Skip to main content

Notes on analytic number theory

Kiran S. Kedlaya

Chapter 2 Dirichlet series and arithmetic functions

In this chapter, we put the definition of the Riemann zeta function in the broader context of Dirichlet series.
🔗

Section 2.1 Dirichlet series

Definition 2.1.

A Dirichlet series is a formal series of the form \(\sum_{n=1}^\infty a_n n^{-s}\) with \(a_n \in \CC\text{.}\) Note that it makes sense to add and multiply such formal sums, and in fact they form a ring under these operations.
🔗
This class of objects includes both the Riemann zeta function and ordinary formal power series: given a formal power series \(\sum_{m=0}^\infty b_m T^m\text{,}\) for any prime \(p\) we may formally evaluate at \(T = p^{-s}\) to obtain the Dirichlet series \(\sum_{m=0}^\infty b_m p^{-ms}\text{.}\) More precisely, the latter is the Dirichlet series whose coefficient of \(n^{-s}\) equals \(b_m\) if \(n = p^{-m}\) for some nonnegative integer \(m\) and 0 otherwise.
🔗
🔗
Our first order of business is to understand when a Dirichlet series converges absolutely.
🔗

Proof.

Definition 2.3.

The quantity \(L\) introduced in Lemma 2.2 is called the abscissa of absolute convergence of the Dirichlet series. If we view a power series in \(T\) as a Dirichlet series as in Definition 2.1, then the abscissa of absolute convergence coincides with the base-\(p\) logarithm of the radius of convergence of the original power series.
🔗
🔗
Recall that an ordinary power series in a complex variable must have a singularity at the boundary of its radius of convergence; for a series with nonnegative real coefficients, we can even ensure a singularity at the intersection of this boundary with the positive real axis (Abel’s theorem). While the first statement does not extend to Dirichlet series (see Exercise 2.4.2), the second statement does extend as follows.
🔗

Proof.

Suppose on the contrary that \(f\) extends to a holomorphic function on the disc \(|s-L| \lt \epsilon\text{.}\) Pick a real number \(c \in (L, L+\epsilon/2)\text{,}\) and write
\begin{align*} f(s) \amp= \sum_{n=1}^\infty a_n n^{-c} n^{c-s}\\ \amp= \sum_{n=1}^\infty a_n n^{-c} \exp((c-s) \log n)\\ \amp= \sum_{n=1}^\infty \sum_{i=0}^\infty \frac{a_n n^{-c} (\log n)^i}{i!} (c-s)^i\text{.} \end{align*}
Since all coefficients in this double series are nonnegative, everything must converge absolutely in the disc \(|s-c| \lt \epsilon/2\text{.}\) In particular, when viewed as a power series in \(c-s\text{,}\) this must give the Taylor series for \(f\) around \(s=c\text{.}\) Since \(f\) is holomorphic in the disc \(|s-c| \lt \epsilon/2\text{,}\) the Taylor series converges there; in particular, it converges for some real number \(L' \lt L\text{.}\) But now we can run the argument backwards to deduce that the original Dirichlet series converges absolutely for \(s=L'\text{,}\) w hich implies that the abscissa of absolute convergence is at most \(L'\text{.}\) This contradicts the definition of \(L\text{.}\)
🔗
🔗
🔗

Section 2.2 Multiplicative functions and Euler products

Recall from (1.2.1) that the Riemann zeta function factors as a product over primes:
\begin{equation*} \zeta(s) = \sum_{n=1}^\infty n^{-s} = \prod_p (1 - p^{-s})^{-1}\text{.} \end{equation*}
In fact, a number of natural Dirichlet series admit such factorizations; they are the ones corresponding to multiplicative functions.
🔗

Definition 2.5.

