=1}}^{{\infty }}{\frac {a_{n}}{n^{s}}}

where ζ(s) is the Riemann zeta function, has the ordinary generating function:

\sum \limits _{{n=1}}^{{\infty }}a_{n}x^{n}=x+{m \choose 1}\sum \limits _{{a=2}}^{{\infty }}x^{{a}}+{m \choose 2}\sum \limits _{{a=2}}^{{\infty }}\sum \limits _{{b=2}}^{{\infty }}x^{{ab}}+{m \choose 3}\sum \limits _{{a=2}}^{{\infty }}\sum \limits _{{b=2}}^{{\infty }}\sum \limits _{{c=2}}^{{\infty }}x^{{abc}}+{m \choose 4}\sum \limits _{{a=2}}^{{\infty }}\sum \limits _{{b=2}}^{{\infty }}\sum \limits _{{c=2}}^{{\infty }}\sum \limits _{{d=2}}^{{\infty }}x^{{abcd}}+...

References

↑ Cashwell, E.D.; Everett, C.J. (1959). "The ring of number-theoretic functions". Pacific J. Math. 9: 975–985. doi:10.2140/pjm.1959.9.975. ISSN 0030-8730. MR 0108510. Zbl 0092.04602.

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Hardy, G.H.; Riesz, Marcel (1915). The general theory of Dirichlet's series. Cambridge Tracts in Mathematics. 18. Cambridge University Press.
The general theory of Dirichlet's series by G. H. Hardy. Cornell University Library Historical Math Monographs. {Reprinted by} Cornell University Library Digital Collections
Gould, Henry W.; Shonhiwa, Temba (2008). "A catalogue of interesting Dirichlet series". Miss. J. Math. Sci. 20 (1).
Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT].
Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge Studies in Advanced Mathematics. 46. Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.
"Dirichlet series". PlanetMath.

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