Quick viewing(Text Mode)

Dirichlet Series

Dirichlet Series

Dirichlet

Fabian Gundlach January 25, 2019

Goal The goal of these notes is to briefly introduce the reader to the beautiful subject of . In particular, we will answer some questions of the following form: given a (number theoretic) a1, a2,... of nonnegative integers, what is the asymptotic behavior of nďN an as N goes to infinity? More precisely, we want to find a “simple” F pNq so that nďN an „ F pNq, by which we mean ř ř an lim nďN “ 1. NÑ8 F pNq ř Here are just a few of the many nice results that can be shown using Dirichlet series:

• nďN dn „ N log N if dn denotes the number of divisors of n. 2 • řnďN σn „ CN for some constant C “ 0.8224 ... if σn denotes the sum of the divisors of n. ř ρ • nďN ∆n „ CN for some constants C “ 0.3181 ... and ρ “ 1.728 ... if ∆n denotes the number of ordered factorizations of n into any number of řfactors bigger than 1.

N • nďN Pn „ log N if Pn is 1 whenever n is a and 0 otherwise. (This is the famous prime number .) ř A familiarity with basic facts of (holomorphic functions, meromorphic functions, poles) is assumed in the later parts of these notes (the analytic statements).

Dirichlet series In combinatorics and additive , it is often useful to associate to a sequence a0, a1,... of complex numbers the ordinary 8 n F pa, Xq “ n“0 anX , which is viewed as a formal . This means that we define the usual operations (addition, multiplication, ř

1 , . . . ) on power series in the obvious way even if the power series don’t actually converge:

8 8 8 n n n F pa, Xq ` F pb, Xq “ anX ` bnX :“ pan ` bnqX “ F pa ` b, Xq n“0 n“0 n“0 ÿ8 ÿ 8 ÿ 8 n n n F pa, Xq ¨ F pb, Xq “ anX ¨ bnX :“ axby X “ F pa ˚ b, Xq ˜n“0 ¸ ˜n“0 ¸ n“0 x,yě0 s.t. ÿ ÿ ÿ ` x`ÿy“n ˘ d d 8 8 8 F pa, Xq “ a Xn :“ na Xn´1 “ pn ` 1qa Xn “ F pa1,Xq dX dX n n n`1 n“0 n“0 n“0 ÿ ÿ ÿ In multiplicative number theory, it is more useful to associate to a sequence 8 ´s a1, a2,... the Dirichlet series Dpa, sq “ n“1 ann . Again, we define the usual operations on Dirichlet series in the obvious way: ř

Dpa, sq ` Dpb, sq “ Dpa ` b, sq where pa ` bqn “ an ` bn

Dpa, sq ¨ Dpb, sq “ Dpa ˚ b, sq where pa ˚ bqn “ axby x,yě1 s.t. x¨ÿy“n d Dpa, sq “ Dpa1, sq where a1 “ ´a logpnq ds n n The addition and multiplication of Dirichlet series gives them the structure of a ring. The multiplicative identity is

1 “ Dpδ, sq where δ “ p1, 0, 0,... q.

A Dirichlet series Dpa, sq is invertible if and only if a1 ‰ 0. Let us look at another very simple sequence. Its Dirichlet series is called the zeta function:

8 1 “ p1, 1,... q ÝÑ Dp1, sq “ n´s “ ζpsq n“1 ÿ

2 Despite the stupidity of this sequence, its Dirichlet series is very useful. Here are some examples of more interesting number-theoretic . Note how each of the corresponding Dirichlet series can be written just in terms of the :

d “ 1 ˚ 1 dn “ 1 ¨ 1 xy“n , ÝÑ Dpd, sq “ Dp1 ˚ 1, sq “ Dp1, sq2 “ ζpsq2 “ #tpx, yq : xy “ nu ř / “ #(pos.) divisors of n. dpkq 1 1 / “ ˚ ¨ ¨ ¨ ˚ -/ k times pkq k k pkq , ÝÑ Dpd , sq “ Dp1 ˚ ¨ ¨ ¨ ˚ 1, sq “ Dp1, sq “ ζpsq dn “ 1 ¨ ¨ ¨ 1 loooomoooonx1¨¨¨xk“n / “ # px1,...,xkq: . ř x1¨¨¨xk“n / 8 8 id “ p1, 2, 3,...( q/ s s 1 - ÝÑ Dpid, sq “ n ¨ n´ “ n´p ´ q “ ζps ´ 1q idn “ n n“1 n“1 * ÿ ÿ σ “ id ˚ 1 σn “ x ¨ 1 ÝÑ Dpσ, sq “ Dpid, sqDp1, sq “ ζps ´ 1qζpsq xy“n , “ divisors of n ř . ř Complex analysis Let’s- now compare the complex analysis of power series and Dirichlet series.

