Comparative Prime Number Theory — Problem 2

Definitions and mathematical statement

Under the usual interpretation of the notation, let \[ \psi(x)=\sum_{\substack{p\text{ prime},\ k\in{\mathbb{Z}}_{\ge1}\\p^k\le x}}\log p, \qquad E(x)=\psi(x)-x\qquad(x\in{\mathbb{R}},\ x\ge1), \] where $\log$ denotes the natural logarithm. The supplied record asks for progress on sign changes of $E(x)$ beyond repeated-averaging techniques. It describes an expectation that large-scale intervals of positive and negative values alternate at approximately the rate suggested by repeated averaging, while each detected transition contains many smaller-scale sign changes. It further suggests that a quantity denoted $W^\psi(T)$, and related sign-change counting functions, should grow roughly like $\sqrt T$, up to logarithmic factors, rather than like $\log T$. These are supplied heuristic expectations, not asserted bounds. The record does not define $W^\psi(T)$, the parameter $T$, the sign-change counting convention, the scales involved, or the repeated-averaging benchmark. Consequently, this portion does not specify a precise quantified conjecture.

The same record also contains a question attributed to Alexandre Bailleul, who attributes its origin to Nilotpal Sinha. For an integer $n\ge2$, define its largest prime factor by \[ P^+(n)=\max\{p:p\text{ is prime and }p\mid n\}. \] Let the real Dirichlet character modulo $4$ be \[ \chi_{-4}(m)= \begin{cases} 0,&m\equiv0,2\pmod4,\\ 1,&m\equiv1\pmod4,\\ -1,&m\equiv3\pmod4 \end{cases} \qquad(m\in{\mathbb{Z}}). \] For real $x\ge2$, the well-defined part of the displayed sum is \[ S(x)=\sum_{\substack{n\in{\mathbb{Z}}\\2\le n\le x}}\chi_{-4}\left(P^+(n)\right). \] Equivalently, \[ S(x)=\#\{n\in{\mathbb{Z}}:2\le n\le x,\ P^+(n)\equiv1\pmod4\} -\#\{n\in{\mathbb{Z}}:2\le n\le x,\ P^+(n)\equiv3\pmod4\}. \] The question in this passage is to determine the asymptotic behaviour of $S(x)$ as $x\to\infty$. The record reports numerical evidence of a bias in favour of the second set, but specifies neither a quantitative conjecture nor a precise meaning of bias. Its original notation $\sum_{n\le x}$ leaves the lower endpoint implicit and does not define $P^+(1)$; the displayed definition of $S(x)$ explicitly restricts to $n\ge2$. The record does not clarify whether this largest-prime-factor question belongs to the titled problem or was included from a separate source item.

Short English statement

The record asks for improved understanding of sign changes of $\psi(x)-x$, but omits definitions needed to state its suggested growth law precisely. It also includes a possibly separate question about the asymptotic difference between counts of integers whose largest prime factor is congruent to $1$ or $3$ modulo $4$. Both questions require clarification before a single precise conjecture can be extracted.

Sources

• Local UnsolvedMath registry, permanent ID $800145$, source code catalogue entry, title "Comparative Prime Number Theory --- Problem 2." The supplied registry labels the record open; this label is not an independently verified mathematical assessment.
• Hamieh, Kadiri, Martin, and Ng, Comparative Prime Number Theory Problem List (2024), supplied source item "Problem 2 (#14)," https://arxiv.org/abs/2407.03530. The registry lists these authors as proposers and gives proposed year 2024. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access or source verification.
• The largest-prime-factor passage is attributed in the supplied text to Alexandre Bailleul, Université Paris--Saclay, and says that the question was raised by Nilotpal Sinha, with reference label [Sinha]. No bibliographic details or link for [Sinha] are supplied. The claim that the asymptotic behaviour remains unknown and the report of numerical bias are supplied source assessments, not independent literature conclusions.