A003420 -id:A003420 - cp's OEIS frontend

A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.

1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 446, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81626, 162686

Offset: 1

Hugo Pfoertner, Feb 04 2020

This sequence is almost identical to A003420. However, there is an additional term 446 and after 30014 the number 81626 follows, while in A003420, 81149 is present between 30014 and 81626. With
C(m) = Product_{p=primes} 1 - Kronecker(-4*m,p)/(p - 1) (Hardy-Littlewood)
L1(m) = Sum_{j>0} Kronecker(-4*m,j)/j (L-function of the Dirichlet series)
the following table shows the differences:
              Criterion
          decrease increase
      k      C        L1
     341  0.28309  2.38177
     446  0.28272  2.38014 not in A003420 because L1(446) < L1(341)
     689  0.28193  2.39370
   ...
   30014  0.21541  3.08274
   81149  0.21560  3.08792 not in this sequence because C(81149) > C(30014)
   81626  0.20883  3.17785
  162686  0.20478  3.24017

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)

Cf. A003420, A003521, A331940, A331941.

\\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2min=oo; for(add=1,500,my(hl=HardyLittlewood2(n^2+add));if(hl

A003419 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>0} Kronecker(D,k)/k.

Original entry on oeis.org

1, 2, 17, 167, 227, 362, 398, 331427, 430022, 737183, 800663, 821498, 1475858, 2271407, 3009173, 5417453

Offset: 1

N. J. A. Sloane

The terms a(2)-a(7) are given in Shanks's Table 4 "Lochamps, 4M = Discriminant". This table gives some values of L(1) for larger discriminants, e.g., L(1) = 0.2510... for D = 4*4813372912697. In comparison, L(1) = 0.28422 for D = 4*a(16) = 4*5417453. - Hugo Pfoertner, Feb 07 2020

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).

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

Cf. A003420, A003421, A003521.

New title, a(1) prepended, and a(8)-a(13) from Hugo Pfoertner, Feb 04 2020
a(14)-a(15) from Hugo Pfoertner, Feb 05 2020
a(16) from Hugo Pfoertner, Feb 07 2020

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.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

			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).

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).

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)

Cf. A003420.

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

A003421 Nonsquare values of m in the discriminant D = 4*m leading to a new maximum of the L-function of the Dirichlet series L(1) = Sum_{k>0} Kronecker(D,k)/k.

Original entry on oeis.org

2, 3, 6, 7, 10, 19, 31, 34, 46, 79, 106, 151, 211, 214, 274, 331, 394, 631, 751, 919, 991, 1054, 1486, 1654, 2146, 2479, 2599, 3826, 5014, 5251, 7459, 8551, 9454, 10651, 13666, 18379, 22234, 32971, 39274, 45046, 48799, 61051, 62386, 74299, 78439, 84319, 111094

Offset: 1

N. J. A. Sloane

nonn

The terms a(1)-a(24) are given in Shanks's Table 6 "Hichamps, 4M = Discriminant". After the term 1654, this table is incomplete and only gives selected values. - Hugo Pfoertner, Feb 07 2020

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).

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

Cf. A003419, A003420, A003521.

New title, a(25)-a(47) from Hugo Pfoertner, Feb 07 2020

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A331949 Addends k > 0 such that x^2 + k produces a new minimum of its Hardy-Littlewood Constant.

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A003419 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>0} Kronecker(D,k)/k.

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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.

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A003421 Nonsquare values of m in the discriminant D = 4*m leading to a new maximum of the L-function of the Dirichlet series L(1) = Sum_{k>0} Kronecker(D,k)/k.

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