3 First approach: Wiener-Ikehara Tauberian theorem
3.1 A Fourier-analytic proof of the Wiener-Ikehara theorem
The Fourier transform of an absolutely integrable function \(\psi : \mathbb {R}\to \mathbb {C}\) is defined by the formula
where \(e(\theta ) := e^{2\pi i \theta }\).
Let \(f: \mathbb {N}\to \mathbb {C}\) be an arithmetic function such that \(\sum _{n=1}^\infty \frac{|f(n)|}{n^\sigma } {\lt} \infty \) for all \(\sigma {\gt}1\). Then the Dirichlet series
is absolutely convergent for \(\sigma {\gt}1\).
If \(\psi : \mathbb {R}\to \mathbb {C}\) is integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)
By the definition of the Fourier transform, the left-hand side expands as
while the right-hand side expands as
Since
the claim then follows from Fubini’s theorem.
If \(\psi : \mathbb {R}\to \mathbb {C}\) is absolutely integrable and \(x {\gt} 0\), then for any \(\sigma {\gt}1\)
The left-hand side expands as
so by Fubini’s theorem it suffices to verify the identity
Now let \(A \in \mathbb {C}\), and suppose that there is a continuous function \(G(s)\) defined on \(\mathrm{Re} s \geq 1\) such that \(G(s) = F(s) - \frac{A}{s-1}\) whenever \(\mathrm{Re} s {\gt} 1\). We also make the Chebyshev-type hypothesis
for all \(x \geq 1\) (this hypothesis is not strictly necessary, but simplifies the arguments and can be obtained fairly easily in applications).
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable then
for all \(u \in \mathbb {R}\). where \(C\) is an absolute constant.
Immediate from the triangle inequality.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable and of bounded variation, then
for all non-zero \(u \in \mathbb {R}\).
By Lebesgue–Stiejtes integration by parts we have
and the claim then follows from the triangle inequality.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then
for all non-zero \(u \in \mathbb {R}\).
Should follow from previous lemma.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is absolutely integrable, absolutely continuous, and \(\psi '\) is of bounded variation, then
for all \(u \in \mathbb {R}\).
Should follow from previous lemmas.
If \(\psi :\mathbb {R}\to \mathbb {C}\) is \(C^2\) and obeys the bounds
for all \(t \in \mathbb {R}\), then
for all \(u \in \mathbb {R}\), where \(C\) is an absolute constant.
From two integration by parts we obtain the identity
Now apply the triangle inequality and the identity \(\int _{\bf R} \frac{dt}{1+t^2}\ dt = \pi \) to obtain the claim with \(C = \pi + 1 / 4 \pi \).
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported and \(x \geq 1\), then
By Lemma 3.1.1 and Lemma 3.1.2, we know that for any \(\sigma {\gt}1\), we have
Now take limits as \(\sigma \to 1\) using dominated convergence together with 1 and Lemma 3.1.7 to obtain the result.
With the hypotheses as above, we have
as \(x \to \infty \).
Immediate from the Riemann-Lebesgue lemma, and also noting that \(\int _{-\infty }^{-\log x} \hat\psi (\frac{u}{2\pi })\ du = o(1)\).
If \(I\) is a closed interval contained in an open interval \(J\), then there exists a smooth function \(\Psi : \mathbb {R}\to \mathbb {R}\) with \(1_I \leq \Psi \leq 1_J\).
A standard analysis lemma, which can be proven by convolving \(1_K\) with a smooth approximation to the identity for some interval \(K\) between \(I\) and \(J\). Note that we have “SmoothBumpFunction”s on smooth manifolds in Mathlib, so this shouldn’t be too hard...
The previous corollary also holds for functions \(\psi \) that are assumed to be in the Schwartz class, as opposed to being \(C^2\) and compactly supported.
For any \(R{\gt}1\), one can use a smooth cutoff function (provided by Lemma 3.1.9 to write \(\psi = \psi _{\leq R} + \psi _{{\gt}R}\), where \(\psi _{\leq R}\) is \(C^2\) (in fact smooth) and compactly supported (on \([-R,R]\)), and \(\psi _{{\gt}R}\) obeys bounds of the form
where the implied constants depend on \(\psi \). By Lemma 3.1.7 we then have
Using this and 1 one can show that
(with implied constants also depending on \(A\)), while from Lemma 3.1.1 one has
Combining the two estimates and letting \(R\) be large, we obtain the claim.
