2.1 Resultants and discriminants

Although the material in this subsection is classical, it is surprisingly difficult to find it all stated in one convenient reference. Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0$ and $g(x) = b_m x^m + b_{m-1} x^{m-1} + \cdots + b_0$ be complex polynomials of positive degrees n and m, respectively. Their resultant $R(f(x),g(x);x)$ is the determinant of the $(m+n)\times (m+n)$ Sylvester matrix

(2-1) $$ \begin{align} {\begin{bmatrix} a_{n}&0&\cdots &0&b_{m}&0&\cdots &0\\ a_{n-1}&a_{n}&\cdots &0&b_{m-1}&b_{m}&\cdots &0\\[-3pt] a_{n-2}&a_{n-1}&\ddots &0&b_{m-2}&b_{m-1}&\ddots &0\\ \vdots &\vdots &\ddots &a_{n}&\vdots &\vdots &\ddots &b_{m}\\[-3pt] a_{0}&a_{1}&\cdots &\vdots &b_{0}&b_{1}&\cdots &\vdots \\[-3pt] 0&a_{0}&\ddots &\vdots &0&b_{0}&\ddots &\vdots \\ \vdots &\vdots &\ddots &a_{1}&\vdots &\vdots &\ddots &b_{1}\\ 0&0&\cdots &a_{0}&0&0&\cdots &b_{0} \end{bmatrix}. } \end{align} $$

We often suppress the variable x and write $R(f,g)$ instead of $R(f(x),g(x);x)$ . Since (2-1) has m columns with the coefficients of f and n columns with the coefficients of g, it follows that $R( \alpha f, \beta g) = \alpha ^m \beta ^n R(f,g)$ for $\alpha , \beta \in \mathbb {C}$ . The definition ensures that $R(f,g) \in \mathbb {Z}$ whenever $f,g \in \mathbb {Z}[x]$ .

We now need an important fact about polynomials. Suppose that $f\in \mathbb {Z}[x]$ and $f = AB$ , in which $A,B\in \mathbb {Q}[x]$ . Gauss’ lemma ensures that $f = ab$ , where $a,b \in \mathbb {Z}[x]$ are of the form $a = \mu A$ and $b = \mu ^{-1} B$ for some $\mu \in \mathbb {Q}$ [Reference Dummit and Foote5, Proposition 5, p. 303].

For $f,g \in \mathbb {C}[x]$ with leading coefficients $a_n$ and $b_m$ , respectively,

(2-2) $$ \begin{align} R(f,g) = a_n^m b_m^n \prod_{ \substack{1 \leq i \leq m \\ 1 \leq j \leq n}} (\lambda_i - \mu_j), \end{align} $$

in which $\lambda _1,\lambda _2,\ldots ,\lambda _n$ and $\mu _1,\mu _2,\ldots ,\mu _m$ are the roots of f and g in $\mathbb {C}$ , counted by multiplicity [Reference Dummit and Foote5, Example 31c, p. 621]. For $c \in \mathbb {C}$ and $h \in \mathbb {C}[x]$ , (2-2) implies that

(2-3) $$ \begin{align} R(fg,h) = R(f,h) R(g,h) \quad \text{and}\quad R(f(cx),g(cx)) = c^{mn} R(f,g). \end{align} $$

The discriminant

$$ \begin{align*} D(f;x) = \frac{(-1)^{n(n-1)/2}}{a_n} R(f(x),f'(x);x) \end{align*} $$

of $f \in \mathbb {C}[x]$ with degree n and leading coefficient $a_n$ is a homogeneous polynomial of degree $2n-2$ in the coefficients of f. The choice of sign ensures that $D(f;x) \geq 0$ if the roots of f are real. We often write $D_f$ or $D_{f(x)}$ instead of $D(f;x)$ .

If $\deg f = 1$ , then $D_f = 1$ . If $f \in \mathbb {Z}[x]$ and $\deg f \geq 2$ , then $D_f \in \mathbb {Z}$ . Lemma 3 ensures that $D_f = 0$ if and only if f and $f'$ share a common divisor in $\mathbb {Z}[x]$ of positive degree; that is, if and only if f has a repeated root in $\mathbb {C}$ . This yields the following.

The definition and (2-2) yield

$$ \begin{align*} D_f = a_n^{2n-2} \prod_{1 \leq i < j\leq n} (\lambda_i - \lambda_j)^2, \end{align*} $$

in which $\lambda _1,\lambda _2,\ldots ,\lambda _n$ are the roots of f, repeated according to multiplicity. For $\alpha \in \mathbb {C}$ , (2-3) implies that

(2-4) $$ \begin{align} D_{f(\alpha x)} = \alpha^{n(n-1)} D_{f(x)} \quad \text{and} \quad D_{gh} = D_g D_h R_{g,h}^2. \end{align} $$

If $f,g \in \mathbb {Z}[x]$ and $g|f$ , then $D_g | D_f$ . If, in addition, $D_f \neq 0$ , then $|D_g| \leq |D_f|$ .

