18.785: Analytic Number Theory, MIT, spring 2007 (K.S. Kedlaya)
Dirichlet series and arithmetic functions

1Dirichlet series

The Riemann zeta function ζ is a special example of a type of series
P we will−sbe considering
often in this course. A Dirichlet series is a formal series of the form ∞with an ∈ C.
n=1 an n
You should think of these as a number-theoretic analogue of formal power series; indeed, ourfirst order of business is to understand when such a series converges absolutely.
Lemma 1. There isPan extended real number L ∈ R ∪ {±∞} with the following property:−s
the Dirichlet series ∞converges absolutely for Re(s) > L, but not for Re(s) < L.
n=1 an nMoreover, for any  > 0, the convergence is uniform on Re(s) ≥ L+, so the series represents
a holomorphic function on all of Re(s) > L.
Proof. Exercise.
The quantity L is called the abscissa of absolute convergence of the Dirichlet series; it isan analogue of the radius of convergence of a power series. (In fact, if you fix a prime p, andonly allow an to be nonzero when n is a power of p, then you get an ordinary power series inp−s . So in some sense, Dirichlet series are a strict generalization of ordinary power series.)
Recall that an ordinary power series in a complex variable must have a singularity at theboundary of its radius of convergence. For Dirichlet series with nonnegative real coefficients,
we have the following analogous fact.
P−sbe a Dirichlet series with nonnegative real
Theorem 2 (Landau). Let f (s) = ∞n=1 an ncoefficients. Suppose L ∈ R is the abscissa of absolute convergence for f (s). Then f cannot
be extended to a holomorphic function on a neighborhood of s = L.
Proof. Suppose on the contrary that f extends to a holomorphic function on the disc |s−L| <. Pick a real number c ∈ (L, L + /2), and write
f (s) ==∞
X
n=1
∞
X

an n−c nc−s
an n−c exp((c − s) log n)

n=1

=∞
∞ X
X
an n−c (log n)ii!

n=1 i=0(c − s)i .

Since all coefficients in this double series are nonnegative, everything must converge absolutely in the disc |s−c| < /2. In particular, when viewed as a power series in c−s, this must
give the Taylor series for f around s = c. Since f is holomorphic in the disc |s − c| < /2,
the Taylor series converges there; in particular, it converges for some real number L0 < L.
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But now we can run the argument backwards to deduce that the original Dirichlet series
converges absolutely for s = L0 , which implies that the abscissa of absolute convergence isat most L0 . This contradicts the definition of L.

2Euler products

Remember that among Dirichlet series, the Riemann zeta function had the unusual property
that one could factor the Dirichlet series as a product over primes:
ζ(s) =∞Xn−s =Y(1 − p−s )−1 .

n=1pIn fact, a number of natural Dirichlet series admit such factorizations; they are the ones
corresponding to multiplicative functions.
We define an arithmetic function to simply be a function f : N → C. Besides the obvious
operations of addition and multiplication, another useful operation on arithmetic functions
is the (Dirichlet) convolution f ∗ g, defined byX(f ∗ g)(n) =f (d)g(n/d).
d|nJust as one can think of formal power series as P
the generating functions for ordinary se−sas the “arithmetic generating
quences, we may think of a formal Dirichlet series ∞n=1 an nfunction” for the multiplicative function n 7→ an . In this way of thinking, convolution ofmultiplicative functions corresponds to ordinary multiplication of Dirichlet series:! ∞!
∞∞XX
X(f ∗ g)(n)n−s =f (n)n−sg(n)n−s .
n=1

n=1

n=1

In particular, convolution is a commutative and associative operation, under which thearithmetic functions taking the value 1 at n = 1 form a group. The arithmetic functions
taking all integer values (with the value 1 at n = 1) form a subgroup (see exercises).
We say f is a multiplicative function if f (1) = 1, and f (mn) = f (m)f (n) wheneverm, n ∈ N are coprime. Note that an arithmetic function f is multiplicative if and only if its
Dirichlet series factors as a product (called an Euler product):!
∞∞
XY X
f (pi )p−is .
f (n)n−s =n=1

pi=0

In particular, the property of being multiplicative is clearly stable under convolution, andunder taking the convolution inverse.

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We say f is completely multiplicative if f (1) = 1, and f (mn) = f (m)f (n) for anym, n ∈ N. Note that an arithmetic function f is multiplicative if and only if its Dirichlet
series factors in a very special way:
∞Xf (n)n−s =n=1Y
(1 − f (p)p−s )−1 .
pIn particular, the property of being completely multiplicative is not stable under convolution.

