A003521 Values of m in the discriminant D = -4*m leading to a new minimum of the L-function of the Dirichlet series L(1) = Sum_{k>=1} Kronecker(D,k)/k. (Formerly M4418)

1, 7, 37, 58, 163, 4687, 30178, 30493, 47338, 83218, 106177, 134773, 288502, 991027

Offset

1,2

Comments

In Shanks's Table 3 "Lochamps, -4N = Discriminant", N = 1 is omitted. Shanks describes the table as being tentative after N = 47338. In Buell's Table 7 "Successive minima of L(1) for even discriminants" several omissions and extra terms are present for N < 30178, but the terms above are confirmed by an independent computation. - Hugo Pfoertner, Feb 03 2020

References

D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), pp. 267-283 of Analytic Number Theory, ed. H. G. Diamond, Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..14.

Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796 (Table 7, page 791).

D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)

Example

With L1(k) = L(1) for D=-4*k:
a(1) = 1: L1(1) ~= 0.785398... = Pi/4;
L1(2) = 1.1107, L1(3) = 0.9069, L1(4) = 0.7854, L1(5) = 1.4050, L1(6) = 1.2825, all >= a(1);
a(2) = 7 because L1(7) = 0.5937 < a(1);
a(3) = 37 because L1(k) > a(2) for 8 <= k <= 36, L1(37) = 0.51647 < a(2).

Crossrefs

Cf. A003420.

Sequence in context: A235463 A127313 A217561 * A155943 A078626 A093389

Adjacent sequences: A003518 A003519 A003520 * A003522 A003523 A003524

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

New title, a(1) prepended and a(10)-a(14) from Hugo Pfoertner, Feb 03 2020

Status

approved