The final lemma of this subsection (an easy consequence of Lagrange’s theorem) concerns the number of solutions $\omega _f(p)$ to $f(x) \equiv 0 \,\,(\operatorname {mod}p)$ for a rational prime p. Note that if $f\in \mathbb {Z}[x]$ is reducible, then $\omega _f(p)> \deg f$ is possible. For example, consider $3x-6$ modulo $3$ .

2.2 Dedekind zeta residues

Let $\mathbb {K}$ be a number field of degree $n_{\mathbb {K}}$ with ring of algebraic integers $\mathcal {O}_{\mathbb {K}}$ and let $\Delta _{\mathbb {K}} $ denote the discriminant of $\mathbb {K}$ . In what follows, $\mathfrak {p}\subset \mathcal {O}_{\mathbb {K}}$ denotes a prime ideal, $\mathfrak {a}\subset \mathcal {O}_{\mathbb {K}}$ an ideal (not necessarily prime), and $N(\mathfrak {a})$ the norm of $\mathfrak {a}$ . The Dedekind zeta function $\zeta _{\mathbb {K}}(s) = \sum _{\mathfrak {a}}N(\mathfrak {a})^{-s}$ of $\mathbb {K}$ is analytic on $\operatorname {Re}{s}> 1$ and extends meromorphically to $\mathbb {C}$ , except for a simple pole at $s=1$ . The analytic class number formula asserts that the residue of $\zeta _{\mathbb {K}}(s)$ at $s=1$ is

$$ \begin{align*} \kappa_{\mathbb{K}} = \frac{2^{r_1}(2\pi)^{r_2}h_{\mathbb{K}}R_{\mathbb{K}}}{w_{\mathbb{K}}\sqrt{| \Delta_{\mathbb{K}} |}}, \end{align*} $$

in which $r_1$ is the number of real places of $\mathbb {K}$ , $r_2$ is the number of complex places of $\mathbb {K}$ , $w_{\mathbb {K}}$ is the number of roots of unity in $\mathbb {K}$ , $h_{\mathbb {K}}$ is the class number of $\mathbb {K}$ , and $R_{\mathbb {K}}$ is the regulator of $\mathbb {K}$ [Reference Lang9]. The nontrivial zeros of $\zeta _{\mathbb {K}}(s)$ lie in the critical strip $0 < \operatorname {Re} s < 1$ , in which there might also exist an exceptional, real zero $0 < \beta < 1$ . The generalized Riemann hypothesis, which remains unproven, asserts that the nontrivial zeros of $\zeta _{\mathbb {K}}(s)$ satisfy $\operatorname {Re} s = \tfrac 12$ and that no exceptional zero exists.

We require unconditional bounds for the residue $\kappa _{\mathbb {K}}$ of $\zeta _{\mathbb {K}}(s)$ at $s=1$ . If $d=1$ , then $\mathbb {K} = \mathbb {Q}$ and hence $\zeta _{\mathbb {Q}}(s)$ is the Riemann zeta function, for which $\kappa _{\mathbb {Q}} = 1$ . Consequently, we restrict our attention to the case $d \geq 2$ .

Lemma 7. Let $g \in \mathbb {Z}[x]$ be irreducible with degree $d \geq 2$ and leading coefficient c. If $g(\alpha ) = 0$ and $\mathbb {K} = \mathbb {Q}(\alpha )$ , then

(2-5) $$ \begin{align} \frac{0.362\,32}{\sqrt{\mathbf{D}_g}} \,\leq\, \kappa_{\mathbb{K}} \,\leq\, \bigg(\frac{e\log \mathbf{D}_g}{2(d-1)} \bigg)^{d - 1}, \end{align} $$

in which $\mathbf {D}_g = |c|^{(d-1)(d-2)}|D_g|$ .

