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Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function. Define $$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$ Is it consistent with current …

Consider the subset of odd positive integers defined and constructed as follows by these rules : A) $1$ is in the set. B) if $x$ is in the set , then $2x + 1$ is in the set. C) if $x$ and $y$ are in t …

Consider prime constellations $p,p+2s$ where both $p,p+2s$ are prime. For instance for $s=1$ we get the twin primes. We define the counting function $\pi_{2s}(n)$ to count the number of such pairs $p, …

Let $r(a,b)$ be a rational number depending on positive integers $a,b$ and $r(a,b)$ being nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$. Let $C(b)$ be a squarefree positive …