Section 4.3 Nonvanishing of \(L\)-functions on \(\Real(s) = 1\)
Much as we used nonvanishing of \(\zeta\) on the line \(\Real(s) = 1\) to study the prime number theorem, we will use nonvanishing of \(L\)-functions on that line to study the prime number theorem in arithmetic progressions. An additional wrinkle, though, is that we have to do some extra work to understand what is going on at \(s=1\) itself; see next section.
Lemma 4.6.
Let \(f(s)\) be a meromorphic function on a neighborhood of \(\Real(s) \geq L\text{,}\) with at worst a simple pole at \(s=L\) and no other poles. Suppose that \(\log f(s)\) is represented by a Dirichlet series with abscissa of convergence \(\leq L\) and nonnegative real coefficients. Then \(f(s) \neq 0\) for \(\Real(s) \geq L\text{.}\)Proof.
Theorem 4.7.
Let \(N\) be a positive integer. Let \(f_N(s)\) be the product of all of the Dirichlet \(L\)-series of level \(N\text{.}\) Then \(f_N(s) \neq 0\) for \(s \in \CC\) with \(\Real(s) = 1\text{.}\)Proof.
Note that for \(\Real(s) > 1\text{,}\) we have
\begin{equation}
\log f_N(s) = \sum_{p: (p,N) = 1} \sum_{n=1}^\infty \left( \sum_\chi \chi(p^n) \right) p^{-ns},\tag{4.3.1}
\end{equation}
which is a Dirichlet series with nonnegative real coefficients. (The sum over \(\chi\) is invariant under multiplication by \(\chi(p^n)\) for any single \(\chi\text{,}\) so either the sum is zero or all of the summands are equal to 1.) We may thus apply Lemma 4.6.
This tells us a lot about nonvanishing of individual \(L\)-functions, but not quite everything.
Theorem 4.8.
For any Dirichlet character \(\chi\text{,}\) \(L(s,\chi) \neq 0\) when \(\Real(s) = 1\) and \(s \neq 1\text{.}\)Proof.
Let \(N\) be the level of \(\chi\text{.}\) Then \(f_N(s)\) is a product of functions, one of which is \(L(s,\chi)\text{,}\) all of which are holomorphic at \(s\text{.}\) By Theorem 4.7, \(f_N(s)\) has no zero at \(s\text{,}\) so none of the factors can either.
It will take a bit more work to deal with
\(s=1\text{;}\) see
Section 4.4.
Section 4.4 Nonvanishing for \(L\)-functions at \(s=1\)
At
\(s=1\) (the so-called
critical point for Dirichlet
\(L\)-functions), life is a bit more complicated; to deduce that none of the
\(L(1, \chi)\) vanish, we need to know that the function
\(f_N(s)\) in
Theorem 4.7 has a simple pole, rather than being holomorphic, at
\(s=1\text{.}\)
Definition 4.9.
We say a Dirichlet character is real if it takes values in \(\pm 1\text{,}\) and nonreal (or complex) otherwise.
Theorem 4.10.
For any nonreal Dirichlet character \(\chi\text{,}\) \(L(1,\chi) \neq 0\text{.}\)Proof.
Let \(N\) be the level of \(\chi\text{.}\) If \(L(1,\chi) = 0\text{,}\) then also \(L(1, \overline{\chi}) = 0\text{,}\) where \(\overline{\chi}\) denotes the complex conjugate character. But then \(f_N(s)\) is the product of one factor with a simple pole at \(s=1\) (coming from the principal character), two factors with zeroes at \(s=1\) (coming from \(\chi\) and \(\overline{\chi}\)), and a bunch of factors which are holomorphic at \(s=1\text{.}\) This would force \(f_N(s)\) to have a zero at \(s=1\text{,}\) contradicting Theorem 4.7.
For the real characters, the above argument fails because
\(\overline{\chi}\) and
\(\chi\) are the same character, so they don't give two different contributions to
\(f_N(s)\text{.}\) Instead, we use a different trick. (There are a number of proofs of this result; see
Exercise 4.6.5 for a second approach.)
Theorem 4.11.
For any real nonprincipal Dirichlet character \(\chi\text{,}\) \(L(1,\chi) \neq 0\text{.}\)Proof.
