The constant in Hardy-Littlewood's Conjecture F

CRGreathouse · 2010-07-06, 15:44

The Hardy-Littlewood Conjecture F includes calculation of the infinite product
$\prod_{\varpi\ge3,\varpi\not|a}\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$
where (I believe!) $\varpi$ ranges over the primes and $\left(\frac D\varpi\right)$ is the Jacobi symbol.

Is there a good way to calculate this? Or, how can one calculate a reasonable number of decimal places of
$f(D)=\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$
where $\varpi$ ranges over the odd primes? (Recovering the original problem for any factorable a is easy.)

CRGreathouse · 2010-07-06, 16:47

The problem can be solved if there are good ways to accelerate the calculation of sums over primes in congruence classes:
$\prod_{\varpi\equiv m\pmod n}1-\frac{1}{\varpi-1}$
and
$\prod_{\varpi\equiv m\pmod n}1+\frac{1}{\varpi-1}$
for $\varpi$ prime and integer m,n. Can this be done?

ccorn · 2010-07-06, 17:33

Quote:

Originally Posted by CRGreathouse

The Hardy-Littlewood Conjecture F includes calculation of the infinite product
$\prod_{\varpi\ge3,\varpi\not|a}\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$
where (I believe!) $\varpi$ ranges over the primes and $\left(\frac D\varpi\right)$ is the Jacobi symbol.

Is there a good way to calculate this? Or, how can one calculate a reasonable number of decimal places of
$f(D)=\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$
where $\varpi$ ranges over the odd primes? (Recovering the original problem for any factorable a is easy.)

Have you checked Shanks' paper?

ccorn · 2010-07-06, 17:34

Quote:

Originally Posted by CRGreathouse

The problem can be solved if there are good ways to accelerate the calculation of sums over primes in congruence classes:
$\prod_{\varpi\equiv m\pmod n}1-\frac{1}{\varpi-1}$
and
$\prod_{\varpi\equiv m\pmod n}1+\frac{1}{\varpi-1}$
for $\varpi$ prime and integer m,n. Can this be done?

(Logs of) These would diverge, I'm afraid.

CRGreathouse · 2010-07-06, 17:47

Thanks! I'll look it over.

Quote:

Originally Posted by ccorn

(Logs of) These would diverge, I'm afraid.

(facepalm)

Of course if I had a Mertens' Theorem-like estimate for
$\prod_{p\equiv m\pmod n,p\le x}1-\frac{1}{p-1}$
and
$\prod_{p\equiv m\pmod n,p\le x}1+\frac{1}{p-1}$
that would suffice.

2010-07-06, 15:44	#1
CRGreathouse Aug 2006 3·1,987 Posts	The constant in Hardy-Littlewood's Conjecture F The Hardy-Littlewood Conjecture F includes calculation of the infinite product $\prod_{\varpi\ge3,\varpi\not\|a}\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$ where (I believe!) $\varpi$ ranges over the primes and $\left(\frac D\varpi\right)$ is the Jacobi symbol. Is there a good way to calculate this? Or, how can one calculate a reasonable number of decimal places of $f(D)=\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$ where $\varpi$ ranges over the odd primes? (Recovering the original problem for any factorable a is easy.)

2010-07-06, 16:47	#2
CRGreathouse Aug 2006 3×1,987 Posts	The problem can be solved if there are good ways to accelerate the calculation of sums over primes in congruence classes: $\prod_{\varpi\equiv m\pmod n}1-\frac{1}{\varpi-1}$ and $\prod_{\varpi\equiv m\pmod n}1+\frac{1}{\varpi-1}$ for $\varpi$ prime and integer m,n. Can this be done? Last fiddled with by CRGreathouse on 2010-07-06 at 16:56

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