The Hardy-Littlewood Conjecture F [1] involves the infinite product
$$\prod\left(1-\frac{1}{\varpi-1}\left(\frac D\varpi\right)\right)$$
where $\varpi$ ranges over the odd primes and $\left(\frac D\varpi\right)$ is the Legendre symbol.


Is there a good way to calculate this? The product converges very slowly, and none of the standard methods (Cohen-Villegas-Zagier, Wynn, etc.) seem to work because of the unpredictable sign changes.


Given that *D* is fixed, it suffices to calculate the partial products in various congruence classes; I don't know if this is a viable approach.


Another possibility: I've seen almost magical series acceleration with the zeta function, it may work here.


[1] G. H. Hardy, J. E. Littlewood. "Some of the problems of partitio numerorum III: On the expression of a large number as a sum of primes". *Acta Mathematica* **44** (1923), pp. 1-70.



Calculating the infinite product from the Hardy-Littlewood Conjecture F
This problem was studied by a few, and the ideas involve too much latex to write here. Mainly there are ideas of transforming to crazy weighted sums and then use ERH to bound errors from crazier integrals. It suffices to say that the culmination of this research is the freely available paper:



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> [New Quadratic Polynomials With High Densities Of Prime Values](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.3281), Jacobson and Williams 1999
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In that year, they used their ideas to compute the constant for discriminants of up to 72 digits! (assuming ERH) The bottleneck of the process seems to be the calculation of the algebraic invariants class number and regulator. Over a decade has passed, the technology today should be able use the same methods to get up to 100 digits, within reasonable time, and up to 110 digits with a bit more time (apparently, two weeks on a cluster):



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> [Practical Improvements to Class Group and Regulator Computation of Real Quadratic Fields](http://arxiv.org/abs/1005.0205), Biasse and Jacobson 2010
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> [Improvements in the Computation of Ideal Class Groups of Imaginary Quadratic Number Fields](http://hal.inria.fr/docs/00/39/74/08/PDF/biasse.pdf), Biasse 2009
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