A Hardy-Ramanujan type inequality for shifted primes and sifted sets
aa r X i v : . [ m a t h . N T ] J a n A HARDY-RAMANUJAN TYPE INEQUALITY FOR SHIFTED PRIMES AND SIFTED SETS
KEVIN FORDA
BSTRACT . We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted setswith a given number of distinct prime factors. In particular, we establish a bound for the number of shiftedprimes p + a below x with k distinct prime factors, uniformly for all positive integers k .
1. I
NTRODUCTION
The distribution of the number, ω ( n ) , of distinct prime factors of a positive integer n has been well studiedduring the past century. In 1917, Hardy and Ramanuajan [ ] proved the inequality(1.1) π k ( x ) := X n xω ( n )= k C x log x (log log x + C ) k − ( k − , where C , C are certain absolute constants. An asymptotic π k ( x ) , valid for each fixed k , had earlier beenproved by Landau in 1900. The chief importance of (1.1) lies in the uniformity in k , and it is this featurewhich allowed Hardy and Ramanujan deduced from (1.1) that ω ( n ) has normal order log log n . An as-ymptotic for π k ( x ) , uniform for k C log log x and arbitrary fixed C , was proved by Sathe and Selbergin 1954. Thanks to subsequent work of a number of authors, notably Hildebrand and Tenenbaum [ ], auniform asymptotic for π k ( x ) is known in a much wider range k c log x (log log x ) , c > some constant. Theright side of (1.1) represents the correct order of magnitude of π k ( x ) when k = O (log log x ) , but is slightlytoo large when k/ log log x → ∞ as x → ∞ . See Ch. II.6 in [ ] for a more detailed history of the problemand concrete formulas for π k ( x ) .The Hardy-Ramanujan inequality (1.1) has been extended and generalized in many ways, such as replac-ing the summation over n x with a restricted sum over shifted primes [
3, 15 ], replacing the summand 1with a multiplicative function [ ], counting integers with a prescribed number of integers in disjoint sets[ , Theorem 3], counting the prime factors of polynomials at integer arguments [ ], counting integers with ω ( n ) = k and ω ( n + 1) = k simultaneously [ , Th. 18] or replacing ω ( n ) with an arbitrary additivefunction [
10, 2 ].In this note we establish an analog of the Hardy-Ramanujan theorem, with complete uniformity in k , forprime factors of integers restricted by a sieve condition. The main theorem is rather technical and we deferthe precise statement to Section 2. Here we describe some corollaries which are easier to digest.1.1. Notation conventions.
Constants implied by the O - and ≪ -symbols are independent of any parameterexcept when noted by a subscript, e.g. O ε () means an implied constant that depends on ε . We denote π ( x ) the number of primes which are x . Date : January 14, 2021. For a set of integers n with counting function x − o ( x ) as x → ∞ , we have ω ( n ) = (1 + o (1)) log log n as n → ∞ . Application: prime factors of shifted primes.
Let a be a nonzero integer. The distribution of theprime factors of numbers p + a , p being prime, plays a central role in investigations of Euler’s totientfunction, the sum of divisors function, orders and primitive roots modulo primes, and primality testingalgorithms (for these applications, a = ± ). It is expected that the distribution of the prime factors of arandom shifted prime p + a x behaves very much like the distribution of the prime factors of a randominteger in [1 , x ] . A complicating factor is that the distribution of the large prime factors of p + a , say those > √ p , is poorly understood. For example, Baker and Harman [ ] showed that infinitely often, p + a has aprime factor at least p . , and this is not known with . replaced by a larger number.In 1935, Erd˝os [ ] proved that the function ω ( p − has normal order log log p over primes p . Toshow this, Erd˝os proved an upper bound of Hardy-Ramanujan type for the number of primes p x with ω ( p −
1) = k in a restricted range of k . The bounds were sharpened by Timofeev [ ], who proveda conjecturally best possible upper bound when k = O (log log x ) . Here we extend this bound to holduniformity for all k , uniformity in a , and correct a small error in Timofeev’s bound when a is odd. Corollary 1.