The Fourier transform is a bijection on the Schwartz class. [Note: only surjectivity is actually used.]
This is a standard result in Fourier analysis. It can be proved here by appealing to Mellin inversion, Theorem ??. In particular, given \(f\) in the Schwartz class, let \(F : \mathbb {R}_+ \to \mathbb {C}: x \mapsto f(\log x)\) be a function in the “Mellin space”; then the Mellin transform of \(F\) on the imaginary axis \(s=it\) is the Fourier transform of \(f\). The Mellin inversion theorem gives Fourier inversion.
If \(\Psi : (0,\infty ) \to \mathbb {C}\) is smooth and compactly supported away from the origin, then,
as \(x \to \infty \).
By Lemma 3.1.11, we can write
for all \(y{\gt}0\) and some Schwartz function \(\psi \). Making this substitution, the claim is then equivalent after standard manipulations to
and the claim follows from Lemma 3.1.10.
Now we add the hypothesis that \(f(n) \geq 0\) for all \(n\).
For any closed interval \(I \subset (0,+\infty )\), we have
Use Lemma 3.1.9 to bound \(1_I\) above and below by smooth compactly supported functions whose integral is close to the measure of \(|I|\), and use the non-negativity of \(f\).
We have
Apply the preceding proposition with \(I = [\varepsilon ,1]\) and then send \(\varepsilon \) to zero (using 1 to control the error).
3.2 Weak PNT
We have
Already done by Stoll, assuming Wiener-Ikehara.
3.3 Removing the Chebyshev hypothesis
In this section we do *not* assume the bound 1, but instead derive it from the other hypotheses.
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, and \(0 {\lt} x\), then
Repeat the proof of Lemma 3.3.1, but use monotone convergence instead of dominated convergence. (The proof should be simpler, as one no longer needs to establish domination for the sum.)
If \(\psi : \mathbb {R}\to \mathbb {C}\) is \(C^2\) and compactly supported with \(f\) and \(\hat\psi \) non-negative, then there exists a constant \(B\) such that
for all \(x {\gt} 0\).
This readily follows from the previous lemma and the triangle inequality.
One has
for all \(x \geq 1\).
By applying Corollary 3.3.1 for a specific compactly supported function \(\psi \), one can obtain a bound of the form \(\sum _{(1-\varepsilon )x {\lt} n \leq x} f(n) = O(x)\) for all \(x\) and some absolute constant \(\varepsilon \) (which can be made explicit).
If \(C\) is a sufficiently large constant, the claim \(|\sum _{n \leq x} f(n)| \leq Cx\) can now be proven by strong induction on \(x\), as the claim for \((1-\varepsilon )x\) implies the claim for \(x\) by the triangle inequality (and the claim is trivial for \(x {\lt} 1\)).
We have
3.4 The prime number theorem in arithmetic progressions
If \(q ≥ 1\) and \(a\) is coprime to \(q\), and \(\mathrm{Re} s {\gt} 1\), we have
From the Fourier inversion formula on the multiplicative group \((\mathbb {Z}/q\mathbb {Z})^\times \), we have
On the other hand, from standard facts about L-series we have for each character \(\chi \) that
Combining these two facts, we obtain the claim.
If \(q ≥ 1\) and \(a\) is coprime to \(q\), the Dirichlet series \(\sum _{n \leq x: n = a\ (q)} \frac{\Lambda (n)}{n^s}\) converges for \(\mathrm{Re}(s) {\gt} 1\) to \(\frac{1}{\varphi (q)} \frac{1}{s-1} + G(s)\) where \(G\) has a continuous extension to \(\mathrm{Re}(s)=1\).
