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

Inflammatory cytokine detection in adenotonsill and peripheral blood mononuclear cells- culture in adenotonsillectomy patients: a comparative study

Background: Tonsils and adenoid hypertrophy is a major respiratory symptom in children which is partly due to recruitment of inflammatory cells in upper airway lymph nodes as a result of the effects of synthesis and release of different inflammatory cytokines. It seems that infections play role in concert with these cytokines leading to tonsilar hypertrophy and other pathologic consequences. It is proposed that cellular infiltrate of tonsils and adenoids may secrete different quantities of these cytokines compared with peripheral blood mononuclear cells (PBMC) cultures.Methods: Among patients who were admitted for adenotonsillectomy to the ENT ward, 37 patients, under 1-12 years old patients with fulfill criteria selected to include the study. Excised adenoid and tonsils cultured and inflammatory cytokines Interferon-γ (INF-γ), Interlukine-1 (IL-1), IL-6, IL-8 and tumor necrosis factor-α (TNF-α) measured in cellular culture supernatant. The same cytokines measured in PBMC cultures.Results: The data shows that there is a significant difference between IFN-γ and IL-8 amounts in adenoid tissue culture supernatant and PBMC culture of our patients. Furth-ermore, the amounts of IFN-γ, IL-1 and IL-8 showed considerable difference between tonsilar tissue culture supernatant and PBMC culture of these patients. Although there is a significant correlation between IL-6 amounts in tissue culture supernatant and PBMC culture (P=0.02), the respective data for TNF is only almost significant.Conclusion: Inflammatory cytokines may have significant role in the early provoke of inflammation occurred in hypertrophied tonsils and adenoid. The majority of these cyt-okines increase the expression of adhesion molecules on epithelial cells and influence the recruitment of leucocytes and inflamed tonsils. On the other hand lack of sufficient cytokine release may lead to persistent infections and may cause chronic inflammation and hypertrophied tissue