A309520 Primes p for which h1(p)/G(p) has a record value.

3, 5, 7, 11, 23, 73, 89, 179, 233, 761, 1451, 2741, 4391, 5231, 42611, 198221, 305741, 6766811, 1326662801, 1979990861, 4735703723, 9697282541, 35285447111, 45169368641, 169684421321, 187946428721

Offset: 1

Author

Michel Marcus, Aug 06 2019

Keywords

• nonn
• more

Comments

h1 is given by A000927, and G(p) = 2*p*(p/(4*Pi^2))^((p-1)/4).

a(1)-a(12) from Fung et al., a(13)-a(14) from Shokrollahi, a(15)-a(17) from Broadhurst, a(18) from Languasco et al. and Broadhurst, a(19)-a(26) from Broadhurst.

Links

• David Broadhurst, Kummer ratio champions with p < 10^12, July 22 2021.
• Gilbert Fung, Andrew Granville and Hugh C. Williams, Computation of the first factor of the class number of cyclotomic fields, Journal of Number Theory, Volume 42, Issue 3, November 1992, Pages 297-312. See Table IV A p. 304.
• Alessandro Languasco, Pieter Moree, Sumaia Saad Eddin and Alisa Sedunova, Computation of the Kummer ratio of the class number for prime cyclotomic fields, arXiv:1908.01152 [math.NT], 2019.
• M. A. Shokrollahi, Relative class number of imaginary Abelian fields of prime conductor below 10000, Math. Comp. 68 (1999), 1717-1728. See Table 4 p. 1726.

Crossrefs

Cf. A000927 (h1), A073010 (value for p=3).

Programs

PARI

h1(p) = if (p<5, 1, abs( matdet(matrix((p-1)/2-2, (p-1)/2-2, i, j, ((i+2)*(j+2))\p - ((i+1)*(j+2))\p)) )); \\ A000927
G(p) = 2*p*(p/(4*Pi^2))^((p-1)/4);
lista(nn) = {my(m = 0, nm); for (n=2, nn, p = prime(n); if ((nm = h1(p)/G(p)) > m, print1(p, ", "); m = nm););}

Extensions

Missing terms 42611, 198221, 305741 and terms larger than 6766811 added by Alessandro Languasco on behalf of David Broadhurst, Jul 24 2021