We expand out the left-hand side using Lemma 3.4.1. The contribution of the non-principal characters \(\chi \) extend continuously to \(\mathrm{Re}(s) = 1\) thanks to the non-vanishing of \(L(s,\chi )\) on this line (which should follow from another component of this project), so it suffices to show that for the principal character \(\chi _0\), that
also extends continuously here. But we already know that
extends, and from Euler product machinery one has the identity
Since there are only finitely many primes dividing \(q\), and each summand \(\frac{\log p}{p^s-1}\) extends continuously, the claim follows.
If \(q ≥ 1\) and \(a\) is coprime to \(q\), we have
3.5 The Chebotarev density theorem: the case of cyclotomic extensions
Throughout this section, \(K\) is a number field, \(m \geq 1\) is a fixed integer, \(L = K(\mu _m)\), and \(G = \mathrm{Gal}(L/K)\). (In particular \(G\) is abelian, and for \(K=\mathbb {Q}\) one recovers \(G \cong (\mathbb {Z}/m\mathbb {Z})^\times \).) Write \(\zeta _K\) and \(\zeta _L\) for the Dedekind zeta functions of \(K\) and \(L\), and for an abelian character \(\chi : G \to \mathbb {C}^\times \) write \(L(\chi ,s)\) for the associated Artin \(L\)-function (equivalently, the Hecke \(L\)-function of the ideal character attached to \(\chi \); cf. Notation 7.1.17 and Proposition 7.1.18 of https://www.math.ucla.edu/ sharifi/algnum.pdf).
The goal of this section is to prove the Chebotarev density theorem in the cyclotomic case (Proposition 3.5.1 below), following the classical argument via Artin \(L\)-functions as in Sharifi, Propositions 7.1.16–7.1.19 and Proposition 7.2.1. The abelian and general cases are then reduced to this one in the subsequent sections.
Let \(\chi : G \to \mathbb {C}^\times \) be an abelian character. For \(\Re (s) {\gt} 1\) one has
where the product runs over nonzero prime ideals of \(\mathcal{O}_K\), with the convention \(\chi (\mathfrak {p}) = 0\) if \(\mathfrak {p}\) ramifies in the fixed field of \(\ker \chi \).
This is the specialisation of the Artin Euler product (Definition 7.1.15 of Sharifi) to abelian characters; see Proposition 7.1.18 of https://www.math.ucla.edu/ sharifi/algnum.pdf. Absolute convergence for \(\Re (s){\gt}1\) follows from comparison with \(\zeta _K(s)\).
For \(\Re (s) {\gt} 1\) one has
where the product runs over all (necessarily one-dimensional) irreducible characters of \(G\).
In general, for a Galois extension \(L/K\) and characters \(\chi \) of irreducible representations of \(\mathrm{Gal}(L/K)\), Proposition 7.1.16 of https://www.math.ucla.edu/ sharifi/algnum.pdf gives \(\zeta _L(s) = \prod _\chi L(\chi ,s)^{\chi (1)}\) for \(\Re (s){\gt}1\). In the present abelian (cyclotomic) setting every irreducible character is one-dimensional, so \(\chi (1)=1\) and the exponents disappear. Alternatively, comparing Euler factors at an unramified prime \(\mathfrak {p}\): the primes of \(L\) above \(\mathfrak {p}\) contribute the factor \((1 - N\mathfrak {p}^{-fs})^{-g}\) to \(\zeta _L\), while the Artin factors multiply to the same quantity by the usual identity \(\prod _\chi (1 - \chi (\varphi _{\mathfrak {p}}) X) = 1 - X^f\) (with \(X = N\mathfrak {p}^{-s}\) and \(f\) the residue degree).
The Dedekind zeta function \(\zeta _L\) admits a meromorphic continuation to a neighbourhood of the line \(\Re (s)=1\) with a single simple pole at \(s=1\) (and is otherwise holomorphic and nonvanishing on that line after removing the pole). In particular \(\log \zeta _L(s) \sim \log (s-1)^{-1}\) as \(s \to 1^+\).
