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

∑n=1∞anns,displaystyle sum _n=1^infty frac a_nn^s,

where s is complex, and andisplaystyle a_n 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:

DwA(s)=∑a∈A1w(a)s=∑n=1∞annsdisplaystyle mathfrak D_w^A(s)=sum _ain Afrac 1w(a)^s=sum _n=1^infty frac a_nn^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:

DwA⊎B(s)=DwA(s)+DwB(s).displaystyle mathfrak D_w^Auplus B(s)=mathfrak D_w^A(s)+mathfrak D_w^B(s).

Moreover, 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),displaystyle 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:

DwA×B(s)=DuA(s)⋅DvB(s).displaystyle mathfrak D_w^Atimes B(s)=mathfrak D_u^A(s)cdot mathfrak D_v^B(s).

This follows ultimately from the simple fact that n−s⋅m−s=(nm)−s.displaystyle n^-scdot m^-s=(nm)^-s.

Examples

The most famous of Dirichlet series is

ζ(s)=∑n=1∞1ns,displaystyle zeta (s)=sum _n=1^infty frac 1n^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:

ζ(s)=DidN(s)=∏p primeDidpn:n∈N(s)=∏p prime∑n∈NDidpn(s)=∏p prime∑n∈N1(pn)s=∏p prime∑n∈N(1ps)n=∏p prime11−p−s,displaystyle beginalignedzeta (s)&=mathfrak D_operatorname id ^mathbb N (s)=prod _ptext primemathfrak D_operatorname id ^p^n:nin mathbb N (s)=prod _ptext primesum _nin mathbb N mathfrak D_operatorname id ^p^n(s)\&=prod _ptext primesum _nin mathbb N frac 1(p^n)^s=prod _ptext primesum _nin mathbb N left(frac 1p^sright)^n=prod _ptext primefrac 11-p^-sendaligned,

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:

1ζ(s)=∑n=1∞μ(n)nsdisplaystyle frac 1zeta (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

1L(χ,s)=∑n=1∞μ(n)χ(n)nsdisplaystyle frac 1L(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

ζ(s−1)ζ(s)=∑n=1∞φ(n)nsdisplaystyle frac zeta (s-1)zeta (s)=sum _n=1^infty frac varphi (n)n^s

where φdisplaystyle varphi (n) is the totient function,

ζ(s−k)ζ(s)=∑n=1∞Jk(n)nsdisplaystyle frac zeta (s-k)zeta (s)=sum _n=1^infty frac J_k(n)n^s

where J_k is the Jordan function, and

ζ(s)ζ(s−a)=∑n=1∞σa(n)nsζ(s)ζ(s−a)ζ(s−2a)ζ(2s−2a)=∑n=1∞σa(n2)nsζ(s)ζ(s−a)ζ(s−b)ζ(s−a−b)ζ(2s−a−b)=∑n=1∞σa(n)σb(n)nsdisplaystyle beginaligned&zeta (s)zeta (s-a)=sum _n=1^infty frac sigma _a(n)n^s\[6pt]&frac zeta (s)zeta (s-a)zeta (s-2a)zeta (2s-2a)=sum _n=1^infty frac sigma _a(n^2)n^s\[6pt]&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^sendaligned

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

ζ2(s)=∑n=1∞d(n)nsζ3(s)ζ(2s)=∑n=1∞d(n2)nsζ4(s)ζ(2s)=∑n=1∞d(n)2ns.displaystyle beginalignedzeta ^2(s)&=sum _n=1^infty frac d(n)n^s\[6pt]frac zeta ^3(s)zeta (2s)&=sum _n=1^infty frac d(n^2)n^s\[6pt]frac zeta ^4(s)zeta (2s)&=sum _n=1^infty frac d(n)^2n^s.endaligned

The logarithm of the zeta function is given by

log⁡ζ(s)=∑n=2∞Λ(n)log⁡(n)1nsdisplaystyle log zeta (s)=sum _n=2^infty frac Lambda (n)log(n),frac 1n^s

for Re(s) > 1.
Similarly, we have that

−ζ′(s)=∑n=2∞log⁡(n)ns, ℜ(s)>1.displaystyle -zeta ^prime (s)=sum _n=2^infty frac log(n)n^s, Re (s)>1.

