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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\...

Let $p_k$ be the $k$-th prime number, $\gamma$ be the Euler-Mascheroni constant and $M$ be the Meissel–Mertens and let $m$ be the integer part of $\log p_n$. We can show that $$ \sum_{r=1}^{m} \frac{...

Bateman & Horn [1], building on Bateman & Stemmler [2], give a conjectured formula for the density of numbers that produce simultaneous primes in a number of fixed polynomials. The constant ...