Let a = 0 and define s = 2 if a is odd, and s = 1 if a is even. Then uniformly for k ∈ N , x > and all a = 0 we have {− a < p x : ω (cid:0) p + as (cid:1) = k } ≪ | a | φ ( | a | ) π ( x ) (log log x + O (1)) k − ( k − x . We remark that Timofeev worked with ω ( p + a ) rather than ω ( p + as ) ; when a is odd and s = 2 , dividingby s is necessary because 2 always divides p + a when p > . The corresponding lower bound is not knownfor any k , although it is conjectured to hold for every k satisfying k = O (log log x ) . The problem of thelower bound is intimately connected with the parity problem in sieve theory. The best lower bound in thisdirection is Theorem 3 of Timofeev [ ] which states (in the case a = 2 ) that { p x : ω ( p + 2) ∈ { k, k + 1 }} ≫ π ( x ) (log log x + O (1)) k − ( k − x uniformly for k ≪ log log x . The case k = 1 is a the celebrate Theorem of J.-R. Chen.1.3. Application: integers with restricted factorization.
Let E be any set of primes and let Q ( E ) be theset of positive integers, all of whose prime factors belong to E . Let(1.2) E ( x ) = X p ∈E p x p . The next corollary was established by Tenenbaum [ , Lemma 1] using a different method. Corollary 2.
Uniformly for all E and all k ∈ N we have { n x, n ∈ Q ( E ) : ω ( n ) = k } ≪ x ( E ( x ) + O (1)) k − ( k − x . We also establish a count of shifted primes p + a with a given number of prime factors, such that p + a only has prime factors from a given set, generalizing Corollaries 1 and 2. Corollary 3.
Let a = 0 , and let s = 1 if a is even and s = 2 is a is odd. Let E be any nonempty set ofprimes, and define E ( x ) by (1.2) . Uniformly for all a , all E and all k ∈ N we have {− a < p x, p + as ∈ Q ( E ) : ω (cid:0) p + as (cid:1) = k } ≪ | a | φ ( | a | ) π ( x ) ( E ( x ) + O (1)) k − ( k − x . HARDY-RAMANUJAN TYPE INEQUALITY FOR SHIFTED PRIMES AND SIFTED SETS 3
Application: the mean of twin primes.
Hardy and Littlewood conjectured in 1922 that the numberof prime p x with p + 2 also prime is asymptotic to Cx/ log x for some constant C . At present, it is notknown that there are infinitely many such twin prime pairs. Here we focus on the number of prime factorsof p + 1 for such primes. Corollary 4.
Uniformly for k ∈ N we have { < n x : n − and n + 1 are both prime , ω (cid:0) n (cid:1) = k } ≪ x (log log x + O (1)) k − ( k − x . Again, we divide by 6 because all such n are divisible by 6.Corollaries 1, 2, 3 and 4 represent only a small sample of the type of bounds attainable using Theorem 1below. For example, we obtain conjecturally best-possible (in the case k = O (log log x ) ) upper bounds onthe number of n x with ω ( n ) = k , and with n − prime, n + 1 prime, n + 5 prime, and such that n hasonly prime factors from a given set.As with (1.1), we expect the left sides in the corollaries to be of smaller order than the right sides when k/ log log x → ∞ as x → ∞ . We will return to this in a subsequent paper.2. S TATEMENT OF THE M AIN T HEOREM
Here we state our main theorem and prove Corollaries 1, 2, 3 and 4.Let G ( A ) denote the set of non-negative multiplicative functions satisfying(2.1) g ( p v ) Ap v ( p prime , v ∈ N ) . An immediate consequence of (2.1) and Mertens’ theorems is(2.2) G ( x ) := X p x g ( p ) A (log log x + O (1)) , ( x > . Theorem 1.