Proof. Let $g \in \mathbb {Z}[x]$ be irreducible with degree $d \geq 2$ and leading coefficient c. Then $c^{d - 1} g(x) = h(cx)$ , in which $h \in \mathbb {Z}[x]$ is monic, irreducible, and has degree d. Let $\alpha ~\in ~\mathbb {C}$ be such that $g(\alpha ) = 0$ . Then $h(c\alpha ) = 0$ and $\mathbb {K} = \mathbb {Q}(\alpha ) = \mathbb {Q}(c\alpha )$ ; the degree $n_{\mathbb {K}}$ of the number field $\mathbb {K}$ is d. Since the discriminant of $c^{d - 1} g(x)$ is $(c^{(d-1)})^{2d-2} D_g$ and the discriminant of $h(cx)$ is $c^{d(d-1)} D_h$ ,

$$ \begin{align*} D_h = c^{(d-1)(d-2)} D_g. \end{align*} $$

From [Reference Neukirch12, Proposition I.2.12], observe that

$$ \begin{align*} 1 \leq [\mathcal{O}_{\mathbb{K}} : \mathbb{Z}[c\alpha]]^2 = \bigg( \frac{ \det \mathbb{Z}[c\alpha] }{ \det \mathcal{O}_{\mathbb{K}} } \bigg)^2 = \frac{|D_h| }{ | \Delta_{\mathbb{K}} |} \end{align*} $$

and hence

(2-6) $$ \begin{align} | \Delta_{\mathbb{K}} | \leq |D_h| = \mathbf{D}_g. \end{align} $$

Since $n_{\mathbb {K}} = d \geq 2$ , Louboutin [Reference Louboutin10, Theorem 1] provided the first inequality in

(2-7) $$ \begin{align} \kappa_{\mathbb{K}} \leq \bigg(\frac{e\log{| \Delta_{\mathbb{K}} |}}{2(n_{\mathbb{K}} - 1)}\bigg)^{n_{\mathbb{K}} - 1} \leq \bigg(\frac{e \log \mathbf{D}_g}{2(d-1)} \bigg)^{d - 1}. \end{align} $$

The lower bound follows from [Reference Garcia and Lee7]; the argument is short and reproduced here. Since $n_{\mathbb {K}} = r_1 + 2r_2\geq 2$ , we have $2^{r_1}(2\pi )^{r_2} \geq 2^{2}(2\pi )^{0} = 4$ . Friedman [Reference Friedman6, Theorem B] proved that $R_{\mathbb {K}}/w_{\mathbb {K}} \geq 0.090\,58$ , improving on Zimmert [Reference Zimmert20] (see [Reference Lang9, Theorem 7, p. 273]). Thus,

(2-8) $$ \begin{align} \kappa_{\mathbb{K}} \geq \frac{2^{r_1}(2\pi)^{r_2}R_{\mathbb{K}}}{w_{\mathbb{K}}\sqrt{ | \Delta_{\mathbb{K}}|}}> \frac{4\times 0.090\,58}{\sqrt{\Delta_{\mathbb{K}}}} = \frac{0.362\,32}{\sqrt{|\Delta_{\mathbb{K}}|}} \geq \frac{0.362\,32}{\sqrt{\mathbf{D}_g}}. \end{align} $$

This concludes the proof.

Remark 8. With respect to $\mathbf {D}_g$ , there are asymptotically superior lower bounds available (with explicit constants). For example, one can obtain

(2-9) $$ \begin{align} \kappa_{\mathbb{K}}> \frac{0.015\,744\,605}{d\,d! \,\mathbf{D}_g^{1/d}} \end{align} $$

from Stark [Reference Stark16] by tracking the constants involved [Reference Garcia and Lee7]. However, the dependence upon $d = n_{\mathbb {K}}$ is poor, so we use the degree-independent lower bound in (2-5) instead. The bound (2-9) is superior to (2-5) only for astronomically large $\mathbf {D}_g$ ; see Table 2.

Table 2

Values of $\mathbf {D}_g$ past which the lower bound (2-9) is greater than (2-5).

2.3 The function $\mathbf {M}_{\mathbb {K}}(x)$

For a number field $\mathbb {K}$ , define the Mertens-type sum

$$ \begin{align*} \mathbf{M}_{\mathbb{K}}(x) = \sum_{ N(\mathfrak{p})\leq x} \frac{1}{N(\mathfrak{p})}, \end{align*} $$

where the sum runs over the prime ideals $\mathfrak {p}$ of $\mathcal {O}_{\mathbb {K}}$ . If $\mathbb {K} = \mathbb {Q}$ , then the sum reduces to the classical Mertens sum

(2-10) $$ \begin{align} \mathbf{M}_{\mathbb{Q}}(x) = \sum_{p\leq x} \frac{1}{p} \leq \log \log x + C_{\mathbb{Q}} + \dfrac{1}{(\log x)^2}, \end{align} $$

in which $C_{\mathbb {Q}} = 0.261\,49\cdots $ is the Meissel–Mertens constant and (2-10) holds unconditionally for ${x> 1}$ [Reference Rosser and Schoenfeld14, formula (3.20)]. For $\mathbb {K} \neq \mathbb {Q}$ , the first two authors [Reference Garcia and Lee7] used an ideal-counting estimate of Sunley [Reference Sunley18, Reference Sunley19] to prove unconditionally for $x \geq 2$ that

