AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM

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1 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Abstract The Siegel-Walfisz Theorem states that for any B > 0, we have p p a mod ϕ log for log B and, a = This only gives an asymptotic formula for the number of primes over an arithmetic progression for quite small moduli compared to However, if we are only concerned about upper bound, we have the Brun-Titchmarsh theorem, namely, for any <, p p a mod ϕ log/ In this article, we prove an etension to the Brun-Titchmarsh theorem on the number of integers, with at most s prime factors, in an arithmetic progression, namely, y<n+y ωns s log log/ + K l ϕ log/ l! for any, y > 0, s and < In particular, for s log log/, we have y<n+y ωns ϕ log/ log log/ + K s log log/ + K s! and for any ε 0, and s ε log log/, we have y<n+y ωns ε ϕ log/ log log/ + K s s! Date: December 4, 2009 Research of KK Choi is supported by NSERC of Canada Research of KM Tsang is fully supported by RGC grant HKU 7042/04P of the Hong Kong SAR, China

2 2 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Introduction Throughout this paper, p, p, p 2, etc denote prime numbers The celebrated Prime Number Theorem, conectured by Gauss and proved by Hadamard and de la Vallée Poussin, asserts that = + o log, p where the summation is over all primes p One can also consider the distribution of primes in arithmetic progressions For a, =, put where Li := 0 p p a mod = Li ϕ + E;, a, log tdt and ϕn is the Euler function The Siegel-Walfisz theorem states that for arbitrary constants A, B > 0 2 ma E;, a log A a,= uniformly for log B and this only gives an asymptotic estimate for quite small moduli compared to Furthermore, if we assume the generalized Riemann hypothesis, we can prove that 2 holds for /2 log A 2 The celebrated Bombieri-Vinogradov theorem has the same strength as the generalized Riemann hypothesis on average, and therefore enables us to deal with various problems It asserts that for every A > 0 there eists a constant B = BA > 0 such that ma a,= /2 log B E;, a log A See [], for instance Although asymptotic estimate can only be proved so far for quite small moduli, if we are only concerned about upper bound inequality for, we have the Brun-Titchmarsh theorem eg see [2], namely, for any <, p p a mod log/ In this article, we consider an etension of the Brun-Titchmarsh theorem and study the number of integers in short intervals with eactly s prime factors counted with multiplicity that lie in the arithmetic progression a mod Let Ωn denote the number of prime factors of n counted with multiplicity and ωn denote the

3 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 3 number of distinct prime factors of n asymptotic formula n ωn=s = + o log p Landau [5] showed that for fied s the log log s s! holds Sathe [8] and Selberg [9] proved a more precise quantitative estimate Let F z := + z z, Γz + p p where the product is taen over all primes p Then s log log s = F log log log s! n ωn=s + O log log holds uniformly for 3 and s C log log for any given fied C > 0 For more discussions, see [4] and [6] Upper bounds for the above sum have been obtained for much wider ranges A classical eample is the Hardy-Ramanuan inequality [3] 3 n ωn=s log log + c 2 s c log s! which is valid, with suitable constants c and c 2, for all 3 and s In [2], Warlimont and Wole etended the Hardy-Ramanuan inequality by estimating the number of such integers in an interval y, + y], namely, 4 y<ny+ ωn=s µ 2 n c ɛ log log + c 2 s log s! for any y 0 and 0 < ɛ < /2, provided that for sufficiently large, log log s ɛ log log log It was mentioned in [2] that, a larger upper bound than 4 is needed if the range of s is etended to s log log A further remar on this will be discussed later Recently, Tudesq in [] showed that the inequality 4 still holds when the squarefree condition is removed but for a larger upper bound He proved that for 2 y and s, y<ny+ ωn=s log log log + c 2 s s!