Power series Dirichlet series

The points X in the The points s in the complex plane where a power series F pa, Xq “ where a Dirichlet series Dpa, sq “ n ´s n anX converges are separated from n ann converges are separated the points where it doesn’t by a circle from the points where it doesn’t by a ř 1 ř 2 t| ¨ | “ rcu centered at the origin: vertical line t= “ σcu: • inside the circle, the power series • to the right of the line, the converges ( ); Dirichlet series converges; • outside the circle, the power se- • to the left of the line, the Dirich- ries doesn’t converge ( ); let series doesn’t converge; • it might converge at some points • it might converge at some points on the circle ( ). on the line.

A power series is holomorphic ( 1) ev- A Dirichlet series is holomorphic ev- erywhere inside its radius of conver- erywhere to the right of its abscissa of gence. convergence.

1 In general, the rc could be 0, meaning that the power series con- verges only at X “ 0, or could be 8, meaning that the power series converges everywhere. 2 In general, the abscissa of convergence σc could be 8, meaning that the Dirichlet se- ries doesn’t converge anywhere, or could be ´8, meaning that the Dirichlet series converges everywhere.

3 A power series converges absolutely For Dirichlet series, the analogue state- (| |) inside its radius of convergence. ment doesn’t hold. In general, there might be a vertical line t= “ σacu sep- arating the points of absolute conver- gence from the others.3 But if an ě 0 for all n, the Dirichlet series converges absolutely to the right of σc.

Another nice feature of power series is For Dirichlet series, this again fails in that they converge until they have a general: the function might have a perfect excuse to stop: the function holomorphic continuation to a wider cannot be extended holomorphically to strip t= ě σc ´ εu for some ε ą 0. any larger circle centered at 0. But if you assume that an ě 0 for So if the power series has a mero- all n, the Dirichlet series must have a morphic continuation, for example, singularity at σc. there must be a pole on the circle of convergence.

(a) Power series (b) Dirichlet series if a ě 0

rc singularity 1 | | | 1| 0 σc

singularity

8 ´s Riemann zeta function What does the Riemann zeta function ζpsq “ n“1 n look like? It is easy to see that ζpsq converges absolutely for

3 pσc ď σac ď σc ` 1q

4 (since the sequences are nonnegative), we even know their respective abscissas of convergence: the series will converge until it “hits” the rightmost pole. This pole will be the abscissa of convergence.

Asymptotics for power series How can we make use of complex analysis of F pa, Xq to study the sequence a? Assume we know the radius of convergence 1 n ´ rc. In terms of the coefficients, rc “ lim infnÑ8 |an| . This produces the ´1 n upper bound |an| ď prc ` εq for all ε ą 0 and sufficiently large n. If we a 1 can find a nice sequence b0, b1,... such that the radius of convergence rc of F pa ´ b, Xq is larger, we can get a (better) upper bound for the difference of 1´1 n the two sequences: |an ´ bn| ď prc ` εq . For example, say F pa, Xq can be meromorphically continued to a circle of 1 radius rc ą rc and the continuation is holomorphic inside this circle except for a R simple pole at X “ z0 with residue R (with |z0| “ rc). Then, F pa, Xq ´ is X´z0 holomorphic inside the circle of radius r1 . But we can write R “ ´ R{z0 “ c X´z0 1´X{z0 n 8 R X n`1 ´ “ F pb, Xq with bn “ ´R{z . We conclude that an “ n“0 z0 z0 0 n`1 1´1 n ´řR{z0 `´Op¯rc ` εq for all ε ą 0. Note that (for small ε and large n) the error term is smaller than the main term. To summarize, the main term of the asymptotics of an comes from singu- larities of F pa, Xq close to the origin. The further away the other singularities are, the smaller the error term.