(2-11) $$ \begin{align} \mathbf{M}_{\mathbb{K}}(x) = \log \log x + C_{\mathbb{K}} + B_{\mathbb{K}}(x), \end{align} $$

in which

(2-12) $$ \begin{align} \gamma + \log \kappa_{\mathbb{K}} - d_{\mathbb{K}} \,\leq\, C_{\mathbb{K}} \,\leq\, \gamma + \log \kappa_{\mathbb{K}} \end{align} $$

and

(2-13) $$ \begin{align} |B_{\mathbb{K}}(x)| \leq \frac{2 \Upsilon_{\mathbb{K}}}{\log x}, \end{align} $$

where

(2-14) $$ \begin{align} \Upsilon_{\mathbb{K}}= \bigg( \frac{(n_{\mathbb{K}}+1)^2}{2\kappa_{\mathbb{K}}(n_{\mathbb{K}}-1)}\Lambda_{\mathbb{K}} + 1\bigg) + \frac{0.55\, \Lambda_{\mathbb{K}}n_{\mathbb{K}}(n_{\mathbb{K}}+1)}{\kappa_{\mathbb{K}}} + n_{\mathbb{K}} + 40.31 \dfrac{\Lambda_{\mathbb{K}}n_{\mathbb{K}}}{\kappa_{\mathbb{K}}} \end{align} $$

and

(2-15) $$ \begin{align} \Lambda_{\mathbb{K}} = e^{28.2n_{\mathbb{K}}+5} (n_{\mathbb{K}}+1)^{ {5(n_{\mathbb{K}}+1)}/{2}} | \Delta_{\mathbb{K}} |^{{1}/({n_{\mathbb{K}}+1})} (\log | \Delta_{\mathbb{K}} |)^{n_{\mathbb{K}}}. \end{align} $$

The Dedekind zeta residue $\kappa _{\mathbb {K}}$ in (2-14) can be explicitly bounded by Lemma 7.

2.4 The function $\mathbf {P}(x)$

For $x\geq 1$ , define

$$ \begin{align*} \mathbf{P}(x) = \sum_{2\leq k\leq x} \frac{P(k)}{k}, \end{align*} $$

in which $P(s) = \sum _p p^{-s}$ is the prime zeta function. Then $\mathbf {P}(1) = 0$ and $\mathbf {P}(n)$ tends monotonically to $0.315\,718\,45\ldots $ as $n \to \infty $ ; see Table 3. That $\lim _{x\to \infty }\mathbf {P}(x)$ exists follows from an old result of Euler (1781) [Reference Srivastava15, formula (2.12), p. 133]:

$$ \begin{align*} \mathbf{P}(x) < \sum_{k \geq 2} \frac{1}{k} \sum_p \frac{1}{p^k} < \sum_{k = 2}^{\infty} \frac{1}{k}\sum_{n= 2}^{\infty} \frac{1}{n^k} = \sum_{k = 2}^{\infty} \frac{\zeta(k)-1}{k} = 1-\gamma. \end{align*} $$

A convenient upper bound is $\mathbf {P}(x) \leq 0.32$ for all $x \geq 2$ .

Table 3

Approximate values of $\mathbf {P}(x)$ .

4.1 Preliminary lemmas

The next two lemmas set up the final estimates needed for the proof of Theorem 2. Recall that $\omega _f(p)$ denotes the number of solutions to $f(x) \equiv 0 \,\,(\operatorname {mod}p)$ . The first lemma concerns the additive structure of $\omega _f(p)$ .

Lemma 11. Suppose that $f =f_1 f_2 \cdots f_k \in \mathbb {Z}[x]$ is a product of distinct, irreducible nonconstant polynomials $f_1, f_2,\ldots ,f_k \in \mathbb {Z}[x]$ . If $p> |D_f|$ , then

$$ \begin{align*} \omega_f(p) = \omega_{f_1}(p) + \cdots + \omega_{f_k}(p). \end{align*} $$

Proof. Each zero of f in $\mathbb {F}_p$ is a zero of some $f_i$ . Thus,

(4-1) $$ \begin{align} \omega_f(p) \leq \omega_{f_1}(p) + \cdots + \omega_{f_k}(p) \end{align} $$

holds for any p. Since every zero of $f_i$ in $\mathbb {F}_p$ is a zero of f, it suffices to show that $f_i$ and $f_j$ have no common zeros in $\mathbb {F}_p$ if $i \neq j$ and $p> |D_f|$ . If $i \neq j$ , then there are $u_{ij}, v_{ij} \in \mathbb {Z}[x]$ such that