Let S be a set of positive integers, and let s be the largest integer dividing every element of S .Suppose that g ∈ G ( A ) , x > s , B > exp {−√ log x } and λ > is a constant so that (2.3) (cid:26) prime q xrs : qrs ∈ S (cid:27) Bx g ( r )log λ ( xrs ) (1 r x/s ) . Then, uniformly for positive integers k , (cid:26) n x, n ∈ S : ω ( n/s ) = k (cid:27) ≪ λ,A Bx ( G ( x ) + O A (1)) k − ( k − λ x . The proof of Theorem 1 will be given in the next section. Here we discuss corollaries.We first recover the original Hardy-Ramanujan inequality (1.1). In this case S = N and the left side of(2.3) is π ( y/r ) ≪ ( y/r ) / log(2 y/r ) by Chebyshev’s estimates for primes. Also, g ( r ) = 1 /r for all r and g ∈ G (1) . Theorem 1 then implies (1.1). Proof of Corollary 3.
Let S = { p + a : p > − a prime , p + as ∈ Q ( E ) } . Provided that r ∈ Q ( E ) , for all y > rs we have by a standard sieve bound (Corollary 2.4.1 in [ ]) that (cid:26) q yrs : qrs ∈ S (cid:27) = (cid:26) q yrs : q, qrs − a both prime (cid:27) ≪ | ars | φ ( | ars | ) yrs log ( yrs ) . KEVIN FORD
When rs
6∈ Q ( E ) , the left side is zero. Thus, defining g ( r ) = 1 /φ ( r ) when r ∈ Q ( E ) and zero otherwise,we see that (2.3) holds with B ≪ | a | φ ( | a | ) . Clearly g ∈ G (2) , and G ( x ) = E ( x ) + O (1) since g ( p ) =1 /p + O (1 /p ) for p ∈ E . Corollary 1 now follows from Theorem 1 when x > and is trivial otherwise. (cid:3) Corollary 1 is a special case of Corollary 3, upon taking E the set of all primes. Proof of Corollary 2.
Let S = Q ( E ) . Here we have s = 1 (in particular, ∈ S ). For any r y we have { q y/r : qr ∈ S} ( if r
6∈ S π ( y/r ) if r ∈ S . By Chebyshev’s bound for π ( x ) , (2.3) holds with λ = 1 , B = O (1) and g defined by g ( p v ) = 1 /p v if p ∈ E , g ( p v ) = 0 if p
6∈ E . Hence g ∈ G (1) and G ( x ) = E ( x ) . The Corollay follows from Theorem 1. (cid:3) Proof of Corollary 4.
Let S = { n > n − , n + 1 both prime } . We have s = 6 . By the sieve (e.g.,Theorem 2.4 of [ ]), for any r x/ , { q x r : 6 rq ∈ S} ≪ xg ( r )log (cid:0) x r (cid:1) , g ( r ) = 1 r Y p | rp> − /p − /p . Thus, (2.3) holds andyn g ∈ G (2) . Since g ( p ) = p + O (cid:16) p (cid:17) , G ( x ) = log log x + O (1) , and the corollaryfollows. (cid:3)
3. P
ROOF OF T HEOREM P + ( r ) is the largest prime factor of r , with P + (1) := 0 . Lemma 3.1.
