bibkey: pintz1984remainder authors: J. Pintz year: 1984 title: On the remainder term of the prime number formula and the zeros of Riemann’s zeta-function doi: 10.1007/BFb0099452 url: https://www.math.ubc.ca/~gerg/teaching/592-Fall2018/papers/1984.Pintz_0.pdf claim: The published sign-change interval for pi minus li supplies quantitatively located positive psi-error points for the actual CA source application. strata_touched: [] license: citation-only triage: anchor

Sign-change intervals and the actual CA source clock

The original scan contains the article in Lecture Notes in Mathematics 1068, pp.186–197, DOI. It has 12 pages and SHA-256 3b2fedd7f1b4aed5938e05eb0d85e9fe3f9884b0cbfa78b0893dad53c9e27741. Printed pp.186, 188–190, 192–193 were read for the following inputs. This checks the cited statements and notation, without an independent whole-proof audit or Lean verification. Section 4, p.193, sketches the proof of Theorem 8 and refers proofs of the other theorems to further papers; the scan is used for Theorem 4’s published statement.

Equation (1.1), p.186, defines $\Delta (x)=\psi (x)-x$, retaining all prime powers. Equation (2.1), p.189, defines ${\Delta }_1(x)=\pi (x)-\mathrm{li}(x)$, with $\mathrm{li}(x)=\mathrm{PV}{\int }_0^{x}dt/\log t$. Theorem 4, p.190, equation (2.4), states unconditionally that ${\Delta }_1$ changes sign in

$$\left[Y\exp \{-500(\log \log Y{)}^3\},Y\right](Y>Y_2),$$

where $Y_2$ is ineffective. In particular the interval contains a point with ${\Delta }_1>0$. The source writes ${\log }_{2}Y$ for $\log \log Y$.

Equation (1.15), p.188, gives the zero abscissa $\Theta$; equation (3.13), p.192, records $\Theta =\inf \{\vartheta :\Delta (x)=O(x^{\vartheta })\}$. For $\Theta <1$, this supplies $|\psi (x)-x|=O(x^{\sigma })$ for every fixed $\sigma >\Theta$. When $\Theta =1$, the application instead uses the prime number theorem’s bound with $\sigma =1$.

Positive psi-error points in power windows

The following application consumes Theorem 4 rather than reconstructing its oscillation proof. Fix $1/2<\sigma \le 1$ and $|E(u)|=O(u^{\sigma })$, where $E(u)=\psi (u)-u$. For every fixed $r>0$ with $r\sigma <1/2$, each sufficiently large $[Y^r,Y]$ contains a $t$ with $\psi (t)>t$.

To see the parameter correspondence, retain every prime power in $\Pi (x)={\sum }_{p^k\le x}1/k$. Finite partial summation, with weak upper endpoints, gives the exact normalization

$$\Pi (x)-\mathrm{li}(x)=\frac{E(x)}{\log x}+{\int }_2^x\frac{E(u)}{u{\log }^{2}u}du+c_0,c_0=\frac{2}{\log 2}-\mathrm{li}(2).$$

Also $\Pi(x)-\pi(x)=\sum_{k=2}^{\lfloor\log x/\log2\rfloor} \pi(x^{1/k})/k\ge\pi(\sqrt x)/2$exactly.Put$B=Y^r$ and suppose $E\le 0$ throughout $[B,Y]$. The published sign-change interval supplies a $y\in [Y\exp \{-500(\log \log Y{)}^3\},Y]$ with $\pi (y)-\mathrm{li}(y)>0$; eventually $y>B$. The endpoint term and the integral over $[B,y]$ are nonpositive. For suitable fixed positive constants $C_1,C_2,c_3$, the error bound and the prime number theorem therefore give

$$0<\pi (y)-\mathrm{li}(y)\le C_1+C_2{B}^{\sigma }-c_3\frac{\sqrt{y}}{\log y}.$$

But $\sqrt y/\log y\ge \sqrt Y\exp{-250(\log\log Y)^3}/\log Y$ eventually, and this dominates $Y^{r\sigma }$ since $r\sigma <1/2$. The upper bound becomes negative, a contradiction. All discarded higher prime powers have the required upper-bound direction. No effective starting value is obtained.

This is a paper-level application used by the actual CA source construction, without a priority claim or Lean certification. A pointwise positive $\psi (t)-t$ is not a lower bound for the original Robin signed integral at the final selected integer’s clock.