(4-2) $$ \begin{align} u_{ij} f_i + v_{ij} f_j = R(f_i,f_j), \end{align} $$

in which $R(f_i,f_j) \in \mathbb {Z}$ is the resultant of $f_i$ and $f_j$ [Reference Cox, Little and O’Shea3, Proposition 5, Section 3.6]. Since $f_i,f_j \in \mathbb {Z}[x]$ are distinct and irreducible, they share no common factors over $\mathbb {C}$ , so $R(f_i,f_j) \neq 0$ [Reference Cox, Little and O’Shea3, Corollary 4, Section 3.6]. If $f_i,f_j$ have a common zero in $\mathbb {F}_p$ , then (4-2) ensures that $p \mid R(f_i,f_j)$ . Consequently, $p \mid D_f$ (Lemma 5) and hence $p \leq |D_f|$ since $D_f \neq 0$ (Lemma 4). Thus, $p> |D_f|$ implies that (4-1) is an equality.

The second lemma is an asymptotic estimate for a special sum over $\omega _f(p)$ .

Lemma 12. Suppose that $f =f_1 f_2 \cdots f_k \in \mathbb {Z}[x]$ is a product of distinct, irreducible, nonconstant polynomials $f_1, f_2, \ldots , f_k \in \mathbb {Z}[x]$ of degrees $d_1, d_2, \ldots , d_k \geq 1$ , respectively. For $x \geq \sqrt { \mathbf {D}_f}$ ,

(4-3) $$ \begin{align} \bigg|\sum_{p \leq x} \frac{\omega_f(p)}{p} - k \log \log x \bigg| \leq d\, \mathbf{M}_{\mathbb{Q}}( | D_f| ) + \mathbf{A} + \mathbf{B}(x) + \mathbf{C}, \end{align} $$

in which

(4-4) $$ \begin{align} d = d_1+d_2+\cdots+d_k \end{align} $$

and

$$ \begin{align*} \mathbf{A} = \sum_{i=1}^k |A_{f_i}| ,\quad \mathbf{B}(x) = \sum_{i=1}^k |B_{\mathbb{K}_i}(x)| ,\quad \text{and}\quad \mathbf{C} = \sum_{i=1}^k |C_{\mathbb{K}_i}|. \end{align*} $$

Here $A_{f_i}$ , $B_{\mathbb {K}_i}(x)$ , and $C_{\mathbb {K}_i}$ are given by (1-1), (2-13), and (2-11), respectively.

Proof. For each $i=1,2,\ldots ,k$ , let $\alpha _i$ be a root of $f_i$ and define $\mathbb {K}_i =\mathbb {Q}(\alpha _i)$ , which has degree $d_i$ since each $f_i$ is irreducible. Observe that $D_f \neq 0$ (since f has no repeated roots in $\mathbb {C}$ ),

$$ \begin{align*} k \leq d, \quad\text{and}\quad 1\leq |D_{f_1}|, |D_{f_2}|,\ldots, |D_{f_k}| \leq |D_f|. \end{align*} $$

Without loss of generality, let $c\geq 1$ denote the leading coefficient of f and let $c_1,c_2,\ldots ,c_k \geq 1$ denote the leading coefficients of $f_1,f_2,\ldots ,f_k$ , respectively. Then $c = c_1 c_2 \cdots c_k$ and $c_1,c_2,\ldots ,c_k \leq c$ .

If $x> |D_f|$ , then Lemma 11 implies that

$$ \begin{align*} \sum_{p \leq x} \frac{\omega_f(p)}{p} &= \sum_{p \leq |D_f|} \frac{\omega_f(p)}{p} + \sum_{|D_f| < p \leq x} \frac{ \omega_f(p)}{p} \\ &= \sum_{p \leq |D_f|} \frac{\omega_f(p)}{p} + \sum_{|D_f| < p \leq x}\sum_{i=1}^k \frac{ \omega_{f_i}(p)}{p} \\ &= \sum_{p \leq |D_f|} \frac{\omega_f(p)}{p} + \sum_{i=1}^k \bigg(\sum_{p \leq x} \frac{ \omega_{f_i}(p)}{p} - \sum_{p \leq |D_f|} \frac{ \omega_{f_i}(p)}{p}\bigg)\\ &= \underbrace{\sum_{p \leq |D_f|} \frac{\omega_f(p)- \sum_{j=1}^{k}\omega_{f_j}(p)}{p} }_{I} + \underbrace{\sum_{i=1}^k \sum_{p \leq x} \frac{ \omega_{f_i}(p)}{p} }_{II}. \end{align*} $$

