Comparative Prime Number Theory — Problem 1

Definitions and mathematical statement

The supplied record does not define the sign-change counting function $W^\psi(T)$, its parameter $T$, or the constant $\gamma_1$. Consequently, it does not determine a complete mathematical formulation. Under the usual convention, the Chebyshev function is \[ \psi(x)=\sum_{\substack{p\text{ prime},\ k\geq 1\\p^k\leq x}}\log p\qquad(x\geq 1), \] and $\gamma_1$ commonly denotes the smallest positive imaginary part of a nontrivial zero of the Riemann zeta function $\zeta(s)$. These interpretations require confirmation from the source; they do not supply the missing definition of $W^\psi$.

Reading the source's unindexed limit inferior as $T\to\infty$ through its intended, unspecified parameter domain, write formally \[ L=\liminf_{T\to\infty}\frac{W^\psi(T)}{\log T}. \] The source attributes to Morrill, Platt, and Trudgian, improving work of Kaczorowski, the bound \[ L\geq\frac{\gamma_1}{\pi}+1.867\cdot10^{-30}. \] The requested problem is to improve this lower bound, with an improvement conditional on the Riemann hypothesis also permitted. Here the Riemann hypothesis means that every nontrivial zero of $\zeta(s)$ has real part $1/2$. Suggested targets, presented separately, are \[ \text{(a)}\quad L\geq1.01\frac{\gamma_1}{\pi},\qquad \text{(b)}\quad L\geq2\frac{\gamma_1}{\pi},\qquad \text{(c)}\quad L=+\infty. \] The assumptions of the quoted existing bound are not specified in the supplied excerpt.

As heuristic motivation, the source expects large-scale positive and negative intervals of $\psi(x)-x$ to alternate at approximately the rate suggested by a repeated-averaging argument, while each detected transition consists of many sign changes on a smaller scale, by analogy with random walks. It therefore suggests growth of $W^\psi(T)$, and related unspecified counting functions, closer to $\sqrt T$, up to logarithmic factors, than to $\log T$. This is qualitative motivation, not a specified asymptotic conjecture.

Short English statement

Improve the supplied lower bound for the limit inferior of $W^\psi(T)/\log T$, allowing the Riemann hypothesis as an additional assumption. The suggested targets range from a factor of $1.01$ times $\gamma_1/\pi$ to an infinite limit inferior. A complete statement requires the source's definitions of $W^\psi$, $T$, and $\gamma_1$.

Sources

• Local UnsolvedMath registry, permanent ID $800144$, exact source code catalogue entry, title "Comparative Prime Number Theory --- Problem 1." The supplied record labels the problem open; this label is supplied metadata, not an independently verified assessment.
• Hamieh, Kadiri, Martin, and Ng, Comparative Prime Number Theory Problem List (2024), source item "Problem 1 (#13)": https://arxiv.org/abs/2407.03530. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; no independent source visit or verification is asserted here.
• The excerpt attributes the displayed lower bound to Morrill, Platt, and Trudgian, reference label [MPT], improving work of Kaczorowski, reference label [K2]. Full bibliographic entries and the hypotheses of those results are not supplied. The description of this bound as the best improvement known to the source speaker is a source-specific awareness statement.