Prime Generating Polynomials(Hardy-Littlewood constant)

[Valeryi Kuryshev-3]

Jacobson found polynomial of the form x^2 + x + c with largest Hardy-
Littlewood constant C(D)~5.36 for the least D=1-4c=-
13598858514212472187,where

 

(1)C(D)=Product(1-(D/p)/p-1),

 

p is prime>2,(D/p) denotes the Legendre symbol

(Math.Comp., 2003, 72, 499-519).

In this paper Jacobson and Williams found polynomial of the same form with
larger C(D)~5.65 and 72-digit D. But in this case D is not the least of
possible magnitudes of D.

But if wanted not least D, there is simple idea to find stronger results.

For the first time D=5(mod8).Then according to (1) we have D=2(mod3), D=2
or 3(mod5) and so on. Thus we generate two sets: s1=(5,2,2,…) and
s2=(8,3,5,…,pmax). Since all numbers from set s2 are coprime, then
according to Chinese Remainder Theorem there exist D such that (D/p)=-1 for
all pmax+1>p>2.

Using the idea, were found the next results with m-digit D`s

(Table 1):

Table1.

C(D)______m______pmax
6.01477…__276____659
7.01289…__1007___2377
8.01071…__3234___7559
9.00134…__10006__23173
10.00130…_31680__73291
11.00010…_98670__227651

For example, for pmax=659 we have:
D=262508778659464115096185323435905718621126859589721245815217424331\58852200686713735757701335690265711480595015201939640121478280530259\08516600030845284564979968834904572433379987818950634053387670481251\59694554959443856960907671060549826406564583370499654905232954370723070397.

Whether C(D) may be infinitely large or not?

The proposed procedure allows simple to answer on the question. Let us
consider C(q):

(2)C(q)=Product(1+1/p-1),

where the product is taken over the primes q=>p>2. Clear that with growing
q, C(q) infinitely increase. Well known that in proposed procedure for any
member of set s2 there exist at least one candidate for set s1 and then we
can always find D for any pmax. If q=pmax than C(D)>=C(q). And consequently
C(D) may be infinitely
large.

I would like to know any information about Hardy-Littlewood constant
estimations.

[Michael Jacobson-2]

Dear Valeryi,

Yes, you can certainly use CRT to find examples for which C(D) is likely as
large as you want.  The problem is finding a provably correct approximation
of C(D) for these large D values.  The point of our Math Comp paper was
constructing sufficiently small examples with large C(D) values that we
could efficiently (and provably, at least assuming the Extended Riemann
Hypothesis) approximate to 8 decimal digits.  To do this, we had to use the
expressions for C(D) found on page 501 that involve the class number and
regulator of the quadratic field Q(sqrt(D)) - computing these invariants is
the limiting factor of this method (around 100 decimal digits for D is
possible today).  As far as I know, it is unknown how to produce such
accurate approximations of C(D) for D values of the sizes that you've
produced via CRT, but it may be possible to prove that they have C(D) > 5.65
by some other means.

It is known that C(D) can be infinitely large, and in fact that
   C(D) >> log(log(D))
This was proved by Lukes, Patterson, and Williams (Nieuw Archif voor
Wiskunde (4) 13 (1995), p.113-139) assuming the Extended Riemann Hypothesis.
 The discussion on p.511-512 in the Math Comp paper mentioned above states
that this can be made unconditional by appealing to a result of Joshi for
bounding L(s,chi).

Best, Mike