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.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

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

Examples

			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).
		

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

• 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)

Crossrefs

Cf. A003420.

Extensions

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