Divisors of shifted primes

Let p range over the set of primes and let a be a non-zero integer. Here we prove that many properties of the divisors of the natural numbers which can be expressed by inequalities are true for the set {p + a} of shifted primes, too. We obtain estimates for the quantities \{p : p + a less than or equal to x, d(m)(p + a) less than or equal to (log x)((1 - alpha)log m) or d(m)(p + a) greater than or equal to (log x)((1 + alpha)log m)}\ 0 < alpha < 1, (see Theorem 1) and \{p : p + a less than or equal to x, p + a has at least one divisor d such that y < d less than or equal to z}\, (see Corollary 2) where d(m)(n) denotes the number of representations of n as the product of m positive integers. Erdos conjectured that almost all integers have two divisors d, d' such that d < d' < 2d and this has recently been confirmed by H. Maier and G. Tenenbaum. We prove that this conjecture is true for the shifted primes, too (see Theorem 7). Let f be a multiplicative function and define M(n, f) = max(y) \Sigma(d\ndless than or equal toy) f(d)\, Delta(n, f) = max(y) \Sigma(d\ny<dless than or equal toey) f(d)\. Under some conditions on f we prove estimates for \{p : p + a less than or equal to x, rho(p + a, f) is an element of [K, S)}\, where rho(n, f) = Delta(n, f) or M(n, f). In particular we show Corollary 12. Let alpha < alpha(0) = -log 2/log (1 - 1/log 3) and let phi(x) --> infinity as x --> infinity. Then lim(x-->infinity) 1/pi(x)\{p : p + a less than or equal to x, (log(2) p)(alpha) < M(p + a, mu) < phi(p) log(2) p}\ = 1, where log(2) x = log log x and mu denotes the Mobius function. Corollary 10. Let x greater than or equal to x(0). Then c(1) log(2) x less than or equal to 1/pi(x) Sigma(p+aless than or equal tox) Delta(p + a, 1) less than or equal to c(2) exp ((1 + 30/log(3) x) rootlog(2) xlog(3) x), where c(1) > 0 and log(3) x = log(log(2) x).