This is the standard analytic continuation of Dedekind zeta (Theorem 7.1.12 of https://www.math.ucla.edu/ sharifi/algnum.pdf): absolute convergence of the ideal Dirichlet series and Euler product for \(\Re (s){\gt}1\), meromorphic continuation across \(\Re (s){\gt}1-[L:\mathbb {Q}]^{-1}\), and a simple pole at \(s=1\). The logarithmic asymptotic is immediate from the simple pole.
Let \(\chi : G \to \mathbb {C}^\times \) be a nontrivial character. Then \(L(\chi ,s)\) extends to an analytic function on the half-plane \(\Re (s) {\gt} 1 - [K:\mathbb {Q}]^{-1}\), and in particular is holomorphic at \(s=1\).
Following Proposition 7.1.19 of https://www.math.ucla.edu/ sharifi/algnum.pdf: let \(n\) be the order of \(\chi \). Geometry of numbers supplies the estimate that, for each \(n\)th root of unity \(\zeta \), the number of ideals \(\mathfrak {a}\) with \(N\mathfrak {a} \leq N\) and \(\chi (\mathfrak {a})=\zeta \) is \(CN + O\bigl(N^{1-[K:\mathbb {Q}]^{-1}}\bigr)\) with \(C\) independent of \(\zeta \). Summing against \(\zeta \) cancels the main terms, so \(\sum _{N\mathfrak {a}\leq N} \chi (\mathfrak {a}) = O\bigl(N^{1-[K:\mathbb {Q}]^{-1}}\bigr)\). Lemma 7.1.5 of Sharifi then yields absolute uniform convergence of \(\sum _{\mathfrak {a}} \chi (\mathfrak {a})\, N\mathfrak {a}^{-s}\) on compact subsets of \(\Re (s) {\gt} 1 - [K:\mathbb {Q}]^{-1}\).
For any nontrivial character \(\chi \) of \(G = \mathrm{Gal}(L/K)\), one has \(L(\chi ,1) \neq 0\). Moreover \(L(\chi ,s)\) does not vanish for \(\Re (s)=1\).
For the value at \(s=1\): write \(\log \zeta _L(t) = \sum _\chi \log L(\chi ,t)\) for real \(t{\gt}1\). As \(t\to 1^+\), the left side is \(\sim \log (t-1)^{-1}\) by Lemma 3.5.3. If some nontrivial \(L(\chi ,\cdot )\) had a zero of order \(m_\chi \geq 1\) at \(s=1\), the right side would behave like \(\bigl(1 - \sum _\chi m_\chi \bigr)\log (t-1)^{-1}\) up to a bounded error (the trivial character contributes the pole of \(\zeta _K\), absorbed into the factorisation), forcing \(1-\sum m_\chi \leq 0\), a contradiction. Hence \(L(\chi ,1)\neq 0\) for all nontrivial \(\chi \) (Sharifi, Proposition 7.1.19).
For the rest of the line \(\Re (s)=1\), \(s\neq 1\): adapt the classical nonvanishing argument for Dirichlet \(L\)-functions (comparison of \(\zeta _L(s)\), \(\zeta _L(s)^3\, |L(\chi ,s)|^4\), or the \(3+4\operatorname {Re}\) inequality) using the Euler product of Lemma 3.5.1 and the absence of poles of nontrivial \(L(\chi ,\cdot )\) on \(\Re (s)=1\) from Lemma 3.5.4.
Fix \(\sigma \in G\). For every prime ideal \(\mathfrak {p}\) of \(\mathcal{O}_K\) unramified in \(L\),
Equivalently, if \(\sigma (\zeta _m) = \zeta _m^a\) with \(\gcd (a,m)=1\), the sum equals \(|G|\) precisely when \(N\mathfrak {p} \equiv a \pmod{m}\).
Standard character orthogonality on the finite abelian group \(G\). The reformulation in terms of \(N\mathfrak {p} \bmod m\) uses that \(\varphi _{\mathfrak {p}}(\zeta _m) = \zeta _m^{N\mathfrak {p}}\) for the cyclotomic character.
For every \(\sigma \in G\), the set of prime ideals \(\mathfrak {p}\) of \(K\) unramified in \(L\) with Frobenius \(\varphi _{\mathfrak {p}} = \sigma \) has Dirichlet density \(1/|G|\).
