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.
Class groups of real cyclotomic fields
