Sum of Euler-Kronecker constants over consecutive cyclotomic fields
Let \(K\) be an algebraic number field and \(\zeta_K(s)\) be its Dedekind zeta-function. This function, meromorphic over \(\mathbb C\), with a simple pole at \(s=1\), has for \(s\not=1\) an expansion in Laurent series \(\zeta_K(s)=c_{-1}(s-1)^{-1}+c_0+c_1(s-1)+\dots\), with a nonzero coefficient \(c_{-1}\). Following \textit{Y. Ihara} [Prog. Math. 253, 407--451 (2006; Zbl 1185.11069)], the \textit{Euler-Kronecker constant associated to} \(K\) is defined by \[ \gamma_K:=c_0/c_{-1}. \] Let \(q\geq 1\) be an integer, \(\zeta_q:=\exp(\frac{2\pi i}{q})\) and \(\gamma_q\) the Euler-Kronecker constant associated with the cyclotomic field \(\mathbb Q(\zeta_q)\). A conjecture due to \textit{Y. Ihara} [Séminaires et Congrès 21, 79--102 (2009; Zbl 1242.11091)] is: there exist absolute constants \(C_1\) and \(C_2\) satisfying \(0 <C_1\leq C_2<2\), such that, for every \(\varepsilon>0\), there exists real \(Q(\varepsilon)>0\) such that \((C_1-\varepsilon)\log q\leq \gamma_q\leq (C_2+\varepsilon)\log q\) for every integer \(q\geq Q(\epsilon)\). The purpose of this paper is to continue the study of the order of magnitude of \(\gamma_q\), when \(q\) is a natural number, not necessarily prime, without appealing to any classical conjecture in analytic number theory (Elliot-Halberstam Conjecture, Extended Riemann Hypothesis,\(\dots\)) which are for instance used in [\textit{Y. Ihara} et al., Acta Arith. 137, No. 3, 253--276 (2009; Zbl 1213.11159); \textit{A. I. Badyan}, Math. Notes 87, No. 1, 31--42 (2010); translation from Mat. Zametki 87, No. 1, 35--47 (2010; Zbl 1209.11098); \textit{K. Ford} et al., ``Values of the Euler \(\varphi\)-function not divisible by a given odd prime and the distribution of Euler-Kronecker constants for cyclotomic fields'', \url{arXiv:1108.3805}]. The main result of this paper describes the average value of \(\gamma_q\), when \(q\) is a natural integer. Uniformly for \(Q\geq 3\), one has the equality \[ \frac{1}{Q}\sum_{Q\leq q\leq 2Q} \gamma_q=\log Q+\mathcal O(\log \log Q). \] The deduced inequalities \(\underline{\lim}\frac{\gamma_q}{\log q}\leq 1\leq \overline{\lim}\frac{\gamma_q}{\log q}\) gives some information about the values of the hypothetical constants \(C_1\) and \(C_2\) appearing in Ihara's conjecture. The strategy of the proof of this theorem follows the method of [\textit{V. K. Murty}, Ann. Sci. Math. Qué. 35, No. 2, 239--247 (2011; Zbl 1259.11105)].
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