Here, Λ(n) is the von Mangoldt function. The logarithmic derivative is then

ζ′(s)ζ(s)=−∑n=1∞Λ(n)ns.displaystyle frac zeta ^prime (s)zeta (s)=-sum _n=1^infty frac Lambda (n)n^s.

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

Given the Liouville function λ(n), one has

ζ(2s)ζ(s)=∑n=1∞λ(n)ns.displaystyle frac zeta (2s)zeta (s)=sum _n=1^infty frac lambda (n)n^s.

Yet another example involves Ramanujan's sum:

σ1−s(m)ζ(s)=∑n=1∞cn(m)ns.displaystyle frac sigma _1-s(m)zeta (s)=sum _n=1^infty frac c_n(m)n^s.

Another pair of examples involves the Möbius function and the prime omega function]:^[1]

ζ(s)ζ(2s)=∑n=1∞|μ(n)|ns≡∑n=1∞μ2(n)ns.displaystyle frac zeta (s)zeta (2s)=sum _n=1^infty frac n^sequiv sum _n=1^infty frac mu ^2(n)n^s.

ζ2(s)ζ(2s)=∑n=1∞2ω(n)ns≡∑n=1∞μ2(n)ns.displaystyle frac zeta ^2(s)zeta (2s)=sum _n=1^infty frac 2^omega (n)n^sequiv sum _n=1^infty frac mu ^2(n)n^s.

Analytic properties of Dirichlet series

Given a sequence a_{nn ∈ N} of complex numbers we try to consider the value of

f(s)=∑n=1∞annsdisplaystyle f(s)=sum _n=1^infty frac a_nn^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_{nn ∈ 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 ∑n=1∞ann−s0displaystyle textstyle sum _n=1^infty a_nn^-s_0 converges for some s0∈C,Re⁡(s0)>0displaystyle textstyle s_0in mathbb C ,operatorname Re (s_0)>0.

• Then A(N)=∑n=1Nan=o(Ns0)displaystyle textstyle A(N)=sum _n=1^Na_n=o(N^s_0). Proof: note that (n+1)s−ns=∫nn+1sxs−1dx=O(ns−1)displaystyle textstyle (n+1)^s-n^s=int _n^n+1sx^s-1,dx=mathcal O(n^s-1). Let B(N)=∑n=1Nanns0=ℓ+o(1)displaystyle textstyle B(N)=sum _n=1^Nfrac a_nn^s_0=ell +o(1) where ℓ=∑n=1∞ann−s0displaystyle ell =sum _n=1^infty a_nn^-s_0, by summation by parts we have

A(N)=∑n=1Nanns0ns0=B(N)Ns0+∑n=1N−1B(n)(ns0−(n+1)s0)=(B(N)−L)Ns0+∑n=1N−1(B(n)−L)(ns0−(n+1)s0)=o(Ns0)+∑n=1N−1o(ns0−1)=o(Ns0)displaystyle beginaligned&A(N)=sum _n=1^Nfrac a_nn^s_0n^s_0=B(N)N^s_0+sum _n=1^N-1B(n)(n^s_0-(n+1)^s_0)\=&(B(N)-L)N^s_0+sum _n=1^N-1(B(n)-L)(n^s_0-(n+1)^s_0)\=&o(N^s_0)+sum _n=1^N-1mathcal o(n^s_0-1)=o(N^s_0)endaligned