This is Proposition 7.2.1 of https://www.math.ucla.edu/ sharifi/algnum.pdf. For \(\Re (s){\gt}1\) one has \(\log L(\chi ,s) \sim \sum _{\mathfrak {p}} \chi (\mathfrak {p})\, N\mathfrak {p}^{-s}\). Summing against \(\chi (\sigma )^{-1}\) and applying Lemma 3.5.6 yields
On the other hand, by Lemmas 3.5.2 and 3.5.5 the left side is \(\sim \log \zeta _K(s) \sim \log (s-1)^{-1}\) as \(s\to 1^+\) (only the trivial character contributes a pole). Comparing the two asymptotics gives the asserted Dirichlet density.
For any nontrivial character \(\chi \) of \(G\),
(Equivalently, writing \(\Lambda _\chi \) for the von Mangoldt-type coefficients of \(-\frac{L'}{L}(\chi ,s)\), one has \(\sum _{n\leq x} \Lambda _\chi (n) = o(x)\).)
Remark on prime powers. Expanding \(-\frac{L'}{L}(\chi ,s)\) as a Dirichlet series selects coefficients supported on prime powers \(\mathfrak {p}^j\) with weight \(\chi (\varphi _{\mathfrak {p}})^j\log N\mathfrak {p}\). Thus an exact identity expressing \(-\sum _\chi \chi (\sigma )^{-1}\frac{L'}{L}(\chi ,s)\) as a sum over primes with Frobenius \(\sigma \) should use the condition \(\varphi _{\mathfrak {p}}^j=\sigma \) rather than \(\varphi _{\mathfrak {p}}=\sigma \). For the density statement in Proposition 3.5.1 this distinction is immaterial: the \(j=1\) terms agree with the Frobenius condition, while the \(j\geq 2\) terms converge absolutely for \(\Re (s){\gt}1/2\) and do not affect the residue at \(s=1\).
By Lemmas 3.5.4 and 3.5.5, \(L(\chi ,s)\) is holomorphic and nonvanishing on \(\Re (s)\geq 1\), so \(-\frac{L'}{L}(\chi ,s)\) extends continuously to \(\Re (s)\geq 1\). The claimed prime-sum estimate then follows by the same Wiener–Ikehara / Ingham contour argument used for Dirichlet \(L\)-functions in the prime-number theorem in arithmetic progressions (cf. the material already formalised for Dirichlet \(L\)-functions earlier in this file). Passing through \(\Lambda _\chi \) (rather than the bare prime sum) makes the \(j\geq 2\) prime-power contributions explicit and shows they are \(O(x^{1/2}\log x)\), hence absorbable in the \(o(x)\) error; cf. the remark after the statement.
3.6 The Chebotarev density theorem: the case of abelian extensions
Now let \(L/K\) be an arbitrary finite abelian extension with Galois group \(G\), and fix \(\sigma \in G\). The goal is to show that the primes of \(K\) with Frobenius \(\sigma \) still have Dirichlet density \(1/|G|\), by reducing to the cyclotomic case already treated. (Cf. Theorem 7.2.2, Step 2, of https://www.math.ucla.edu/ sharifi/algnum.pdf; cyclic extensions already suffice for the later reduction to the general case.)
Choose an integer \(m\geq 1\) not dividing the discriminant of \(L/K\), large enough that \(H := \mathrm{Gal}(L(\mu _m)/L) \cong (\mathbb {Z}/m\mathbb {Z})^\times \) via the cyclotomic character and \(\mathrm{Gal}(L(\mu _m)/K) \cong G \times H\). For \(\sigma \in G\) and \(\tau \in H\) write \(S_{\sigma ,\tau }\) for the set of primes of \(K\) unramified in \(L(\mu _m)\) with Frobenius \((\sigma ,\tau )\in G\times H\), and \(S_\sigma \) for the set of primes of \(K\) unramified in \(L\) with Frobenius \(\sigma \) in \(G\). Then
Every prime counted in \(S_\sigma \) lifts to primes in the various \(S_{\sigma ,\tau }\) according to the Frobenius in the cyclotomic layer; the identity of lower densities is the usual additivity of Dirichlet densities over a finite partition (up to the finitely many ramified primes).
