Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.Comment: 20 pages, 1 figure, survey, to appear in `Irregularities in the Distribution of Prime Numbers - Research Inspired by Maier's Matrix Method', Eds. J. Pintz and M. Th. Rassia

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Let L(s,χ) be either log L(s,χ) or L'/L(s,χ), associated with an (abelian) L-function L(s,χ) of a global field K. For any quasi-character ψ: ℂ→ℂ× of the additive group of complex numbers, consider the average “Avgfχ=f” of ψ(L(s,χ)) over all Dirichlet characters χ on K with a given prime conductor f. This paper contains (i) study of the limit as N(f)→∞ of this average, (ii) basic studies of the analytic function Ms(z1,z2) in 3 complex variables arising from (i) (here, (z1,z2)∈ℂ2 is the natural parameter for ψ), and (iii) application to value-distribution theory for {L(s,χ)}χ. Our base field K is either a function field over a finite field, or a special type of number field: the rational number field ℚ or an imaginary quadratic field. But in the number field case, the Generalized Riemann Hypothesis is assumed in (i) and (iii).journal articl

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Towards a new direction of urban revitalization after the completion of Three Connecting Bridges in Seto Inland Sea : Studies on urban policies

1999年度香川大学公開講座「〈中小企業経営セミナー〉瀬戸内三橋時代のまちづくりの方向を探る―都市政策の検討―」（1999年11月1日～12月6日、全5回、高松商工会議所との共同主催）の内容を論稿として取りまとめたもの。departmental bulletin pape