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

Author

Hugo Pfoertner, Feb 04 2020

Keywords

• nonn
• more

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

• 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)

Crossrefs

Cf. A003420, A003521, A331940, A331941.

Programs

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