• Let L=∑n=1∞andisplaystyle L=sum _n=1^infty a_n if it converges, L=0displaystyle L=0 otherwise. Then the number σ=limsupN→∞ln⁡|A(N)−L|ln⁡N=infσ,A(N)−L=O(Nσ)displaystyle sigma =lim sup _Nto infty frac A(N)-Lln N=inf leftsigma ,A(N)-L=mathcal O(N^sigma )right is called the abscissa of convergence of the Dirichlet series :
  ∑n=1∞ann−sdisplaystyle sum _n=1^infty a_nn^-s converges for Re⁡(s)>σdisplaystyle operatorname Re (s)>sigma and diverges for Re⁡(s)<σdisplaystyle operatorname Re (s)<sigma
  From the definition ∀ε>0displaystyle forall varepsilon >0, A(N)−L=O(Nσ+ε)displaystyle textstyle A(N)-L=mathcal O(N^sigma +varepsilon ) so that

∑n=1Nann−s=A(N)N−s+∑n=1N−1A(n)(n−s−(n+1)−s)=(A(N)−L)N−s+∑n=1N−1(A(n)−L)(n−s−(n+1)−s)=O(Nσ+ε−s)+∑n=1N−1O(nσ+ε−s−1)displaystyle beginalignedsum _n=1^Na_nn^-s&=A(N)N^-s+sum _n=1^N-1A(n)(n^-s-(n+1)^-s)\&=(A(N)-L)N^-s+sum _n=1^N-1(A(n)-L)(n^-s-(n+1)^-s)\&=mathcal O(N^sigma +varepsilon -s)+sum _n=1^N-1mathcal O(n^sigma +varepsilon -s-1)endaligned

which converges as N→∞displaystyle textstyle Nto infty whenever Re⁡(s)>σdisplaystyle textstyle operatorname Re (s)>sigma . Hence, for every sdisplaystyle s such that ∑n=1∞ann−sdisplaystyle sum _n=1^infty a_nn^-s diverges, we have σ≥Re⁡(s)displaystyle sigma geq operatorname Re (s), and this finishes the proof.

• If ∑n=1∞andisplaystyle sum _n=1^infty a_n converges then f(σ+it)=o(1σ)displaystyle f(sigma +it)=oleft(frac 1sigma right) as σ→0+displaystyle sigma to 0^+ and where it is meromorphic f(s)displaystyle f(s) has no poles on Re⁡(s)=0displaystyle operatorname Re (s)=0
  

Proof

Since n−s−(n+1)−s=sn−s−1+O(n−s−2)displaystyle n^-s-(n+1)^-s=sn^-s-1+O(n^-s-2) and A(N)−f(0)→0displaystyle A(N)-f(0)to 0 we have by summation by parts, for Re⁡(s)>0displaystyle operatorname Re (s)>0 :

f(s)=limN→∞∑n=1Nann−s=limN→∞A(N)N−s+∑n=1N−1A(n)(n−s−(n+1)−s)=s∑n=1∞A(n)n−s−1+O(∑n=1∞A(n)n−s−2)⏟=O(1)displaystyle beginalignedf(s)&=lim _Nto infty sum _n=1^Na_nn^-s\&=lim _Nto infty A(N)N^-s+sum _n=1^N-1A(n)(n^-s-(n+1)^-s)\&=ssum _n=1^infty A(n)n^-s-1+underbrace mathcal Oleft(sum _n=1^infty A(n)n^-s-2right) _=mathcal O(1)endaligned

Now find N such that for n > N, |A(n)−f(0)|<ε<varepsilon

s∑n=1∞A(n)n−s−1=sf(0)ζ(s+1)+s∑n=1N(A(n)−f(0))n−s−1⏟=O(1)+s∑n=N+1∞(A(n)−f(0))n−s−1⏞<ε|s|∫N∞x−Re⁡(s)−1dxdisplaystyle beginaligned&ssum _n=1^infty A(n)n^-s-1\=&underbrace sf(0)zeta (s+1)+ssum _n=1^N(A(n)-f(0))n^-s-1 _textstyle =mathcal O(1)\[6pt]&quad +quad overbrace ssum _n=N+1^infty (A(n)-f(0))n^-s-1 ^int _N^infty x^-operatorname Re (s)-1,dxendaligned

and hence, for every ε>0displaystyle varepsilon >0 there is a Cdisplaystyle C such that for σ>0displaystyle sigma >0 : |f(σ+it)|<C+ε|σ+it|1σ<C+varepsilon