Term I. Observe that

$$ \begin{align*}\quad |I| &= \bigg|\sum_{p \leq |D_f|} \frac{\omega_f(p)-( \omega_{f_1}(p)+\omega_{f_2}(p) +\cdots + \omega_{f_k}(p))}{p} \bigg| \\ &\leq \sum_{p \leq |D_f|} \frac{ \omega_{f_1}(p)+\omega_{f_2}(p)+\cdots + \omega_{f_k}(p)}{p} && \text{by (4-1)} \\[-12pt] \\ &\leq \sum_{p \leq |D_f|} \frac{d_1+d_2+\cdots + d_k }{p} && \text{by Lemma 6} \\ &\leq \sum_{p \leq |D_f|} \frac{d}{p} && \text{by (4-4)}\\ &\leq d\,\mathbf{M}_{\mathbb{Q}}(|D_f|). \end{align*} $$

Term II. Since $|c_i| \leq |c|$ and $d_i \leq d$ ,

(4-5) $$ \begin{align} \mathbf{D}_{f_i} \leq \mathbf{D}_f \end{align} $$

for $i=1,2,\ldots ,k$ . For $x \geq \sqrt {\mathbf {D}_f} \geq \max _{1\leq i \leq k} \sqrt {\mathbf {D}_{f_i}}$ , Theorem 1 ensures that

$$ \begin{align*} II &= \sum_{i=1}^k \sum_{p \leq x} \frac{ \omega_{f_i}(p)}{p} \\ &= \sum_{i=1}^k ( \mathbf{M}_{\mathbb{K}_i}(x) + A_{f_i} ) && \text{by Theorem 1}\\ &= k \log \log x + \sum_{i=1}^k (A_{f_i} +B_{\mathbb{K}_i}(x) + C_{\mathbb{K}_i} ) && \text{by (2-11)}\\ &= k \log \log x + \mathbf{A} + \mathbf{B}(x) + \mathbf{C}. \end{align*} $$

Putting this all together, we obtain (4-3).

To complete the proof of Theorem 2, we must estimate $\mathbf {A}$ , $\mathbf {B}(x)$ , and $\mathbf {C}$ .

4.2 Estimating $\mathbf {A}$

Since $\mathbf {M}_{\mathbb {Q}}$ and $\mathbf {P}$ are increasing, Theorem 1 and (4-4) provide

$$ \begin{align*} \mathbf{A} = \sum_{i=1}^k |A_{f_i}| &\leq \sum_{i=1}^k d_i\bigg(\mathbf{M}_{\mathbb{Q}}(|c_i|) + \mathbf{M}_{\mathbb{Q}}\bigg(\sqrt{\mathbf{D}_{f_i}}\bigg) + 0.64\bigg)\\ &\leq \bigg(\sum_{i=1}^k d_i\bigg)\bigg(\mathbf{M}_{\mathbb{Q}}(|c|) + \mathbf{M}_{\mathbb{Q}}\bigg(\sqrt{\mathbf{D}_{f}}\bigg) + 0.64\bigg) \\ &\leq d\,\bigg( \mathbf{M}_{\mathbb{Q}}(|c|) + \mathbf{M}_{\mathbb{Q}}\bigg( \sqrt{\mathbf{D}_f}\bigg) + 0.64 \bigg). \end{align*} $$

4.3 Estimating $\mathbf {B}(x)$

The degree $d_i$ of each irreducible factor $f_i$ of f equals the degree $n_{\mathbb {K}_i}$ of the number field $\mathbb {K}_i$ generated by a root of $f_i$ ; that is, $n_{\mathbb {K}_i} = d_i$ for $i=1,2,\ldots ,k$ . Suppose for now that each $d_i \geq 2$ . Then (2-6) and (2-15) imply that

$$ \begin{align*} \Lambda_{\mathbb{K}_i} &= e^{28.2 n_{\mathbb{K}_i}+5}(n_{\mathbb{K}_i}+1)^{({5n_{\mathbb{K}_i}+5})/{2}} | \Delta_{\mathbb{K}_i}|^{{1}/({n_{\mathbb{K}_i}+1})}(\log |\Delta_{\mathbb{K}_i} | )^{n_{\mathbb{K}_i}} \\ &\leq e^{28.2 d_i+5}(d_i+1)^{({5d_i+5})/{2}}|D_{f_i}|^{{1}/({d_i+1})}(\log |D_{f_i}|)^{d_i} \\ &\leq e^{28.2 d+5}(d+1)^{({5d+5})/{2}}|D_f|(\log |D_f|)^d \end{align*} $$

since $D_{f_i} | D_f$ and $D_f \neq 0$ (see Lemma 4 and the discussion below (2-4)). Define

$$ \begin{align*} \Lambda = e^{28.2 d+5}(d+1)^{({5d+5})/{2}}|D_f|(\log |D_f| )^d, \end{align*} $$

which satisfies $0 \leq \Lambda _{\mathbb {K}_i} \leq \Lambda $ for $i=1,2,\ldots ,k$ and vanishes if $D_f=1$ .

