There is a long history of interest in primes of a special form.
Writing for a prime and
for the next larger prime,
some of the famous examples are: twin primes (
),
primes represented by polynomials (the simplest case is
),
Sophie Germaine primes (
is prime), and many others.
None of these problems has been connected to the zeros of the Riemann
-function is a satisfactory way. Turan [38 #127] related
twin primes to zeros of
-functions near
, 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
-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 -functions which implies that there are infinitely
many primes of one of the special forms described above.
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