Relative class numbers and Euler-Kronecker constants of maximal real cyclotomic subfields

TL;DR

This paper investigates the Euler--Kronecker constants of maximal real cyclotomic subfields, concentrating on γ_q^+ and the difference γ_q^+-γ_q via the normalized κ(q). It develops both average-case results and conditional distributional statements by relating κ(q) to prime-pattern conjectures (EH and HL) and to Kummer's conjecture through the Kummer ratio r(q), employing Dirichlet $L$-functions and methods inspired by Granville and Croot–Granville. The authors establish sharp average bounds for the Kummer ratio and for γ_q^+-γ_q, derive explicit κ(q) bounds under RH for odd characters, and demonstrate deep distributional parallels between κ(q) and r(q), including the existence of rich limit-point sets under strong conjectures. They further extend the analysis with CG-style methods to obtain refined distributional results, including conditions under which κ(q) attains prescribed limit points and connections between κ(q) and the infinitude of primes in linear patterns. Numerical experiments corroborate the theoretical predictions, illustrating the symmetry and spike phenomena in the observed distributions and supporting the proposed conjectural picture of κ(q) and its relation to Kummer-type questions in cyclotomic fields.

Abstract

The Euler--Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_K(s)$ at $s=1$. We study the distribution of the Euler--Kronecker constant $γ_q^+$ of the maximal real subfield of $\mathbb Q(ζ_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $γ_q^+-γ_q$, with $γ_q$ the Euler--Kronecker constant of $\mathbb Q(ζ_q)$ and show how it is connected with Kummer's conjecture, which predicts the asymptotic growth of the relative class number of $\mathbb Q(ζ_q)$. We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for $γ_q^+-γ_q$. The methods employed are partly inspired by those used by Granville (1990) and Croot and Granville (2002) to investigate Kummer's conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.

Figures (6)

• Figure 1: On the left: the values of $r(q)$; on the right: the values of $\kappa(q)$, $q$ prime, $3\leqslant q\leqslant 10^7$. The red dashed lines represent the mean values.
• Figure 2: On the left: the histogram of $r(q)$; on the right: the histogram of $\kappa(q)$, $q$ prime, $3\leqslant q\leqslant 10^7$. The red dashed lines represent the mean values, and $\pm1/4$.
• Figure 3: The same histograms of Figure \ref{['fig3']} but the contributions of the primes $q$ such that $2q+1$ is prime or $2q-1$ is prime (the "spikes") are superimposed.
• Figure 4: The same histograms of Figure \ref{['fig3']} but the contributions of the primes $q$ such that $4q+1$ is prime or $4q-1$ is prime (the "spikes") are superimposed.
• Figure 5: On the left: the histogram for $\kappa(q)$, $q$ prime, $5\leqslant q\leqslant 10^7$, such that $2q\pm1$ are composite; on the right: idem with both $2q\pm1$ and $4q\pm1$ that are composite numbers. The red dashed lines represent the mean values.

Theorems & Definitions (60)

• Theorem
• Theorem 1.1
• Theorem 1.2
• Remark 1.1
• Definition 1.1
• Conjecture 1.1
• Theorem 1.3
• Conjecture 1.2: Euler--Kronecker constant analogue of Kummer's conjecture
• Theorem 1.4
• Theorem 1.5