Asymptotics for Dirichlet series Here is a theorem of a similar spirit con- cerning Dirichlet series:

Theorem 1 (Wiener–Ikehara). Assume an ě 0 for all n and σc ą 0 and that k ϕpsq “ ps ´ σcq Dpa, sq has a holomorphic continuation to (a neighborhood of) t< ě σcu with ϕpσcq ‰ 0. Then

ϕ σ p cq σc k´1 an „ N log N. σ pk ´ 1q! nďN c ÿ Let’s apply this theorem to find asymptotics for the sequences we’ve seen before. We just need to find the rightmost pole σc of Dpa, sq, its order k and its coefficient ϕpσcq.

1 ÝÑ ζpsq ÝÑ simple pole at 1, coefficient 1 ÝÑ 1 „ N nďN ÿ So the number of positive integers up to N is asymptotically roughly N.— Exciting, isn’t it?

5 Let’s see what happens for the other sequences:

d ÝÑ ζpsq2 ÝÑ double pole at 1, coefficient 1

ÝÑ dn „ N log N nďN ÿ

dpkq ÝÑ ζpsqk ÝÑ order k pole at 1, coefficient 1 1 ÝÑ dpkq „ N logk´1 N n pk ´ 1q! nďN ÿ

σ ÝÑ ζps ´ 1qζpsq ÝÑ simple pole at 2, coefficient ζp2q ζp2q ÝÑ σ „ N 2 n 2 nďN ÿ These results can be easily shown directly without the use of Dirichlet series (and we can even get a nice error bound). 4

Ordered factorizations Here is a simple sequence whose asymptotics can be analyzed substantially more easily using Dirichlet series. Let ∆n “ tpx1, . . . , xkq : k ě 0, x1, . . . , xk ě 2, x1 ¨ ¨ ¨ xk “ nu be the number of ordered factorizations of n into integers bigger than 1. Note that the number of factors is not fixed. 5 We can write 8 8

∆n “ 1 “ p1 ´ δx1 q ¨ ¨ ¨ p1 ´ δxk q. k“0 x1,...,xkě2: k“0 x1,...,xkě1: ÿ x1¨¨¨ÿxk“n ÿ x1¨¨¨ÿxk“n This translates to 8 ∆ “ p1 ´ δq ˚ ¨ ¨ ¨ ˚ p1 ´ δq, k“0 ÿ k times so loooooooooooomoooooooooooon 8 8 1 1 Dp∆, sq “ Dp1 ´ δ, sqk “ pζpsq ´ 1qk “ “ . 1 ´ pζpsq ´ 1q 2 ´ ζpsq k“0 k“0 ÿ ÿ 4For example, 1 N 2 N 2 N σn “ a “ a “ a “ ` Op1q “ ` O 2 b 2b2 b nďN nďN ab“n abďN bďN a N b bďN bďN ÿ ÿ ÿ ÿ ÿ ďÿt { u ÿ ˆ ˙ ÿ ˆ ˆ ˙˙ N b´2 ζp2q “ b“1 N 2 ` OpN log Nq “ N 2 ` OpN log Nq. 2 2 ř 5This sequence’s asymptotic behavior was studied by Laszl´ o´ Kalmar´ in the Hungarian article A “factorisatio numerorum” probl´em´aj´ar´ol (Matematikai ´esFizikai Lapok) and in the German article Uber¨ die mittlere Anzahl der Produktdarstellungen der Zahlen (Acta Litt. Sci. Szeged).

