The Euler-Kronecker constant of a cyclotomic field (Q2888165)

Let \(K\) be a number field and let \(\zeta_K(s)\) be its Dedekind \(\zeta\)-function, which has a simple pole at \(s = 1\). The Laurent expansion at \(s = 1\) therefore has the form NEWLINE\[NEWLINE\zeta_K(s) = c_{-1}(s - 1)^{-1} + c_0 + c_1(s - 1) + \cdotsNEWLINE\]NEWLINE with \(c_{-1} \not= 0\). The Euler-Kronecker constant of \(K\) is the real number \(\gamma_K = c_0/c_{-1}\). The article under review explores the behaviour of \(\gamma_K\) as \(K\) varies over cyclotomic fields of prime conductor. See below for a more precise description of the main result.NEWLINENEWLINESome bounds on \(\gamma_K\) have been proven under the Generalized Riemann Hypothesis (GRH). For example, Ihara has shown that the GRH implies the existence of positive constants \(c_1\) and \(c_2\) such that, for any number field \(K \not= \mathbb{Q}\), NEWLINE\[NEWLINE -c_1 \log d_K \leq \gamma_K \leq c_2 \log \log d_K ,NEWLINE\]NEWLINE where \(d_K\) is the absolute value of the discriminant of \(K\).NEWLINENEWLINEWe consider a conjecture of Ihara on Euler-Kronecker constants for cyclotomic fields. For a positive integer \(m\), let \(\gamma_m\) be the Euler-Kronecker constant of the field \(K_m = \mathbb{Q}(\zeta_m)\) where \(\zeta_m\) is a primitive \(m\)th root of unity. Then Ihara conjectures that there are constants \(c_3\) and \(c_4\) lying in the interval \((0,2]\) such that, for any \(\varepsilon > 0\), there is a positive integer \(N\) with the property that NEWLINE\[NEWLINE (c_3 - \varepsilon) \log m < \gamma_m < (c_4 + \varepsilon) \log m NEWLINE\]NEWLINE holds for \(m \geq N\).NEWLINENEWLINEThe main result of the present article may be stated as follows: NEWLINE\[NEWLINE \sum_{\frac{1}{2} Q < q \leq Q} |\gamma_q| \ll \pi^*(Q) \log Q ,NEWLINE\]NEWLINE where the sum runs over primes in the interval \((\frac{1}{2}Q,Q]\) and \(\pi^*(Q)\) denotes the number of primes in that interval. Thus, the article demonstrates Ihara's upper bound on average (in the prime-conductor situation) and does so unconditionally.NEWLINENEWLINENEWLINEWe conclude with some remarks on the proof. A key idea is that \(\zeta_{K_q}(s)\) is the product of the Dirichlet \(L\)-functions mod \(q\). (Of course, \(q\) need not be prime for there to be such a factorization.) This permits the result to be attacked via a character-by-character analysis. If \(\chi\) is a Dirichlet character mod \(q\), then one considers a function \(\Phi_\chi(x)\), which is, very roughly speaking, a sum involving the von Mangoldt function weighted by \(\chi\). Proposition 2.1 of the article concerns the average behaviour of \(\sum_{\chi \not= \chi_0} \Phi_\chi(x)\) as \(q\) varies, where \(\chi_0\) is the principal character mod \(q\). On the other hand, Proposition 3.1 concerns the average behaviour of \(\sum_{\chi \not= \chi_0} (\frac{L'}{L}(1,\chi) + \Phi_\chi(x))\). Since \(\gamma_q\) can be expressed in a simple way in terms of \(\sum_{\chi \not= \chi_0} \frac{L'}{L}(1,\chi)\), which is the difference of the functions studied in Propositions 2.1 and 3.1, the author is able to deduce the main result.