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

Abstract

AbstractWe introduce a new algorithm, which is faster and requires less computing resources than the ones previously known, to compute the Euler–Kronecker constants $${\mathfrak {G}}_q$$ G q for the prime cyclotomic fields $$ {\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , where q is an odd prime and $$\zeta _q$$ ζ q is a primitive q-root of unity. With such a new algorithm we evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}_q^+$$ G q + , where $${\mathfrak {G}}_q^+$$ G q + is the Euler–Kronecker constant of the maximal real subfield of $${\mathbb {Q}}(\zeta _q)$$ Q ( ζ q ) , for some very large primes q thus obtaining two new negative values of $${\mathfrak {G}}_q$$ G q : $${\mathfrak {G}}_{9109334831}= -0.248739\dotsc $$ G 9109334831 = - 0.248739 ⋯ and $${\mathfrak {G}}_{9854964401}= -0.096465\dotsc $$ G 9854964401 = - 0.096465 ⋯ We also evaluated $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + for every odd prime $$q\le 10^6$$ q ≤ 10 6 , thus enlarging the size of the previously known range for $${\mathfrak {G}}_q$$ G q and $${\mathfrak {G}}^+_q$$ G q + . Our method also reveals that the difference $${\mathfrak {G}}_q - {\mathfrak {G}}^+_q$$ 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 $$M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi ) \vert $$ M q = max χ ≠ χ 0 | L ′ / L ( 1 , χ ) | for every odd prime $$q\le 10^6$$ q ≤ 10 6 , where $$L(s,\chi )$$ L ( s , χ ) are the Dirichlet L-functions, $$\chi $$ χ run over the non trivial Dirichlet characters mod q and $$\chi _0$$ χ 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.