# Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used?

In their paper,

Some problems of 'Partitio numerorum'; III - On the expression of a number as a sum of primes, (Acta Math. Vol.44, pp.1–70, 1922)

Hardy and Littlewood postulated 16 conjectures labeled Conjecture A through Conjecture P. Conjecture O is not explicitly labeled as such, and is commonly referred to as the second Hardy-Littlewood conjecture.

This question concerns Conjectures B and D which are:

Conjecture B (Prime Pairs Conjecture [page 42]) For every even $k$, there are infinitely many prime pairs, $p,p' = p + k$. If, $P_k(n)$, is the number of such pairs less than $n$, then \begin{equation*} P_k(n) \sim 2 C_2 \frac{n}{\left( \log n \right)^2} \prod\limits_{\substack{ p > 2 \\ p \mid k}} \left( \frac{p-1}{p-2}\right) \end{equation*} and \begin{equation*} C_2 = \prod\limits_{p = 3}^{\infty} \left( 1 - \frac{1}{\left( p - 1\right)^2} \right) \end{equation*}

Conjecture D (Sophie Germain Primes Conjecture [page 45]) If $(a, b) = 1$ and $P(n)$ is the number of prime pairs which are the solution of, $ap - bp' = k$, then, if $(k, a) = 1$ , $(k, b) = 1$ , and one and only one of $k$, $a$, and $b$ is even then: \begin{equation*} P(n) \sim \frac{2 C_2}{a} \frac{n}{\left( \log n \right)^2} \prod\limits_{p} \left( \frac{p-1}{p-2}\right) \end{equation*} where the product extends over all odd primes $p$ which divide $k$, $a$, or $b$.

Hardy and Littlewood derived these asymptotic densities using the circle method and a generalization of the Riemannn Hypothesis.

I am wondering does the converse of the two above conjectures imply the generalized RH used by Hardy-Littlewood?

If both: \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \sum\limits_{n=1}^{N} \Lambda\left(n\right)\Lambda\left(n + 2h\right) = 2 C_2 \prod\limits_{\substack{p \\ p \mid 2 h}} \left( \frac{p-1}{p-2}\right) \end{equation*} and \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \sum\limits_{n=1}^{N} \Lambda\left(n\right) \Lambda\left(\frac{bn + k}{a}\right) \left( \frac{1}{a} \sum\limits_{j=0}^{a-1} e^{2 \pi i \frac{bn + k}{a} j}\right) = \frac{2 C_2}{a} \prod\limits_{p} \left( \frac{p-1}{p-2}\right) \end{equation*} are true, then is the generalized hypothesis used by Hardy and Littlewood also true?

In other words does proving that the mean values of the two sieves converge to the same constants found by Hardy-Littlewood via the circle method, give any insight on whether their version of the Riemannn hypothesis is true?

P.S. Since \begin{equation*} \frac{1}{a} \sum\limits_{j=0}^{a-1} e^{2 \pi i \frac{bn + k}{a} j} = \begin{cases} 1 & a \mid \left( bn + k \right) \\ 0 & \text{ otherwise} \end{cases} \end{equation*} the expression for the second sieve avoids the messy question of what is the value of the von Mangoldt function, $\Lambda\left( x \right)$, for values which are not integers.

• Your question is interesting, but please keep in mind that the right spelling is Riemann, not Reimann. – Sylvain JULIEN Jul 22 '15 at 16:15
• -1 for spelling Riemann wrong 3 times in a single post. – M.G. Jul 22 '15 at 16:27
• 6 times if you count the 'ei' and the single 'n' as separate errors. – Stopple Jul 22 '15 at 19:04
• Now that we're done commenting on spelling errors without editing to correct them ... no, I don't know of any result that shows that these conjectures imply the relevant Riemann hypothesis. (Of course, both statements are likely to be proved true someday, so logically speaking, any statement will imply them.) – Greg Martin Jul 22 '15 at 19:09
• I suspect the answer is no. The ternary Goldbach conjecture is presented on page 3 and is a result of the their GRH and the circle method. The ternary Goldbach conjecture though is now a theorem and this result had no implication on the veracity of the GRH used by Hardy and Littlewood. – John Washburn Jul 23 '15 at 19:25