Primes of a special form

There is a long history of interest in primes of a special form. Writing $p$ for a prime and $p'$ for the next larger prime, some of the famous examples are: twin primes ($p'-p=2$), primes represented by polynomials (the simplest case is $p=n^2+1$), Sophie Germaine primes ($2p+1$ is prime), and many others.

None of these problems has been connected to the zeros of the Riemann $\zeta$-function is a satisfactory way. Turan [38 #127] related twin primes to zeros of $L$-functions near $\frac12$, and it is possible that recent ideas on the distribution of low-lying zeros in a family will shed some light on that problem. Bogolmony and Keating [Nonlinearity 8, 1115-1131] [Nonlinearity 9, 911-935] derive all $n$-correlation functions by assuming the Hardy-Littlewood conjectures and ignoring error terms, However, one cannot deduce the Hardy-Littlewood conjectures from the correlation functions.

Problem: Devise a believable conjecture about the zeros of one or more $L$-functions which implies that there are infinitely many primes of one of the special forms described above.

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