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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)=∑n=1∞a(n)n−s displaystyle D(a,s)=sum _n=1^infty a(n)n^-s

with addition and multiplication defined by

D(a,s)+D(b,s)=∑n=1∞(a+b)(n)n−s displaystyle D(a,s)+D(b,s)=sum _n=1^infty (a+b)(n)n^-s

D(a,s)⋅D(b,s)=∑n=1∞(a∗b)(n)n−s displaystyle D(a,s)cdot D(b,s)=sum _n=1^infty (a*b)(n)n^-s

where

(a+b)(n)=a(n)+b(n) displaystyle (a+b)(n)=a(n)+b(n)

is the pointwise sum and

(a∗b)(n)=∑k∣na(k)b(n/k) displaystyle (a*b)(n)=sum _kmid na(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.^[2]

Derivatives

Given

F(s)=∑n=1∞f(n)nsdisplaystyle F(s)=sum _n=1^infty frac f(n)n^s

it is possible to show that

F′(s)=−∑n=1∞f(n)log⁡(n)nsdisplaystyle 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

F′(s)F(s)=−∑n=1∞f(n)Λ(n)nsdisplaystyle 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)=∑n=1∞f(n)n−sdisplaystyle F(s)=sum _n=1^infty f(n)n^-s

and

G(s)=∑n=1∞g(n)n−s.displaystyle 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

12T∫−TTF(a+it)G(b−it)dt=∑n=1∞f(n)g(n)n−a−b as T∼∞.displaystyle frac 12Tint _-T^T,F(a+it)G(b-it),dt=sum _n=1^infty f(n)g(n)n^-a-btext as Tsim infty .

If a = b and ƒ(n) = g(n) we have

12T∫−TT|F(a+it)|2dt=∑n=1∞[f(n)]2n−2a as T∼∞.^2,dt=sum _n=1^infty [f(n)]^2n^-2atext as Tsim infty .

Integral and series transformations

The inverse Mellin transform of a Dirichlet series, divided by s, is given by Perron's formula.
Additionally, if F(z):=∑n≥0fnzndisplaystyle F(z):=sum _ngeq 0f_nz^n is the (formal) ordinary generating function of the sequence of fnn≥0displaystyle f_n_ngeq 0,
then an integral representation for the Dirichlet series of the generating function sequence, fnznn≥0displaystyle f_nz^n_ngeq 0, is given by
^[3]

∑n≥0fnzn(n+1)s=(−1)s−1(s−1)!∫01logs−1⁡(t)F(tz)dt, s≥1.displaystyle sum _ngeq 0frac f_nz^n(n+1)^s=frac (-1)^s-1(s-1)!int _0^1log ^s-1(t)F(tz)dt, sgeq 1.

Another class of related derivative and series-based generating function transformations on the ordinary generating function of a sequence which effectively produces the left-hand-side expansion in the previous equation are respectively defined in.^[4][5]

Relation to power series

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

ζ(s)m=∑n=1∞annsdisplaystyle zeta (s)^m=sum _n=1^infty frac a_nn^s

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

∑n=1∞anxn=x+(m1)∑a=2∞xa+(m2)∑a=2∞∑b=2∞xab+(m3)∑a=2∞∑b=2∞∑c=2∞xabc+(m4)∑a=2∞∑b=2∞∑c=2∞∑d=2∞xabcd+⋯displaystyle beginaligned&sum _n=1^infty a_nx^n\=&x+m choose 1sum _a=2^infty x^a+m choose 2sum _a=2^infty sum _b=2^infty x^ab\&+m choose 3sum _a=2^infty sum limits _b=2^infty sum _c=2^infty x^abc+m choose 4sum _a=2^infty sum _b=2^infty sum limits _c=2^infty sum _d=2^infty x^abcd+cdots endaligned

See also

• General Dirichlet series


• Zeta function regularization


• Euler product

References

• 
  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). Archived from the original on 2011-10-02.<-link dead


• 
  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.