In the notation of Lemma 3.6.1, if the order of \(\tau \in H\) is divisible by \(|G|\), then \(\langle (\sigma ,\tau )\rangle \cap (G\times \{ 1\} ) = \{ 1\} \), so \(L(\mu _m)\) is obtained by adjoining \(\mu _m\) to the fixed field \(F := L(\mu _m)^{\langle (\sigma ,\tau )\rangle }\). Applying Proposition 3.5.1 to the cyclotomic extension \(F(\mu _m)/F\) and transporting densities as in the reduction step of Theorem 7.2.2 yields \(\delta (S_{\sigma ,\tau }) = 1/(|G|\, |H|)\).
See Sharifi, Theorem 7.2.2, Step 2 (display after “Now suppose that \(|G|\) divides the order of \(\tau \)”).
Let \(L/K\) be finite abelian with Galois group \(G\). For every \(\sigma \in G\), the Dirichlet density of primes of \(K\) with Frobenius \(\sigma \) exists and equals \(1/|G|\).
Let \(H_n \subset H\) be the set of elements whose order is divisible by \(n\). Summing the densities of Lemma 3.6.2 over \(\tau \in H_n\) gives \(\delta _{\mathrm{inf}}(S_\sigma ) \geq |H_n|/(|G|\, |H|)\). Choosing \(m\) so that \(|H_n|/|H|\) is arbitrarily close to \(1\) (possible since there are primes \(m\equiv 1\pmod{n^j}\) by the cyclotomic case already proved, and \(|H_n|/|H|\) tends to \(1\) as \(j\to \infty \)), one obtains \(\delta _{\mathrm{inf}}(S_\sigma ) \geq 1/|G|\). Summing over \(\sigma \in G\) forces equality throughout, so each density exists and equals \(1/|G|\).
3.7 The Chebotarev density theorem: the general case
Finally let \(L/K\) be an arbitrary finite Galois extension with group \(G\), and let \(C\subset G\) be a conjugacy class. The theorem reduces to the abelian (indeed cyclic) case already proved, by passing to the fixed field of an element of \(C\).
Fix \(\sigma \in C\) and let \(E\) be the fixed field of \(\langle \sigma \rangle \), so \(L/E\) is cyclic of degree \(f = |\langle \sigma \rangle |\). Write \(S\) for the set of primes of \(K\) unramified in \(L\) whose Frobenius class equals \(C\), and \(T_\sigma \) for the set of primes \(\mathfrak {P}\) of \(E\) unramified in \(L\) (and lying over \(K\)) with Frobenius equal to \(\sigma \). Then
whenever either density exists. In particular, the Chebotarev statement for \(L/K\) and \(C\) follows from the abelian statement for the cyclic extension \(L/E\) and the element \(\sigma \).
If \(\mathfrak {P}\in T_\sigma \), then \(\varphi _{\mathfrak {P}}=\sigma \) fixes \(E\), so \(\mathfrak {P}\) has residue degree one over \(K\). There are exactly \(|G|/f\) primes of \(L\) over \(\mathfrak {P}\cap K\), and their Frobenii are equidistributed among the \(|C|\) elements of the conjugacy class \(C\); exactly \(|G|/(f\, |C|)\) of them have Frobenius \(\sigma \). Comparing the Dirichlet series \(\sum N\mathfrak {p}^{-s}\) over \(S\) with \(\sum N\mathfrak {P}^{-s}\) over \(T_\sigma \) (and using \(\sum _{\mathfrak {p}} N\mathfrak {p}^{-s} \sim \sum _{\mathfrak {P}} N\mathfrak {P}^{-s}\)) yields the displayed identity. See Sharifi, Theorem 7.2.2, Step 1.
Let \(L/K\) be a finite Galois extension of number fields with Galois group \(G\), and let \(C\subset G\) be a conjugacy class. The set of prime ideals \(\mathfrak {p}\) of \(K\) unramified in \(L\) whose Frobenius conjugacy class equals \(C\) has Dirichlet density \(|C|/|G|\).