We define an arithmetic function to simply be a function \(f\colon \NN \to \CC\text{.}\) Besides the obvious operations of addition and multiplication, another useful operation on arithmetic functions is the (Dirichlet) convolution \(f \ast g\text{,}\) defined by
\begin{equation*} (f \ast g)(n) = \sum_{d | n} f(d) g(n/d)\text{.} \end{equation*}
Just as one can think of formal power series as the generating functions for ordinary sequences, we may think of a formal Dirichlet series \(\sum_{n=1}^\infty a_n n^{-s}\) as the “arithmetic generating function” for the multiplicative function \(n \mapsto a_n\text{.}\) In this way of thinking, convolution of multiplicative functions corresponds to ordinary multiplication of Dirichlet series:
\begin{equation*} \sum_{n=1}^\infty (f \ast g)(n) n^{-s} = \left( \sum_{n=1}^\infty f(n) n^{-s} \right) \left( \sum_{n=1}^\infty g(n) n^{-s} \right)\text{.} \end{equation*}
In particular, convolution is a commutative and associative operation, under which the arithmetic functions taking the value 1 at \(n=1\) form a group. The arithmetic functions whose values are all integers, again with the value 1 at \(n=1\text{,}\) form a subgroup (see Exercise 2.4.3).
🔗
🔗

Definition 2.6.

An arithmetic function \(f\) is multiplicative if \(f(1) = 1\) and \(f(mn) = f(m)f(n)\) whenever \(m,n \in \NN\) are coprime. Note that \(f\) is multiplicative if and only if its Dirichlet series factors as an Euler product:
\begin{equation*} \sum_{n=1}^\infty f(n) n^{-s} = \prod_p \left( \sum_{i=0}^\infty f(p^i) p^{-is} \right)\text{.} \end{equation*}
In particular, the property of being multiplicative is stable under convolution and under taking the convolution inverse.
🔗
🔗

Definition 2.7.

An arithmetic function \(f\) is completely multiplicative if \(f(1) = 1\text{,}\) and \(f(mn) = f(m) f(n)\) for any \(m,n \in \NN\text{.}\) Note that \(f\) is completely multiplicative if and only if its Dirichlet series factors in a very special way:
\begin{equation*} \sum_{n=1}^\infty f(n) n^{-s} = \prod_p (1 - f(p) p^{-s})^{-1}\text{.} \end{equation*}
In particular, the property of being completely multiplicative is not stable under convolution.
🔗
🔗
🔗

Section 2.3 Examples of multiplicative functions

Here are some examples of multiplicative functions, some of which you may already be familiar with. All assertions in this section are left as exercises (Exercise 2.4.4).
🔗

Definition 2.8.

Define the following arithmetic functions, all of which are multiplicative.
  • The unit function \(\varepsilon\text{:}\) \(\varepsilon(1) = 1\) and \(\varepsilon(n) = 0\) for \(n \gt 1\text{.}\) This is the identity under \(\ast\text{.}\)
    🔗
  • The constant function \(1\text{:}\) \(1(n) = 1\text{.}\)
    🔗
  • The Möbius function \(\mu\text{:}\) if \(n\) is squarefree with \(d\) distinct prime factors, then \(\mu(n) = (-1)^d\text{,}\) otherwise \(\mu(n) = 0\text{.}\) This is the inverse of \(1\) under \(*\text{.}\)
    🔗
  • The identity function \(\id\text{:}\) \(\id(n) = n\text{.}\)
    🔗
  • The \(k\)-th power function \(\id^k\text{:}\) \(\id^k(n) = n^k\text{.}\)
    🔗
  • The Euler totient function \(\phi\text{:}\) \(\phi(n)\) counts the number of integers in \(\{1, \dots, n\}\) coprime to \(n\text{.}\) Note that \(1 * \phi = \id\text{,}\) so \(\id * \mu = \phi\text{.}\)
    🔗
  • The divisor function \(d\) (or \(\tau\)): \(d(n)\) counts the number of integers in \(\{1,\dots,n\}\) dividing \(n\text{.}\) Note that \(1 * 1 = d\text{.}\)
    🔗
  • The divisor sum function \(\sigma\text{:}\) \(\sigma(n)\) is the sum of the divisors of \(n\text{.}\) Note that \(1 * \id = d * \phi = \sigma\text{.}\)
    🔗
  • For \(k\) a nonnegative integer, the divisor power sum function \(\sigma_k\text{:}\) \(\sigma_k(n) = \sum_{d|n} d^k\text{.}\) Note that \(\sigma_0 = d\) and \(\sigma_1 = \sigma\text{.}\) Also note that \(1 * \id^k = \sigma_k\text{.}\)
    🔗
Of these, only \(\varepsilon, 1, \id, \id^k\) are completely multiplicative. We will introduce some more completely multiplicative functions, the Dirichlet characters, in Chapter 3.
🔗
🔗

Definition 2.9.

