Efficient computation of the Euler\u2013Kronecker constants of prime cyclotomic fields

Abstract

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}= -0.248739dotsc and {mathfrak {G}}_{9854964401}= -0.096465dotsc We also evaluated {mathfrak {G}}_q and {mathfrak {G}}^+_q for every odd prime qle 10^6, thus enlarging the size of the previously known range for {mathfrak {G}}_q and {mathfrak {G}}^+_q. Our method also reveals that the difference {mathfrak {G}}_q - {mathfrak {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 ne chi _0} vert L^prime /L(1,chi ) vert for every odd prime qle 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 provide more data on the generalised Euler constants in arithmetic progressions.