In 2002, M. A. Tsfasman and S. G. Vladut [13] formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.2010 Mathematics Subject Classification. 11M41.

In 2002, the second author introduced a class of [Formula: see text]-functions [Formula: see text], which contains the Selberg class and forms a ring. In this paper, we study this class and prove that the invariant [Formula: see text], which is the generalization of degree in the Selberg class cannot take non-integer values between [Formula: see text] and [Formula: see text]. We also study the ring structure of [Formula: see text] showing that it is non-Noetherian.

We study lower bounds of a general family of L-functions on the $1$ -line. More precisely, we show that for any $F(s)$ in this family, there exist arbitrarily large t such that $F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$ , where m is the order of the pole of $F(s)$ at $s=1$ . This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the $1$ -line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type $L(s,f\times f)$ on the $1$ -line.

As a natural generalization of the Euler-Mascheroni constant γ, Ihara [6] introduced the Euler-Kronecker constant γ K attached to any number field K. In this paper, we prove that a certain bound on γ K in a tower of number fields K implies the generalized Brauer-Siegel conjecture for K as formulated by Tsfasman and Vlǎduţ. Moreover, we use known bounds on γ K for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function ζ K (s) in the critical strip.

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