6 (You can also verify this without using a summation over k by multiplying by 2 ´ ζpsq and using a direct bijective argument.) The poles of Dp∆, sq are the places where ζpsq “ 2. It is clear that the 8 ´s function ζpsq “ n“1 n is strictly decreasing on the real interval p1, 8q. On this interval, the place ρ where the function is 2, is ρ “ 1.7286 .... Furthermore, you can easily showř that on each vertical line t< “ σu with σ ą 1, the function ζpsq assumes the maximum value exactly at the intersection with the real axis, so ρ is in fact the (unique) rightmost (simple) pole of Dp∆, sq. The Wiener– Ikehara theorem thus tells us that

1 ρ ∆n „ ´ N . ρζ1pρq nďN ÿ Multiplicative sequences, Euler products The sequences δ, 1, d, dpkq, σ are all multiplicative: a e1 ek “ a e1 ¨ ¨ ¨ a ek whenever p1, . . . , pk are distinct p1 ¨¨¨pk p1 pk primes.6 For a multiplicative sequence a, we can rewrite Dpa, sq as an infinite product (called ):

8 ´es ´s ´2s Dpa, sq “ ape p “ 1 ` app ` ap2 p ` ¨ ¨ ¨ . p prime e“0 p prime ź ÿ ź ` ˘ (Try expanding this infinite product and using the uniqueness of factorizations!) For example, 8 1 Dp1, sq “ p´es “ p1 ` p´s ` p´2s ` ¨ ¨ ¨ q “ 1 ´ p´s p prime e“0 p prime p ź ÿ ź ź 8 1 Dpd, sq “ pe ` 1qp´es “ p1 ` 2p´s ` 3p´2s ` ¨ ¨ ¨ q “ p1 ´ p´sq2 p prime e“0 p prime p ź ÿ ź ź 1 Dpdpkq, sq “ 1 p´s k p p ´ q ź 8 Dpσ, sq “ p1 ` p ` ¨ ¨ ¨ ` peqp´es p prime e“0 ź ÿ 1 “ p1 ` p1 ` pqp´s ` p1 ` p ` p2qp´2s ` ¨ ¨ ¨ q “ 1 p1´s 1 p´s p prime p p ´ qp ´ q ź ź The Let’s find an asymptotic formula for the num- ber of primes up to N. Unfortunately, the Dirichlet series for the sequence

1, n prime Pn “ #0, else

6 Equivalently, a1 “ 1 and anm “ anam whenever n and m are coprime. 2 Note that not necessarily ap2 “ ap.

7 doesn’t have a simple expression in terms of the Riemann zeta function. But the Euler product expansions suggest a different approach. We want a sum over primes, not a product, so it might be helpful to look at the of ζpsq: 1 8 p´ks log ζpsq “ log “ 1 ´ p´s k p prime p prime k“1 ÿ ÿ ÿ In complex analysis, can be somewhat nasty: It’s impossible to define logpsq in a neighborhood of the origin, unless you look at a branched cover of the complex plane. Therefore, if a function fpsq has zeros, log fpsq is defined only on a branched cover of the complex plane. d ζ1psq To circumvent this issue, we look at the derivative ds log ζpsq “ ζpsq . This quotient is of course meromorphic everywhere in the complex plane.

ζ1psq d 8 “ log ζpsq “ p´ log pqp´ks ζpsq ds p prime k“1 ÿ ÿ Thus, we have found an expression for the Dirichlet series of the sequence

log p, n “ pk for some prime p and some k ě 1, Λn “ #0, else.

ζ1psq DpΛ, sq “ ´ ζpsq The poles of this function are at the places where ζpsq has a pole (s “ 1) or a zero. Already knowing that ζpsq has no zero with < ě 1 is enough to apply the Wiener–Ikehara theorem and conclude that

Λn „ N. nďN ÿ The left-hand side is clearly closely related to the number of primes up to N and it is in fact not too hard to deduce the famous prime number theorem:7

N dt N #pprimes ď Nq „ „ . log t log N ż2 By analogy with asymptotics in the case of power series, you should expect that the further away from the line t< “ 1u the zeros of ζpsq (the poles of DpΛ, sq other than s “ 1) are, the smaller the error in this asymptotic should be. In fact, the states that the only zeroes of ζpsq are the negative 1 even integers and some (infinitely many) numbers on the line t< “ 2 u. 7Hint:

N „ Λn « Λn ` log N Pn „ Np1 ´ εq ` log N Pn. nďN n N 1 ε N 1 ε n N N 1 ε n N ÿ ď ÿp ´ q p ´ÿqă ď p ´ÿqă ď

8 Further reading For a deeper introduction to Dirichlet series and far more examples, I highly recommend the book Problems in by M. Ram Murty.

9