We show by an inclusion-exclusion argument that the prime k-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime k-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.

The most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given x. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310, . . . . As a step towards proving this conjecture they introduced a second weaker conjecture that any fixed prime p divides all sufficiently large jumping champions. In this paper we extend a method of P. Erdős and E. G. Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of G. H. Hardy and J. E. Littlewood.2010 Mathematics Subject Classification. Primary 11N05. Secondary 11P32, 11N36.

We consider random analytic functions defined on the unit disk of the complex plane f (z) = ∞ n=0 a n X n z n , where the X n 's are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients a n are chosen so that f (z) is defined on a domain of C carrying a planar or hyperbolic geometry, and Ef (z)f (w) is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virág. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.

We study the irrational factor function I(n) introduced by Atanassov and dened by I(n)is the prime factorization of n. We show that the sequence G(n)/n n 1 , where G(n) = n ν=1 I(ν) 1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).