Let λ > and g ∈ G ( A ) . Uniformly for x > and ℓ > we have X ω ( r )= ℓrP + ( r ) x g ( r )log λ ( x/r ) ≪ A,λ ( G ( x ) + O A (1)) ℓ ℓ ! log λ x . Proof. If ℓ = 0 then the only summand corresponds to r = 1 and the result is trivial. Now suppose ℓ > .Then r x/ . We separately consider r in special ranges. Let Q j = x / j for j > and define T j = (cid:26) r ∈ (cid:20) , x (cid:21) ∩ (cid:20) xQ j − , xQ j (cid:21) : ω ( r ) = ℓ, rP + ( r ) x (cid:27) . For r ∈ T j , we have P + ( r ) x/r Q j − . Also, if T j is nonempty then Q j − > and j > . We have X r ∈ T j g ( r )log λ ( x/r ) λ Q j X r ∈ T j g ( r ) . For the sum on the right side, we use the “Rankin trick” familiar from the study of smooth numbers. Let α =
120 log Q j . Since Q j − > , Q j > √ and thus < α . From the definition (2.1) of G ( A ) we have g ( m ) A ω ( m ) m ≪ A m − / . Hence, when r > x/Q j − , g ( r ) = g ( r ) α g ( r ) − α ≪ A r − α/ g ( r ) − α ≪ x − α/ g ( r ) − α , HARDY-RAMANUJAN TYPE INEQUALITY FOR SHIFTED PRIMES AND SIFTED SETS 5 since ( x/r ) α/ Q α/ j − = Q αj = e / . Thus,(3.1) X r ∈ T j g ( r )log λ ( x/r ) ≪ x − α/ log λ Q j X r ∈ T j g ( r ) − α x − α/ log λ Q j X P + ( r ) Q j − ω ( r )= ℓ g ( r ) − α . Using (2.1) again, X P + ( r ) Q j − ω ( r )= ℓ g ( r ) − α ℓ ! (cid:26) X p Q j − g ( p ) − α + g ( p ) − α + · · · (cid:27) ℓ = 1 ℓ ! (cid:26) O A (1) + X p Q j − g ( p ) − α (cid:27) ℓ . If g ( p ) > /p we have g ( p ) − α p α = 1 + O ( α log p ) when p Q j − . Hence, X p Q j − g ( p ) − α X g ( p ) < /p g ( p ) / + X p Q j − g ( p ) > /p g ( p )(1 + O ( α log p )) O (1) + G ( Q j − ) + O A (1) , using (2.1) again plus Mertens’ theorems. Thus, X P + ( r ) Q j − ω ( r )= ℓ g ( r ) − α ( G ( x ) + O A (1)) ℓ ℓ ! . Inserting the last bound into (3.1), we see that for each j , X r ∈ T j g ( r )log λ ( x/r ) ≪ A x α/ log λ Q j X P + ( r ) Q j − ω ( r )= ℓ g ( r ) − α λj exp (cid:8) − · j (cid:9) log λ x ( G ( x ) + O A (1)) ℓ ℓ ! . Summing over j completes the proof. (cid:3) Proof of Theorem 1.
Let n x, n ∈ S and ω ( n/s ) = k . Define q = P + ( n/s ) and write n = qrs . If q ∤ r then ω ( r ) = k − , and if q | r then ω ( r ) = k . Also, rP + ( r ) rq = n/s x/s . It follows that r ∈ R k − ∪ R k , where R ℓ = { r ∈ N : rP + ( r ) x/s, ω ( r ) = ℓ } . KEVIN FORD
Using (2.3), followed by Lemma 3.1, we have for ℓ ∈ { k − , k } the bounds { n x, n ∈ S : ω ( r ) = ℓ } X r ∈R ℓ { q x/ ( rs ) : qrs ∈ S} Bx X r ∈R ℓ g ( r )log λ ( xrs ) ≪ λ,A Bx ( G ( x ) + O A (1)) ℓ ℓ ! log λ ( x/s ) ≪ λ,A Bx ( G ( x ) + O A (1)) ℓ ℓ ! log λ x , (3.2)using that x > s in the last step.When k > log log x we use the crude estimate G ( x ) ≪ A log log x from (2.2) and deduce from (3.2) that { n x, n ∈ S : ω ( n ) = k } ≪ λ,A Bx ( G ( x ) + O A (1)) k − ( k − λ x (cid:18) G ( x ) + O A (1) k (cid:19) ≪ λ,A Bx ( G ( x ) + O A (1)) k − ( k − λ x . If k log log x , we keep the ℓ = k − term from (3.2) and bound the ω ( r ) = k term in adifferent way. If ω ( r ) = k then q | ( n/s ) . Thus, using a crude version of the main theorem in [ ], and thehypothesized bound B > exp {−√ log x } , we deduce that { n x, n ∈ S : ω ( r ) = k } { m x/s : P + ( m ) | m }≪ x exp {− p log x } ≪ A,λ Bx ( G ( x ) + O A (1)) k − ( k − λ x ) . (cid:3) Acknowledgements . The author thanks R´egis de la Bret`eche for drawing his attention to [ ] and Lemma1 of [ ]. R EFERENCES [1] R. C. Baker and G. Harman,
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EST G REEN S TREET , U
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