A003521 -id:A003521 - 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

A003420 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=1..oo} Kronecker(D,k)/k.

Original entry on oeis.org

1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81149, 81626, 162686, 243374, 644474, 839354, 879941

Offset: 1

N. J. A. Sloane

In Shanks's Table 5 "Hichamps, -4N = Discriminant", N = 1 is omitted, and N = 23201 is missing. Shanks describes the table as being tentative after N = 24569. In Buell's Table 10 "Successive maxima of L(1) for even discriminants", the values N = 11 and N = 1091 are missing in the D/4 column. The further terms 644474, 839354, 879941, provided there require an independent check. - Hugo Pfoertner, Feb 02 2020

			a(1) = 1: L(1) for D=-4*1 ~= 0.785398... = Pi/4.
a(2) = 2: L(1) for D=-4*2 ~= 1.11072073... = Pi/(2*sqrt(2)), a(2) > a(1);
L(1) for D=-4*3 ~= 0.90689..., L(1) for D=-4*4 ~= 0.785398..., both < a(2);
a(3) = 5: L(1) for D=-4*5 = 1.40496..., a(3) > 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 10, page 792).
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. A003521.

Cf. A331949, which has almost identical terms.

New title, a(1) prepended, missing term 23201 and a(29)-a(33) from Hugo Pfoertner, Feb 02 2020

3 further terms < 10^6 added by Hugo Pfoertner, Aug 27 2022

A342547 Addends k > 0 such that the polynomial x^3 + k produces a record of its Hardy-Littlewood constant.

Original entry on oeis.org

2, 3, 17, 74, 165, 205, 2609, 23602

Offset: 1

Hugo Pfoertner, Apr 29 2021

For more information and references see A331950.

Cubic polynomials with no quadratic terms have a poor yield in generating primes compared to quadratic polynomials. This can be seen when comparing the Hardy-Littlewood constants HL for quadratic polynomials of the form x^2 + k (k given in A003521) where HL(x^2 + 1) = 1.3728..., HL (x^2 + 7) = 1.9730..., ..., HL(x^2 + 991027) = 4.1237..., whereas the best known result for the present sequence, a(8) only leads to HL(x^3 + 23602) = 1.7167...

			  n  a(n)   Hardy-Littlewood
            constant (rounded)
  1     2   1.298539558
  2     3   1.390543939
  3    17   1.442297580
  4    74   1.451456320
  5   165   1.589487813
  6   205   1.637173422
  7  2609   1.679828689
  8 23602   1.716729673

Cf. A003521 (records for x^2+k), A331950.

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

cp's OEIS Frontend

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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A003420 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=1..oo} Kronecker(D,k)/k.

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A342547 Addends k > 0 such that the polynomial x^3 + k produces a record of its Hardy-Littlewood constant.

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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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