9 Secondary explicit estimates
9.1 Definitions
In this section we define the basic types of secondary estimates we will work with in the project. Key references:
FKS1: Fiori–Kadiri–Swidninsky arXiv:2204.02588
FKS2: Fiori–Kadiri–Swidninsky arXiv:2206.12557
\(\pi (x)\) is the number of primes less than or equal to \(x\).
\(\mathrm{li}(x) = \int _0^x \frac{dt}{\log t}\) and \(\mathrm{Li}(x) = \int _2^x \frac{dt}{\log t}\).
\(\theta (x) = \sum _{p \leq x} \log p\) where the sum is over primes \(p\).
\(E_\pi (x) = |\pi (x) - \mathrm{Li}(x)| / \mathrm{Li}(x)\)
\(E_\theta (x) = |\theta (x) - x| / x\)
We say that \(E_\theta \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have
Similarly for \(E_\pi \).
We say that \(E_\pi \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have
9.2 The prime number bounds of Rosser and Schoenfeld
In this section we formalize the prime number bounds of Rosser and Schoenfeld [ 11 ] .
\(\vartheta (x) = x + O( x / \log ^2 x)\).
This in principle follows by establishing an analogue of Theorem 5.0.1, using mediumPNT in place of weakPNT.
\(B := \lim _{x \to \infty } \left( \sum _{p \leq x} \frac{1}{p} - \log \log x \right)\).
\(E := \lim _{x \to \infty } \left( \sum _{p \leq x} \frac{\log p}{p} - \log x \right)\).
The function \(\vartheta (x) = \sum _{p \leq x} \log p\) defines a Stieltjes function (monotone and right continuous).
Trivial
\(\sum _{p \leq x} f(p) = \int _{2}^x \frac{f(y)}{\log y}\ d\vartheta (y)\).
This follows from the definition of the Stieltjes integral.
\(\sum _{p \leq x} f(p) = \frac{f(x) \vartheta (x)}{\log x} - \int _2^x \vartheta (x) \frac{d}{dy}( \frac{f(y)}{\log y} )\ dy.\)
Follows from Sublemma 9.2.2 and integration by parts.
Follows from Sublemma 9.2.3 and integration by parts.
Follows from Sublemma 9.2.3 applied to \(f(t) = 1\).
Follows from Sublemma 9.2.3 applied to \(f(t) = 1/t\).
Follows from Sublemma 9.2.3 applied to \(f(t) = 1/t\). One can also use this identity to demonstrate convergence of the limit defining \(B\).
9.3 Tools from BKLNW
In this file we record the results from [ 1 ] . -
9.4 The implications of FKS2
In this file we record the implications in the paper [ 7 ] that allow one to convert primary bounds on \(E_\psi \) into secondary bounds on \(E_\pi \), \(E_\theta \).
\(\operatorname {li}(x) - \operatorname {Li}(x) = \operatorname {li}(2)\).
This follows directly from the definitions of \(\operatorname {li}\) and \(\operatorname {Li}\).
For any \(a,b,c,x \in \mathbb {R}\) we define \(g(a,b,c,x) := x^{-a} (\log x)^b \exp ( c (\log x)^{1/2} )\).
For any \(2 \leq x_0 {\lt} x\) one has
This follows from Sublemma 9.2.7.
We have
This follows from straightforward differentiation.
\(\frac{d}{dx} g(a, b, c, x) \) is negative when \(-au^2 + \frac{c}{2}u + b {\lt} 0\), where \(u = \sqrt{\log (x)}\).
Clear from previous sublemma.
If \(a{\gt}0\), \(c{\gt}0\) and \(b {\lt} -c^2/16a\), then \(g(a,b,c,x)\) decreases with \(x\).
We apply Lemma 9.4.3. There are no roots when \(b {\lt} -\frac{c^2}{16a}\), and the derivative is always negative in this case.
