A heuristic asymptotic formula concerning the distribution of prime numbers

Artin prime producing quadratics

“…Since r(( 1 + 1/p) ^-e 3' logx/^(2) by a result of MERTENS (1874), it then follows from (13) that L(1, XA) = 0. However, L(1, Xo) > 0 by (2). q…”

“…By a special case of a conjecture due to BATEMAN and HORN [2] n f(x), the number of integers 0 < n <x such that f (n) is prime, should satisfy, as x tends to infinity,…”

“…Having a computer at his/her disposal the modern number theorist immediately disposes of Griffin's assertion: the first 16 primes p of the form 10X2 + 7 have indeed decimal period p -1, but this is not true for p = 7297, the 17th such prime. Nevertheless, one can wonder whether there exists a quadratic polynomial such that the prime values amongst f(0), f(l), f (2), • • • are all in J" (g) (we call this Grin's dream). To avoid trivialities one wants f to be such that, assuming the Bateman-Horn conjecture (see next section), f assumes infinitely many distinct prime values.…”

On values of n 2 + 1 free of large prime factors

“…Indeed, there is a more general conjecture on irreducible polynomials without fixed prime divisors, and this has been put into a quantitative form [2,9]. It is also reasonable to suppose that, given > 0, infinitely often one should have P + (n 2 + 1) < n , where P + (m) denotes the greatest prime factor of the positive integer m. Indeed, this has been proved by Schinzel [15].…”

Upper bounds for prime k -tuples of size log N and oscillations

“…, a k } is admissible if there is no trivial congruence obstruction, i.e. if the set A does not contain a complete set of residues modulo any prime p k. A quantitative version of these conjectures has been suggested by Hardy and Littlewood, see [8], and later by Bateman and Horn, see [1] and [2]. Let E A (N ) = |{n N : n + a i prime for i = 1, .…”