2 Basic definitions and notation
Let \(p_n\) denote the \(n^{th}\) prime.
\(P(g)\) is the first prime \(p_n\) for which the prime gap \(p_{n+1}-p_n\) is equal to \(g\), or \(0\) if no such gap exists.
\(\pi (x)\) is the number of primes less than or equal to \(x\).
\(\mathrm{li}(x) = \int _0^x \frac{dt}{\log t}\) (in the principal value sense) and \(\mathrm{Li}(x) = \int _2^x \frac{dt}{\log t}\).
\(E_\psi (x) = |\psi (x) - x| / x\)
We say that \(E_\psi \) satisfies a classical bound with parameters \(A, B, C, R, x_0\) if for all \(x \geq x_0\) we have
We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have
\(E_\pi (x) = |\pi (x) - \mathrm{Li}(x)| / \mathrm{Li}(x)\).
\(E_\theta (x) = |\theta (x) - x| / x\)
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
We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have
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
We say that it obeys a numerical bound with parameter \(\varepsilon (x_0)\) if for all \(x \geq x_0\) we have
If \(A,B,C,R {\gt} 0\) then the classical bound is monotone decreasing for \(x \geq \exp ( R (2B/C)^2 )\).
Differentiate the bound and check the sign.
A classical bound for \(x \geq x_0\) implies a numerical bound for \(x \geq \max (x_0, \exp ( R (2B/C)^2 ))\).
Immediate from previous lemma