Note that all of the Dirichlet series corresponding to the multiplicative functions listed in Definition 2.8 can be written explicitly in terms of the Riemann zeta function (again see Exercise 2.4.4). An important function with the same property, but which is not multiplicative, is the von Mangoldt function \(\Lambda = \mu * \log\text{;}\) see Exercise 2.4.6.
🔗
🔗
🔗

Exercises 2.4 Exercises

1.

Prove Lemma 2.2. Then exhibit examples to show that a Dirichlet series with some abscissa of absolute convergence \(L \in \RR\) may or may not converge absolutely on \(\Real(s) = L\text{.}\)
🔗
🔗

2.

Consider the Dirichlet series
\begin{equation*} \left( 1 - \frac{1}{2^{s-1}} \right) \zeta(s) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^{s}}\text{.} \end{equation*}
Show that this series has abscissa of absolute convergence 1, but the resulting holomorphic function extends all the way to \(\Real(s) \gt 0\text{.}\)
🔗
🔗

3.

Let \(f\colon \NN \to \ZZ\) be an arithmetic function with \(f(1) = 1\text{.}\) Prove that the convolution inverse of \(f\) also has values in \(\ZZ\text{.}\) Deduce that the set of such \(f\) forms a group under convolution. (Likewise with \(\ZZ\) replaced by any subring of \(\CC\text{,}\) e.g., the integers in an algebraic number field.)
🔗
🔗

4.

Prove all assertions in Section 2.3. Then write the Dirichlet series for all of the functions introduced therein in terms of the Riemann zeta function.
🔗
🔗

5.

Here is a non-obvious example of a multiplicative function. Let \(r_2(n)\) be the number of pairs \((a,b)\) of integers such that \(a^2 + b^2 = n\text{.}\) Prove that \(r_2(n)/4\) is multiplicative, using facts you know from elementary number theory.
🔗
🔗

6.

For \(\Lambda = \mu * \log\) the von Mangoldt function (Definition 2.9), prove that
\begin{equation*} \Lambda(n) = \begin{cases} \log(p) \amp n = p^i, i \geq 1 \\ 0 \amp \mbox{otherwise} \end{cases} \end{equation*}
and that the Dirichlet series for \(\Lambda\) is \(-\zeta'/\zeta\text{.}\)
🔗
🔗

7.

For \(t\) a fixed positive real number, verify that the function
\begin{equation*} Z(s) = \zeta^2(s)\zeta(s+it)\zeta(s-it) \end{equation*}
is represented by a Dirichlet series with nonnegative coefficients which does not converge everywhere.
🔗
Hint.
Check \(s=0\text{.}\)
🔗
🔗
🔗

8.

Assuming that \(\zeta(s) - s/(s-1)\) extends to an entire function (we’ll prove this in a subsequent unit), use the previous exercise to give a second proof that \(\zeta(s)\) has no zeroes on the line \(\Real(s) = 1\text{.}\)
🔗
🔗

9. Dirichlet’s hyperbola method.

Suppose \(f,g,h\) are arithmetic functions with \(f = g * h\text{,}\) and write
\begin{equation*} G(x) = \sum_{n \leq x} g(n), \qquad H(x) = \sum_{n \leq x} h(n)\text{.} \end{equation*}
Prove that (generalizing a previous exercise)
\begin{equation*} \sum_{n \leq x} f(n) = \left(\sum_{d \leq y} g(d) H(x/d) \right) + \left(\sum_{d \leq x/y} h(d) G(x/d)\right) - G(y) H(x/y) \end{equation*}
🔗
🔗

10.

Prove that the abscissa of absolute convergence \(L\) of a Dirichlet series \(\sum_{n=1}^\infty a_n n^{-s}\) satisfies the inequality
\begin{equation*} L \leq \limsup_{n \to \infty} \left( 1 + \frac{\log |a_n|}{\log n} \right) \end{equation*}
(where \(\log 0 = -\infty\)). Then show that equality holds if the \(|a_n|\) are bounded away from 0, but not necessarily otherwise.
🔗
🔗
🔗
🔗
PrevTopNext