From (2-8), (2-13), and (2-14),

(4-6) $$ \begin{align} |B_{\mathbb{K}_i}| &\leq \frac{2}{\log x} \bigg[ \bigg( \frac{(n_{\mathbb{K}_i}+1)^2}{2\kappa_{\mathbb{K}}(n_{\mathbb{K}_i}-1)}\Lambda_{\mathbb{K}_i} + 1\bigg) + \frac{0.55\, \Lambda_{\mathbb{K}_i}n_{\mathbb{K}_i}(n_{\mathbb{K}_i}+1)}{\kappa_{\mathbb{K}_i}} + n_{\mathbb{K}_i} + 40.31 \dfrac{\Lambda_{\mathbb{K}_i}n_{\mathbb{K}_i}}{\kappa_{\mathbb{K}_i}} \bigg] \nonumber \\ &\leq \frac{2 }{\log x} \bigg[ \frac{\Lambda}{\kappa_{\mathbb{K}_i}} \bigg( \frac{(n_{\mathbb{K}_i}+1)^2}{2(n_{\mathbb{K}_i}-1)} + 0.55\, n_{\mathbb{K}_i}(n_{\mathbb{K}_i}+1) + 40.31 n_{\mathbb{K}_i} \bigg) + 1 + n_{\mathbb{K}_i} \bigg] \nonumber \\ &= \frac{2 }{\log x} \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}\bigg( \frac{(d_i+1)^2}{2(d_i-1)} + 0.55\, d_i(d_i+1) + 40.31 d_i \bigg) + 1 + d_i \bigg] \nonumber \\ &\leq \frac{2 }{\log x} \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}( 0.5(d_i+7) + 0.55\, d_i(d +1) + 40.31 d_i ) + 1 + d_i \bigg] \end{align} $$

for $d_i \geq 2$ and $x \geq 2$ . In the final inequality, we use the fact that

$$ \begin{align*} \frac{(x+1)^2}{2(x-1)} \leq \frac{x+7}{2} \quad \text{for}\ x \geq 2. \end{align*} $$

This inequality is equivalent to $2-2(x-1)^{-1} \geq 0$ , which holds for $x \geq 2$ . We use this argument to ensure that the choice $d_i=1$ is permissible below.

If $d_i = 1$ , then $\mathbb {K}_i = \mathbb {Q}$ and $D_{f_i} = 1$ (the discriminant of a nonconstant linear polynomial is $1$ ). Recall from (2-10) that $M_{\mathbb {Q}}(x) = \log \log x + C_{\mathbb {Q}} + B_{\mathbb {Q}}(x)$ , in which $C_{\mathbb {Q}} = 0.261\,497\,2\ldots $ is the Meissel–Mertens constant and $|B_{\mathbb {Q}}(x)| \leq 2(\log x)^{-2}$ for ${x>1}$ [Reference Rosser and Schoenfeld14, formulas (3.17) and (3.20)]. Since

$$ \begin{align*} \frac{2}{(\log x)^2} \leq \frac{4 }{\log x} \quad \text{for}\ x \geq 2, \end{align*} $$

it follows that (4-6) holds if $d_i = 1$ (even if $|D_f |=1$ , so $\Lambda = 0$ ). Thus, (4-4) yields

$$ \begin{align*} \mathbf{B}(x) &= \sum_{i=1}^k |B_{\mathbb{K}_i}(x)|\\ &\leq \sum_{i=1}^k \frac{2 }{\log x} \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}( 0.5(d_i+7)+ 0.55\, d_i(d +1) + 40.31 d_i ) + 1 + d_i \bigg] \\ &\leq\frac{2 }{\log x} \bigg(\sum_{i=1}^k \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}( 0.5d_i+3.5+ 0.55\, d_i(d +1) + 40.31 d_i ) \bigg] + k + d \bigg)\\ &\leq\frac{2 }{\log x} \bigg( \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}( 0.5d+3.5k+ 0.55\, d(d +1) + 40.31 d ) \bigg] + 2 d \bigg)\\ &=\frac{2 }{\log x} \bigg( \bigg[ \frac{\Lambda \sqrt{\mathbf{D}_f}}{0.362\,32}( 0.55 d^2 + 44.86 d ) \bigg] + 2 d \bigg). \end{align*} $$

