The Hardy-Ramanujan and Halasz inequalities for shifted primes

Abstract

We denote the number of prime divisors of a number n by co(n), the number of prime divisors of n taking into account their multiplicity by ~(n), and put N(m, x,g) = I{ :n < = m} l, where g(n) = co(n) or g(n) = gt(n), and IAI is the number of elements of a set A.In 1917, Hardy and Ramanujan [1] proved the existence of constants Cl, c2 such that the inequalityholds for x _> 2. In the case where g(n) = co(n), the inequality is valid for any m, and for g(n) = f~(n) it is valid for m < (2 -5) log log x, 5 > 0. In the case where g(n) = f~(n), Hardy and Ramanujan proved, in fact,inequality (1) for m <_ (~q -5)log log x. More recently, Sathe [2] and Selberg [3] found the asymptotics of N(m, x, g) as x --+ ee. This result implies that (i) inequality (1) is valid for the mentioned values of m, and (it) estimate (1) is order-sharp for g(n) = co(n) if m _< bloglogx, where b is a positive constant, and for g(n) = f~(n) if m < (2 -5) log log x.Later on, the Hardy-Ramanujan inequality was applied in different problems and it was investigated to different kinds of generalizations. The most remarkable result was obtained by Halasz.Let E be an arbitrary nonerapty set of primes, p<_x,pEE and let co(N; E) and f~(n; E) be the number of prime divisors of n and the number of prime divisors of n taking into account their multiplicity, respectively, provided the divisors lie in E. Halasz [4] proved that the inequality cj( )xEm( ) N(m,z,g;E) = I{n:n < x,g(n;E) = rn}l < m!exp(E(x))(2) holds for m < (2 -5)E(x), and the inequality g(m, g; E) + g(m + 1, g; E) >_ m! exp(E( )) holds for 5E(x) < m <_ (2 -5)E(z), where 5 > 0, ca(5) > 0, c4(5 ) > 0, and g(n; E) = f~(n; E). Norton [5] proved that inequality (2) also holds for g(n; E) = co(n; E). As was shown in [6], the last estimate for N(m, x, fl; E) holds for the mentioned values of m.

On the range of Carmichael's universal-exponent function

“…That is, we will show that for t in the interval [y 1−1/ log log y , y], the number of primes p ≤ t such that any of the remaining conditions in the definition of P(y) fails is o(t/ log t) as y → ∞. The number of primes p ≤ t where the second item in the definition of P(y) fails is o(t/ log t) as can be seen by the method of Erdős [4], or more explicity by Timofeev [14] (see also Lemmas 2.1 and 2.2 in [5]).…”

“…for an absolute positive constant κ 1 . By a suitable adjustment of κ 1 if necessary, it follows from equation (2) of [14] (due to Halász) and partial summation that…”

Integers divisible by a large shifted prime

“…Proof. This is essentially a special case of part of Theorem 3 of [10], except that in the cited theorem it is stated that we must have z → ∞ as x → ∞. This condition does not make sense in light of the uniformity claimed in the theorem, and in fact this stronger hypothesis on z (which comes into play when dealing with a set E of primes, which in our application is taken to be the set of primes in [2, z]) is never used in the proof.…”

“…The next two lemmas, due to Timofeev [10], state that the prime factors of shifted primes have roughly the same distribution as prime factors of integers taken as a whole.…”

On a multiplicative function on the set of shifted primes

“…Along with the sieve method, we shall use Chang's method used by him to obtain a lower bound for the numbers p + 2, p < x, having at most two prime divisors. We shall apply Chang's method, following the scheme of [5].…”