Extreme Values of Euler-Kronecker Constants

In a family of S_n-fields (n ≤ 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are −(n − 1) log log d_K+ O(log log log d_K) and loglog d_K + O(log log log d_K), resp., where d_K is the absolute value of the discriminant of a number field K.