The value distribution of Artin \(L\)-series and zeros of zeta-functions.
Let \(K/\mathbb{Q}\) be a finite, Galois extension of \(\mathbb{Q}\), with Galois group \(G= \text{Gal}(K/\mathbb{Q})\). The main purpose of this paper is to obtain a generalization of a theorem of Voronin, and others (see below and the references in the paper) on the distribution of the values of the Riemann zeta-function and Dirichlet \(L\)-series to the case of the Artin \(L\)-functions \(L(s,\chi_j,K/\mathbb{Q})\) associated with a set of linearly independent characters \(\chi_1,\dots, \chi_n\) of \(G\). More precisely, the author proves the following theorem, which is the principal result of this paper and whose many applications are explained and developed in it. With the foregoing notation, denote by \(k\) the order of \(G\) and let \(f_1(s),\dots, f_n(s)\) be holomorphic functions on the set of \(s\in\mathbb{C}\) such that \(| s|< r\), which is continuous on the disc \(| s|\leq r\), where \(r\) is some fixed number such that \(0< r<{1\over 4k}\). Suppose that \(f_j(s)\neq 0\) on \(| s|\leq r\). Then the author's main theorem (Theorem 1.1 of the paper) states that: for every \(\varepsilon> 0\), there is a set \(A_\varepsilon\subset \mathbb{R}\) such that \[ \liminf_{T\to\infty}\, {\text{vol} (A_\varepsilon \cap (0,T))\over T}> 0, \] and for \(j= 1,2,\dots, n\) and any \(t\in A_\varepsilon\) and \(| s|\leq r\), \[ \Biggl| L\Biggl(s+ 1-{1\over 4k}+ it,\,\chi_j, K/\mathbb{Q} \Biggr)\Biggr|< \varepsilon, \] where \(L(z,\chi_j, K/\mathbb{Q})\) denotes the Artin \(L\)-functions. Later in the paper, the author also proves a similar result for a single Artin \(L\)-function, \(L(s,\phi,K/L)\) of a normal extension \(K/L\) of number fields, where, in general, \(\phi\) denotes a non-Abelian character of \(\text{Gal}(K/L)\). That last result may be used to derive, by an appeal to class field theory, a theorem of \textit{H. Mishou} [Acta Arith. 98, 395--410 (2001; Zbl 0982.11050)]. The author also refers to a result of \textit{H. Davenport} and \textit{H. Heilbronn} [On the zeros of certain Dirichlet series, J. Lond. Math. Soc. 11, 181--185 (1936; JFM 62.0138.01); ibid. 11, 307--312 (1936; JFM 62.1145.03)] concerning the zeta-functions associated to positive definite quadratic forms of discriminant \(d\) and class number \(h(d)> 1\), which states that those zeta-functions have zeros with \(\text{Re}(s)> 1\). Voronin [see \textit{A. A. Karatsuba} and \textit{S. M. Voronin}, The Riemann zeta-function (de Gruyter Expositions in Mathematics 5, de Gruyter, Berlin) (1992; Zbl 0756.11022)] proved that those functions do also have zeros in the strip \({1\over 2}< \text{Re}(s)< 1\), and the author applies his main theorem to prove that this is also true for every partial zeta-function attached to a ray class of any algebraic number field, provided that the ray class group is non-trivial, a result which holds in particular for the class group of a number field with class number \(>1\). More precisely, he proves the following theorem. Denote by \(O_K\) the ring of integers of the number field \(K\) and by \(H_{\mathfrak f}\) an ideal group with conductor \({\mathfrak f}\); let \(I^{({\mathfrak f})}\) denote the group of fractional ideals of \(O_K\), prime to \({\mathfrak f}\). Then, if \(I^{({\mathfrak f})}/H_{\mathfrak f}\) contains more than one ideal class, the partial zeta-function \(\zeta(s,{\mathcal A})\) attached to any class \({\mathcal A}\in I^{({\mathfrak f})}/H_{\mathfrak f}\) has infinitely many zeros in the strip \({1\over 2}< \text{Re}(s)< 1\). Moreover if \(T> 0\) is sufficiently large, then there exists \(c> 0\) such that there are at least \(cT\) zeros of \(\zeta(s,{\mathcal A})\) in the region defined by \({1\over 2}< \text{Re}(s)< 1\) and \(|\text{Im}(s)|< T\). Those very interesting deductions from the main theorem are developed in the course of the paper, though the proof of the main theorem is complicated and draws on many ideas, which the author sets out very lucidly in the paper.
scientific article; zbMATH DE number 1902879