A003419 -id:A003419

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A331949

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..32.

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)

PROG

(PARI) \\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.

hl2min=oo; for(add=1, 500, my(hl=HardyLittlewood2(n^2+add)); if(hl<hl2min, print1(add, ", "); hl2min=hl))

CROSSREFS

Cf. A003420, A003521, A331940, A331941.

KEYWORD

nonn,more

AUTHOR

Hugo Pfoertner, Feb 04 2020

STATUS

approved

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)

+10
5

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

(list; graph; refs; listen; history; text; internal format)

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.

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

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.
(Formerly M1387)

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

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..36.

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)

EXAMPLE

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).

CROSSREFS

Cf. A003521.

Cf. A331949, which has almost identical terms.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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.
(Formerly M0750)

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

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..47.

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. A003419, A003420, A003521.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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