4.4 Estimating $\mathbf {C}$

Recall from (2-12) that

$$ \begin{align*} \gamma -d_i +\log \kappa_{\mathbb{K}_i} \leq C_{\mathbb{K}_i} \leq \gamma +\log \kappa_{\mathbb{K}_i} \end{align*} $$

for $i=1,2,\ldots ,k$ . Therefore,

$$ \begin{align*} |C_{\mathbb{K}_i}| \leq \gamma + d_i + | \log \kappa_{\mathbb{K}_i} |. \end{align*} $$

For $d_i \geq 2$ , the residue bounds (2-7), (2-8), and the inequality (4-5) provide

$$ \begin{align*} \frac{0.362\,32}{\sqrt{\mathbf{D}_f}} \,\leq\, \frac{0.362\,32}{\sqrt{\mathbf{D}_{f_i}}} \ \,\leq\, \kappa_{\mathbb{K}_i} \,\leq\, \bigg(\frac{e \log \mathbf{D}_{f_i}}{2(d_i-1)}\bigg)^{d_i-1} \,\leq\, \bigg(\frac{e \log \mathbf{D}_{f}}{2}\bigg)^{d-1}. \end{align*} $$

If $d_i = 1$ , then $\mathbb {K}_i = \mathbb {Q}$ , $\kappa _{\mathbb {K}_i} = 1$ , $D_{f_i}=1$ , and $\mathbf {D}_f \geq \mathbf {D}_{f_i} \geq 1$ . Thus,

(4-7) $$ \begin{align} \quad \frac{0.362\,32}{\sqrt{\mathbf{D}_f}} \, \leq \, \kappa_{\mathbb{K}_i}\, \leq\, \max\bigg\{1,\frac{e \log \mathbf{D}_{f}}{2}\bigg\}^{d-1}\quad\text{for}\ d_i \geq 1, \end{align} $$

where the maximum in (4-7) is $1$ for $\mathbf {D}_f =1,2$ only. Using (4-7) for $i=1,2,\ldots ,k$ and $d_i\geq 1$ ,

$$ \begin{align*} | \log \kappa_{\mathbb{K}_i}| &\leq \max\! \bigg\{ \log \frac{\sqrt{\mathbf{D}_f}}{0.362\,32},\, (d-1)\log\max\bigg\{1,\frac{e \log \mathbf{D}_{f}}{2}\bigg\} \bigg\} \nonumber \\[5pt] &\leq \begin{cases} \max\! \big\{ 1.015\,23 + \frac{1}{2}\log \mathbf{D}_f,\, 0 \big\} &\ \text{if}\ \mathbf{D}_f =1,2,\\[6pt] \max\! \bigg\{ 1.015\,23 + \dfrac{1}{2}\log \mathbf{D}_f,\, (d-1)\bigg(\log{\dfrac{e}{2}} + \log\log \mathbf{D}_{f}\bigg) \bigg\} &\ \text{if}\ \mathbf{D}_f \geq 3, \end{cases}\nonumber\\[5pt] &\leq (d-1) \begin{cases} 1.015\,23 + \frac{1}{2}\log \mathbf{D}_f &\ \text{if}\ \mathbf{D}_f =1,2,\\ \max\! \big\{ 1.015\,23 + \frac{1}{2}\log \mathbf{D}_f,\, \big(0.306\,86 + \frac{1}{2}\log \mathbf{D}_f\big) \big\} &\ \text{if}\ \mathbf{D}_f \geq 3, \end{cases}\nonumber\\ &\leq (d-1)\big(1.02 + \tfrac{1}{2} \log\mathbf{D}_f\big) \nonumber \end{align*} $$

since $\log {\mathbf {D}_f} < \sqrt {\mathbf {D}_f}$ because $\mathbf {D}_f \geq 1$ .

Since $k \leq d$ and because of (4-4),

$$ \begin{align*} \mathbf{C} = \sum_{i=1}^k |C_{\mathbb{K}_i}| \leq \sum_{i=1}^k (\gamma + d_i + | \log \kappa_{\mathbb{K}_i} | ) &\leq d\, \gamma + d + (d-1) \sum_{i=1}^k\bigg(1.02 + \frac{1}{2}\log \mathbf{D}_f\bigg) \\ &\leq d\, \bigg(\gamma + 1 + (d-1) 1.02 + \frac{d-1}{2}\log \mathbf{D}_f\bigg) \\ &=d\, \bigg(\gamma + 1.02\,d - 0.02 + \frac{d-1}{2}\log \mathbf{D}_f\bigg). \end{align*} $$

4.5 Wrapping things up

Now return to (4-3) and use the bounds on $\mathbf {A}$ , $\mathbf {B}(x)$ , and $\mathbf {C}$ obtained in Sections 4.2–4.4, respectively. This yields the desired bound (1-2) of Theorem 2 and completes the proof.