Assume on the contrary that \(L(1,\chi) = 0\text{.}\) Define
\begin{equation*}
\psi(s) = \frac{L(s,\chi) L(s,\chi_0)}{L(2s,\chi_0)},
\end{equation*}
where \(\chi_0\) is the principal character of level \(N\text{.}\) Then the numerator of \(\psi\) is holomorphic for \(\Real(s) > 0\text{,}\) because \(L(s,\chi)\) counterbalances the simple pole of \(L(s,\chi_0)\) at \(s=1\text{.}\) On the other hand, the denominator of \(\psi\) is holomorphic and nonzero for \(\Real(s) > 1/2\text{;}\) moreover, it extends meromorphically to a neighborhood of \(s=1/2\) with a simple pole at \(s=1/2\text{.}\) Therefore \(\psi\) is holomorphic for \(\Real(s) > 1/2\text{,}\) and extends holomorphically to a neighborhood of \(1/2\) with a simple zero at \(s=1/2\text{.}\)
However, \(\psi(s)\) admits the formal factorization
\begin{equation*}
\psi(s) = \prod_{p: \chi(p) = 1} \left( \frac{1 + p^{-s}}{1 - p^{-s}} \right)
\end{equation*}
and so expands as a Dirichlet series with nonnegative real coefficients and constant coefficient 1. The product factorization converges absolutely for
\(\Real(s) > 1\text{,}\) so the Dirichlet series does too. But
\(\psi\) is holomorphic for
\(\Real(s) > 1/2\text{,}\) so
Theorem 3.4 implies that the Dirichlet series converges absolutely on
\(\Real(s) > 1/2\text{.}\)
This yields \(\psi(s) \geq 1\) for \(s > 1/2\text{,}\) whereas \(\lim_{s \to (1/2)^+} \psi(s) = 0\text{,}\) contradiction.
Section 4.5 Historical aside: Dirichlet's class number formula
Continuing the thread from
Remark 4.12, we recall Dirichlet's original approach to
Theorem 4.11: for
\(\chi\) a real nonprincipal Dirichlet character, one can express the value
\(L(1,\chi)\) in terms of a important numerical invariant, the
class number of binary quadratic forms of a given discriminant. That number evidently being positive, Dirichlet could not only establish the nonvanishing of
\(L(1,\chi)\text{,}\) but also its sign (but see
Exercise 4.6.5 for another approach to the latter).
Nowadays, one typically expresses this in the language of algebraic number theory. If you are not familiar with this language, feel free to ignore the rest of this section.
Let \(K\) be a quadratic number field, and let \(\chi_K\) be the character such that
\begin{equation*}
\chi_K(p) = \begin{cases} 1 \amp \mbox{ is unramified and split in } \\
-1 \amp \mbox{ is unramified and inert in } \\
0 \amp \mbox{ is ramified in }
\end{cases}
\end{equation*}
One then proves that \(L(1,\chi_K)\) is equal to the class number \(h_K\) of \(K\) times the regulator \(R_K\text{.}\) (The latter equals 1 if \(K\) is imaginary quadratic, and otherwise is equal to a fixed normalization factor times the logarithm of the fundamental unit of \(K\text{.}\))
The point here is that up to multiplication by Euler factors for the ramified primes, \(L(1, \chi_0) L(1, \chi_K)\) is equal to the Dedekind zeta function \(\zeta_K\) of \(K\text{,}\) defined by
\begin{equation*}
\zeta_K(s) := \sum_{\gotha} \Norm(\gotha)^{-s},
\end{equation*}
for \(\gotha\) running over nonzero ideals of the ring of integers \(\gotho_K\text{.}\) For a general number field \(K\text{,}\) \(\zeta_K\) has a simple pole at 1, whose residue is the class number of \(K\) times the regulator of \(K\) times a normalization factor (determined by the number of real and complex places of \(K\)); the point is that each factor in this product is visibly nonzero.
One sometimes turns this around and tries to use analytic information about \(\zeta_K\) to get information about the product \(h_K R_K\text{.}\) It is quite difficult to separate the two factors in this expression; indeed, one can make a good case that they really are simply two separate factors in the computation of the volume of a certain compact topological group, the Arakelov class group of \(R\text{,}\) whose group of components is isomorphic to the usual class group.
A notable exception is that there is no regulator for an imaginary quadratic field, so you can get good bounds in this case. For instance, the Brauer–Siegel theorem says that the class number of an imaginary quadratic field of discriminant
\(D\) is at least
\(c_\epsilon D^{1/2 - \epsilon}\) for any
\(\epsilon>0\text{,}\) though unfortunately the constant
\(c_\epsilon\) cannot be effectively determined from
\(\epsilon\text{.}\) The best effective results are due to Goldfeld
[7], who proves an effective lower bound which is polynomial in
\(\log(D)\text{;}\) this is a far cry from the truth, but can be used to give a new resolution of Gauss's conjecture that there are exactly nine imaginary quadratic fields of class number 1
[26], or to determine all imaginary quadratic number fields with class number up to 100
[27].
