Comparative Prime Number Theory — Problem 3

Definitions and mathematical statement

For a real number $x$, irrationality means $x\in{\mathbb{R}}\setminus{\mathbb{Q}}$, where ${\mathbb{Q}}$ denotes the rational numbers. The supplied source requests:

Show that $\gamma_1$ is irrational.

However, the record does not define $\gamma_1$, specify its domain, or identify the function or other object from which it is obtained. If the intended $\gamma_1$ is a specified real number, the requested conclusion would be \[ \gamma_1\notin{\mathbb{Q}}. \] This conditional interpretation does not identify the intended number; the supplied record is insufficient for a self-contained mathematical formulation.

The surrounding source discusses an undefined condition called "LI" for higher-degree $L$-functions. It cautions about factorizations of Dedekind $\zeta$-functions and Artin $L$-functions into other $L$-functions, vanishing of elliptic curve $L$-functions at the central point, and the possibility of shifting an entire $L$-function vertically when no restriction to the central character is imposed. It calls the irrationality question a concrete instance of an earlier problem, whose formulation is absent. These contextual qualifications do not supply a definition of $\gamma_1$ or precise hypotheses for the requested assertion.

Short English statement

The source asks for a proof that $\gamma_1$ is irrational, but does not identify $\gamma_1$. Its surrounding discussion concerns qualifications to an undefined "LI" condition for higher-degree $L$-functions. The intended number and the missing preceding formulation must be recovered before a precise statement can be completed.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800134$, exact source problem code catalogue entry, title "Comparative Prime Number Theory --- Problem 3." The registry labels the record open; this is supplied status metadata, not an independently verified mathematical assessment.
• Hamieh, Kadiri, Martin, and Ng, Comparative Prime Number Theory Problem List (2024), source item Problem (#3), as identified by the supplied record: https://arxiv.org/abs/2407.03530. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent consultation or verification.
• The registry attributes the proposal to Hamieh, Kadiri, Martin, and Ng (2024). The extracted statement also contains the attribution "Alexandre Bailleul, Université Paris--Saclay" before the concrete-instance sentence and "Nicol Leong, University of New South Wales--Canberra at ADFA" after the irrationality question. Their precise attachment to the surrounding problems is unclear from this extract.