For any \(a{\gt}0\), \(c{\gt}0\) and \(b \geq -c^2/16a\), \(g(a,b,c,x)\) decreases with \(x\) for \(x {\gt} \exp ((\frac{c}{4a} + \frac{1}{2a} \sqrt{\frac{c^2}{4} + 4ab})^2)\).
We apply Lemma 9.4.3. If \(a {\gt} 0\), there are two real roots only if \(\frac{c^2}{4} + 4ab \geq 0\) or equivalently \(b \geq -\frac{c^2}{16a}\), and the derivative is negative for \(u {\gt} \frac{\frac{c}{2} + \sqrt{\frac{c^2}{4} + 4ab}}{2a}\).
If \(c{\gt}0\), \(g(0,b,c,x)\) decreases with \(x\) for \(\sqrt{\log x} {\gt} -2b/c\).
We apply Lemma 9.4.3. If \(a = 0\), it is negative when \(u {\lt} \frac{-2b}{c}\).
If \(B \geq 1 + C^2 / 16R\) then \(g(1,1-B,C/\sqrt{R},x)\) is decreasing in \(x\).
This follows from Lemma 9.4.1 applied with \(a=1\), \(b=1-B\) and \(c=C/\sqrt{R}\).
The Dawson function \(D_+ : \mathbb {R} \to \mathbb {R}\) is defined by the formula \(D_+(x) := e^{-x^2} \int _0^x e^{t^2}\ dt\).
The Dawson function has a single maximum at \(x \approx 0.942\), after which the function is decreasing.
The Dawson function satisfies the differential equation \(F'(x) + 2xF(x) = 1\) from which it follows that the second derivative satisfies \(F''(x) = −2F(x) − 2x(−2xF(x) + 1)\), so that at every critical point (where we have \(F(x) = \frac{1}{2x}\)) we have \(F''(x) = −\frac{1}{x}\). It follows that every positive critical value gives a local maximum, hence there is a unique such critical value and the function decreases after it. Numerically one may verify this is near 0.9241 see https://oeis.org/ A133841.
Suppose that \(E_\theta \) satisfies an admissible classical bound with parameters \(A,B,C,R,x_0\). Then, for all \(x \geq x_0\),
where
Since \(\varepsilon _{\theta ,\mathrm{asymp}}(t)\) provides an admissible bound on \(\theta (t)\) for all \(t \geq x_0\), we have
We perform the substitution \(u = \sqrt{\log (t)}\) and note that \(u^{2B-3} \leq m(x_0, x)\) as defined in (21). Thus the above is bounded above by
Then, by completing the square \(u^2 - \frac{Cu}{\sqrt{R}} = \left( u - \frac{C}{2\sqrt{R}} \right)^2 - \frac{C^2}{4R}\) and doing the substitution \(v = u - \frac{C}{2\sqrt{R}}\), the above becomes
Now we have
Combining the above completes the proof.
Suppose that \(A_\psi ,B,C,R,x_0\) give an admissible bound for \(E_\psi \). If \(B {\gt} C^2/8R\), then \(A_\theta , B, C, R, x_0\) give an admissible bound for \(E_\theta \), where
with
We have an admissible bound for \(E_\theta \) with \(A = 121.0961\), \(B=3/2\), \(C=2\), \(R = 5.5666305\), \(x_0=2\).
For \(x_0,x_1 {\gt} 0\), we define
.
Let \(x_0 {\gt} 2\). We say a (step) function \(ε_{\diamond ,num}(x_0)\) gives an admissible numerical bound for \(E_\diamond (x)\) if \(E_\diamond (x) \leq ε_{\diamond ,num}(x_0)\) for all \(x \geq x_0\).
If
then \(\mu _{num,1}(x_0,x_1,x_2) {\lt} \mu _{num,2}(x_0,x_1)\).
If \(\log x_0 \geq 1000\) then we have an admissible bound for \(E_\theta \) with the indicated choice of \(A(x_0)\), \(B = 3/2\), \(C = 2\), and \(R = 5.5666305\).
