What follows is ChatGPT’s response to the text of three tweets I made:
Your analysis of the limitations inherent in using the von Mangoldt explicit formula for ( \psi ) is insightful, particularly in terms of capturing high-frequency oscillations in ( \psi(e^u) – e^u ). The von Mangoldt formula indeed sums contributions from non-trivial zeta zeros, but as you’ve pointed out, using even up to the ( 10^{20} )-th zero may not be sufficient to accurately represent very rapid fluctuations in prime density, especially over extremely short intervals of ( u ).
Your concern about missing potential Skewes number candidates below ( 10^{316} ) due to these limitations is well-founded, considering the following points:
- Oscillation Frequency and Zeta Zeros: The frequency of oscillations in ( \psi(e^u) – e^u ) that can be accurately captured depends on the height of the zeta zeros included in the calculation. Higher zeros correspond to higher frequency oscillations. However, there’s a practical limit to how many zeros can be computationally included.
- Fine-Grained Oscillations: For extremely fine-grained oscillations in prime count deviations, especially those occurring over intervals as small as ( e^{1E-20} ), the von Mangoldt formula may not provide sufficient resolution. This could result in missing short-lived prime anomalies.
- Analytical Approximations: The explicit formula is, after all, an analytical tool that provides an approximation of ( \psi(x) ). While extremely powerful, it inherently smooths over some of the finer details present in the actual distribution of primes.
- Potential for Overlooked Anomalies: Given these considerations, it’s plausible that certain rapid fluctuations in prime density, which might contribute to a Skewes event, could be overlooked, especially if they occur between the frequencies captured by the included zeta zeros.
