Title: Kummer ratio champions with p < 10^12

I sought to extend, to p < 10^12, the lists of high and low 
champions given for p < 3000 in Table IV of [4] for the ratio 
r(p) = h_1(p)/G(p) of the first class number [1,2] h_1(p) 
of prime cyclotomic fields to Kummer's estimate, 
G(p) = 2*p*(p/4/Pi^2)^((p-1)/4), obtaining:

Table IVA of [4] (continued):
                 p           r(p)
[4391,  1.50777641013105282560036183203]
[5231,  1.55656224754669055462930589411]
[42611,  1.61990657115753239986736117278]
[198221,  1.62347727075119766150086424241]
[305741,  1.66143648590878694868852809641]
[6766811,  1.709379]
[1326662801,  1.709758]
[1979990861,  1.720793]
[4735703723,  1.721656]
[9697282541,  1.724742]
[35285447111,  1.727900]
[45169368641,  1.729699]
[169684421321,  1.732018]
[187946428721,  1.771167]

Table IVB of [4] (continued):
                  p        r(p)
[3169,  0.684480631908289300957901004853]
[3331,  0.642429297634719506688741152270]
[37189,  0.625231255787654795233417601860]
[149119,  0.624149715978401425409347395847]
[401179,  0.621507092276527124572758370990]
[2083117,  0.614280]
[5589169,  0.586973]
[102598099,  0.586137]
[116827429,  0.575675]
[26890996879,  0.567003]

For p < 10^6, I computed products of character sums at 128
bit precision. For p > 10^6, I used the sum over prime
powers in Andrew's paper [3], truncated at x, with x/p
increasing from 3*10^9 to 5*10^9, in 100 steps. In the
tables, I give mean values. The standard deviations were
less than 10^(-6).

Pieter and colleagues [5,6] have given results that confirm
the champions with p < 10^7.

I hope that the tables above are complete for p < 10^12.
For 10^7 < p < 10^12, I considered all primes for which 
2*p+1 and 6*p+1 are both prime, as candidates for high 
champions. As candidates for low champions, I required the 
primality of 2*p-1 and 6*p-1. It seems to me rather unlikely 
that any champion was missed by these screening criteria.

In the third table, below, I compare values for r(p),
obtained by implementing Andrew's formula, with the less
accurate values given in Table 4 of [6], where the authors
used fast Fourier transforms at 80 bits, to estimate
products of character sums. It appears that the latter
method is about two orders of magnitude less accurate than
the former, for primes of this size. I would not recommend
the use of FFT methods for p > 10^10.

Table 4 of [6] (improved accuracy):
	p         r(p)       ref [6]      excess
[4151292581,  1.669702,  1.669735,    0.000033]
[6406387241,  1.625612,  1.625741,    0.000129]
[7079770931,  1.688790,  1.688607,   -0.000183]
[9109334831,  1.658104,  1.657855,   -0.000249]
[9854964401,  1.687950,  1.688033,    0.000083]

All of the results in this letter were obtained in less than 
a week, using two threads on a laptop running Pari-GP, whose 
"ispseudoprimepower" command consumed most of the cycles.

I am grateful to Michael St Clair Oakes, for encouragement,
to Johannes Bluemlein, for providing copies of the very
impressive papers [1] and [2], and to Alessandro Languasco,
for on-line resources relating to [6].

References:

[1] E. E. Kummer, Mémoire sur la théorie des nombres
complexes composés de racines de l'unité et de nombres
entiers, J. Math. Pures Appl. 16 (1851) 377-498,
https://archive.org/details/s1journaldemat16liou/page/376/mode/2up .

[2] G. Schrutka von Rechtenstamm, Tabelle der
(Relativ)-Klassenzahlen der Kreiskorper, deren phi-Funktion
des Wurzelexponenten (Grad) nicht grosser als 256 ist,
Abh. Deutschen Akad. Wiss. Berlin, Kl. Math. Phys. 2 (1964) 1-64,
copy kindly provided by DESY-Zeuthen.

[3] A. Granville, On the size of the first factor of the class
number of a cyclotomic field, Invent. Math. 100 (1990) 321-338.

[4] G. Fung, A. Granville, H. C. Williams, Computation of
the first factor of the class number of cyclotomic fields,
J. Number Theory 42 (1992) 297-312.

[5] P. Moree, Irregular behaviour of class numbers and
Euler-Kronecker constants of cyclotomic fields : the log log
log devil at play, in "Irregularities in the Distribution of
Prime Numbers - Research Inspired by Maier's Matrix Method",
Eds. J. Pintz and M. Th. Rassias, Springer, 2018, 143-163,
arXiv:1711.07996 [math.NT].

[6] Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin,
Alisa Sedunova, Computation of the Kummer ratio of the class
number for prime cyclotomic fields, arXiv:1908.01152 [math.NT],
with tables at http://www.math.unipd.it/~languasc/rq-comput.html .

Best regards,

David Broadhurst,
Open University,
David.Broadhurst@open.ac.uk
22 July 2021