If \(B \geq \max (3/2, 1 + C^2/16 R)\), \(x_0 {\gt} 0\), and one has an admissible asymptotic bound with parameters \(A,B,C,x_0\) for \(E_\theta \), and
then
for all \(x \geq x_1\). In other words, we have an admissible bound with parameters \((1+\mu _{asymp}(x_0,x_1))A, B, C, x_1\) for \(E_\pi \).
Let \(x {\gt} x_0 {\gt} 2\). IF \(E_\psi (x) \leq \varepsilon _{\psi ,num}(x_0)\), then
where
Let \(x_1 {\gt} x_0 \geq 2\), \(N \in \mathbb {N}\), and let \((b_i)_{i=1}^N\) be a finite partition of \([x_0,x_1]\). Then
Assume \(x \geq 6.58\). Then \(Li(x) - \frac{x}{\log x}\) is strictly increasing and \(Li(x) - \frac{x}{\log x} {\gt} \frac{x-6.58}{\log ^2 x} {\gt} 0\).
Let \(x_0 {\gt} 0\) be chosen such that \(\pi (x_0)\) and \(\theta (x_0)\) are computable, and let \(x_1 \geq \max (x_0, 14)\). Let \(\{ b_i\} _{i=1}^N\) be a finite partition of \([\log x_0, \log x_1]\), with \(b_1 = \log x_0\) and \(b_N = \log x_1\), and suppose that \(\varepsilon _{\theta ,\mathrm{num}}\) gives computable admissible numerical bounds for \(x = \exp (b_i)\), for each \(i=1,\dots ,N\). For \(x_1 \leq x_2 \leq x_1 \log x_1\), we define
and for \(x_2 {\gt} x_1 \log x_1\), including the case \(x_2 = \infty \), we define
Then, for all \(x_1 \leq x \leq x_2\) we have
Let \(\{ b'_i\} _{i=1}^M\) be a set of finite subdivisions of \([\log (x_1),\infty )\), with \(b'_1 = \log (x_1)\) and \(b'_M = \infty \). Define
Then \(E_\pi (x) \leq \varepsilon _{\pi ,num}(x_1)\) for all \(x \geq x_1\).
Let \(B \geq \max (\frac{3}{2}, 1 + \frac{C^2}{16R})\). Let \(x_0, x_1 {\gt} 0\) with \(x_1 \geq \max (x_0, \exp ( (1 + \frac{C}{2\sqrt{R}})^2))\). If \(E_\psi \) satisfies an admissible classical bound with parameters \(A_\psi ,B,C,R,x_0\), then \(E_\pi \) satisfies an admissible classical bound with \(A_\pi , B, C, R, x_1\), where
for all \(x \geq x_0\), where
where
and
One has
for all \(x \geq 2\).
\(A_\pi , B, C, x_0\) as in Table 6 give an admissible asymptotic bound for \(E_\pi \) with \(R = 5.5666305\).
We have the bounds \(E_\pi (x) \leq B(x)\), where \(B(x)\) is given by Table 7.
One has
for all \(x \geq 2\).
9.5 Summary of results
Here we list some papers that we plan to incorporate into this section in the future, and list some results that have not yet been moved into dedicated paper sections.
References to add:
Dusart: https://piyanit.nl/wp-content/uploads/2020/10/art_10.1007_s11139-016-9839-4.pdf
PT: D. J. Platt and T. S. Trudgian, The error term in the prime number theorem, Math. Comp. 90 (2021), no. 328, 871–881.
JY: D. R. Johnston, A. Yang, Some explicit estimates for the error term in the prime number theorem, arXiv:2204.01980.
One has
for all \(x \geq \exp (2000)\).
One has
for all \(x \geq 2\).
One has
for all \(x \geq 2\).
TODO: input other results from JY
There exists a constant \(X_0\) (one may take \(X_0 = 89693\)) with the following property: for every real \(x \geq X_0\), there exists a prime \(p\) with
TODO: input other results from Dusart