We introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants Gq\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q$$\end{document} for the prime cyclotomic fields Q(ζq)\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb {Q}}(\zeta _q)$$\end{document}, where q is an odd prime and ζq\documentclass[12pt]{minimal}
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\begin{document}$$\zeta _q$$\end{document} is a primitive q-root of unity. With such a new algorithm we evaluated Gq\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q$$\end{document} and Gq+\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q^+$$\end{document}, where Gq+\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q^+$$\end{document} is the Euler–Kronecker constant of the maximal real subfield of Q(ζq)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Q}}(\zeta _q)$$\end{document}, for some very large primes q thus obtaining two new negative values of Gq\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q$$\end{document}: G9109334831=-0.248739⋯\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$\end{document} and G9854964401=-0.096465⋯\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$\end{document} We also evaluated Gq\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q$$\end{document} and Gq+\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}^+_q$$\end{document} for every odd prime q≤106\documentclass[12pt]{minimal}
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\begin{document}$$q\le 10^6$$\end{document}, thus enlarging the size of the previously known range for Gq\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q$$\end{document} and Gq+\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}^+_q$$\end{document}. Our method also reveals that the difference Gq-Gq+\documentclass[12pt]{minimal}
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\begin{document}$${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$\end{document} can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed Mq=maxχ≠χ0|L′/L(1,χ)|\documentclass[12pt]{minimal}
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\begin{document}$$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$\end{document} for every odd prime q≤106\documentclass[12pt]{minimal}
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\begin{document}$$q\le 10^6$$\end{document}, where L(s,χ)\documentclass[12pt]{minimal}
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\begin{document}$$L(s,\chi )$$\end{document} are the Dirichlet L-functions, χ\documentclass[12pt]{minimal}
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\begin{document}$$\chi $$\end{document} run over the non trivial Dirichlet characters mod q and χ0\documentclass[12pt]{minimal}
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\begin{document}$$\chi _0$$\end{document} is the trivial Dirichlet character mod q. As another by-product of our computations, we will provide more data on the generalised Euler constants in arithmetic progressions.
