Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields

We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler-Kronecker constants $\mathfrak{G}_q$ for the prime cyclotomic fields $\mathbb{Q}(\zeta_q)$, where $q$ is an odd prime and $\zeta_q$ is a primitive $q$-root of unity. With such a new algorithm we evaluated $\mathfrak{G}_q$ and $\mathfrak{G}_q^+$, where $\mathfrak{G}_q^+$ is the Euler-Kronecker constant of the maximal real subfield of $\mathbb{Q}(\zeta_q)$, for some very large primes $q$ thus obtaining two new negative values of $\mathfrak{G}_q$: $\mathfrak{G}_{9109334831}= -0.248739\dotsc$ and $\mathfrak{G}_{9854964401}= -0.096465\dotsc$ We also evaluated $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$ for every odd prime $q\le 10^6$, thus enlarging the size of the previously known range for $\mathfrak{G}_q$ and $\mathfrak{G}^+_q$. Our method also reveals that difference $\mathfrak{G}_q - \mathfrak{G}^+_q$ can be computed in a much simpler way than both its summands, see Section 3.4. Moreover, as a by-product, we also computed $M_q=\max_{\chi\ne \chi_0} \vert L^\prime/L(1,\chi) \vert $ for every odd prime $q\le 10^6$, where $L(s,\chi)$ are the Dirichlet $L$-functions, $\chi$ run over the non trivial Dirichlet characters mod $q$ and $\chi_0$ is the trivial Dirichlet character mod $q$. As another by-product of our computations, we will also provide more data on the generalised Euler constants in arithmetic progressions. The programs used to performed the computations here described and the numerical results obtained are available at the following web address: \url{http://www.math.unipd.it/~languasc/EK-comput.html}.