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Examples of multiplicative functions

Here are some examples of multiplicative functions, some of which you may already befamiliar with. All assertions in this section are left as exercises.
• The unit function ε: ε(1) = 1 and ε(n) = 0 for n > 1. This is the identity under ∗.
• The constant function 1: 1(n) = 1.
• The Möbius function µ: if n is squarefree with d distinct prime factors, then µ(n) =(−1)d , otherwise µ(n) = 0. This is the inverse of 1 under ∗.
• The identity function id: id(n) = n.
• The k-th power function idk : idk (n) = nk .
• The Euler totient function φ: φ(n) counts the number of integers in {1, . . . , n} coprime
to n. Note that 1 ∗ φ = id, so id ∗µ = φ.
• The divisor function d (or τ ): d(n) counts the number of integers in {1, . . . , n} dividing
n. Note that 1 ∗ 1 = d.
• The divisor sum function σ: σ(n) is the sum of the divisors of n. Note that 1 ∗ id =d ∗ φ = σ.
P
• The divisor power sum functions σk : σk (n) = d|n dk . Note that σ0 = d and σ1 = σ.
Also note that 1 ∗ idk = σk .
Of these, only ε, 1, id, idk are completely multiplicative. We will deal with some more
completely multiplicative functions, the Dirichlet characters, in a subsequent unit.
Note that all of the Dirichlet series corresponding to the aforementioned functions can be
written explicitly in terms of the Riemann zeta function; see exercises. An important nonmultiplicative function with the same property is the von Mangoldt function Λ = µ ∗ log; seeexercises.

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Exercises
1. Prove Lemma 1. Then exhibit examples to show that a Dirichlet series with someabscissa of absolute convergence L ∈ R may or may not converge absolutely on Re(s) =L.
2. Give a counterexample to Theorem 2 in case the series need not have nonnegative realcoefficients. (Optional, and I don’t know the answer: must a Dirichlet series have asingularity somewhere on the abscissa of absolute convergence?)
3. Let f : N → Z be an arithmetic function with f (1) = 1. Prove that the convolution
inverse of f also has values in Z; deduce that the set of such f forms a group underconvolution. (Likewise with Z replaced by any subring of C, e.g., the integers in an
algebraic number field.)
4. Prove the assertions involving ∗ in Section 3. Then use them to write the Dirichlet
series for all of the functions introduced there in terms of the Riemann zeta function.
5. Here is a non-obvious example of a multiplicative function. Let r2 (n) be the numberof pairs (a, b) of integers such that a2 + b2 = n. Prove that r2 (n)/4 is multiplicative,
using facts you know from elementary number theory.
6. We defined the von Mangoldt function as the arithmetic function Λ = µ ∗ log. Provethat(log(p) n = pi , i ≥ 1
Λ(n) =0otherwise
and that the Dirichlet series for Λ is −ζ 0 /ζ.
7. For t a fixed positive real number, verify that the function
Z(s) = ζ 2 (s)ζ(s + it)ζ(s − it)is represented by a Dirichlet series with nonnegative coefficients which does not converge everywhere. (Hint: check s = 0.)
8. Assuming that ζ(s) − s/(s − 1) extends to an entire function (we’ll prove this in a
subsequent unit), use the previous exercise to give a second proof that ζ(s) has nozeroes on the line Re(s) = 1.
9. (Dirichlet’s hyperbola method) Suppose f, g, h are arithmetic functions with f = g ∗ h,and writeXXG(x) =g(n),H(x) =h(n).
n≤xn≤x4

Prove that (generalizing a previous exercise)! XXXf (n) =g(d)H(x/d) + h(d)G(x/d) − G(y)H(x/y).
n≤xd≤yd≤x/y

10. Prove that the abscissa of absolute convergence L of a Dirichlet seriessatisfies the inequality
log |an |L ≤ lim sup 1 +log nn→∞

P∞−sn=1 an n(where log 0 = −∞), with equality if the |an | are bounded away from 0. Then exhibit
an example where the inequality is strict. (Thanks to Sawyer for pointing this out.)
Optional (I don’t know the answer): is there a formula that computes the abscissa of
absolute convergence in general? Dani proposedP
log m≤n |am |lim suplog nn→∞but Sawyer found a counterexample to this too.

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