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 G(q) for the prime cyclotomic fields 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 G(q) and G(q)(+), where G(q)(+) is the Euler-Kronecker constant of the maximal real subfield of Q(zeta(q)), for some very large primes q thus obtaining two new negative values of G(q): G(9109334831) = -0.248739 ... and G(9854964401) = -0.096465 ... We also evaluated G(q) and G(q)(+) for every odd prime q <= 10(6), thus enlarging the size of the previously known range for G(q) and B-q(+). Our method also reveals that the difference G(q) - G(q)(+) can be computed in a much simpler way than both its summands, see Sect. 3.4. Moreover, as a by-product, we also computed M-q = max(chi not equal chi 0) vertical bar L'/L(1, chi)vertical bar for every odd prime q <= 10(6), where L(s, chi) are the Dirichlet L-functions,. 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 provide more data on the generalised Euler constants in arithmetic progressions.