4 4 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Here t r := tt + t + r It can be easily shown that t s = t s + + s 2 { } s 2s t s ep t t 2t and hence in the range s log log, Tudesq s result recovers Warlimont and Wole s result 4 By etending Selberg s idea in [9], Spiro proved in [0], among other results, that if we fi B with 0 < B < B B = 2 for η = Ω and B is arbitrary for η = ω, select u > 0, and let a, s, be integers with a, =, and s B log log, < ep log, then for ηn = ωn or Ωn, we have 5 n ηn=s p log u, p = + ogy ϕ log uniformly in a, s and, where y = Gz := g,z,χ0;η Γ+z and gw, z, χ; η := p wherever the product converges log log s s! ϕ y s log log, χ 0 is the principal character modulo, χp m z ηpm p mw χpp w z m=0 For larger values of the situation is much less satisfactory and no simple asymptotic formulae have been obtained so far The purpose of this paper is to obtain an upper bound estimate for the number of positive integers in the interval y, + y] which have eactly s prime factors and lie in the arithmetic progression a mod for large range of In [7], Orazov showed that for fied s, y<n+y ωn=s µ 2 n s τ s log log s ϕ log provided /2s, where τ is the divisor function of Our main Theorem is the following

5 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 5 Theorem Let, y > 0 and s and let a, be coprime positive integers with < We have 6 y<n+y ωns K s log log/ + K l ϕ log/ l! In particular, for s log log/, the right hand side of 6 is K ϕ log/ log log/ + K s log log/ + K ; s! and for any ε 0, and s ε log log/, the right hand side is Kε ϕ log/ For any L, it is not difficult to show that L L l l! l L L log log/ + K s s! L l l! L LL L! e L Hence, when s log log/ + K, the sum in the right hand side of 6 is log/, yielding a trivial bound for the sum in the left hand side Thus the main interest of our result concerns the range s log log/ Clearly Theorem includes the following Theorem 2 Let, y > 0 and let a, be coprime positive integers with < For ηn = ωn or Ωn, we have y<n+y ηn=s K s log log/ + K l ϕ log/ l! In particular, for s log log/, we have y<n+y ηn=s K ϕ log/ log log/ + K s log log/ + K ; s! and for any ε 0, and s ε log log/, we have 7 y<n+y ηn=s Kε ϕ log/ log log/ + K s s!

6 6 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Our Theorem 2 improves Warlimont and Wole s, Tudesq s and Orazov s results We remar that in our range of s ε log log/, 5 shows that our upper bound in Theorem 2 is of the correct order of magnitude, at least when is a sufficiently large function of y It is very tempting to etend 7 in our Theorem 2 to the more natural range s + ε log log/ However, similar to the comment of Warlimont and Wole in [2], this can t be done We prove this in the following proposition Proposition 3 For any A and ɛ > 0, there eist constants c 0 A, ɛ and c A, ɛ with the following property: Let c 0, ep log 2, p p log and 8 log log/ + 2 log log/ log2/ϕ + c log log/ s 2 ɛ log log/ Then for any positive integer a coprime with, there eists y > 0 such that y<ny+ Ωn=s log log/ s A ϕ log/ s! Proof Since ep log 2, we have ep log/ Then we apply Spiro s result in 5 with Y = eplog/ +δ in place of, where δ > 0 is to be determined later We note that cf Lemma 3 in [0] for s satisfying condition 8, we have s / log log Y < 2 ɛ/ + δ < 2 ɛ and s G log log Y = Γ + s s log log Y g, log log Y, χ 0; Ω Hence, by 5 ny Ωn=s c 3 Y ϕ log Y log log Y s s! s / log log Y ϕ, for some constant c 3 ɛ > 0

7 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 7 Partition the interval 0, Y ] into [Y/]+ subintervals of the form 0, ],, 2], 2, 3], Then there eists y = > 0 for some such that y<ny+ Ωn=s c 3 2 [Y/] + ϕ log Y ny Ωn=s log log Y s s! s / log log Y ϕ Comparing the right hand side with the desired lower bound in our proposition, we see that our proposition would follow if s δ log log/ + log2a/c 3 log + δ log log Y log/ϕ holds Using now the inequalities log + δ δ 2 δ2 and log log Y > log log/, the right hand side above is δ log log/ + log2a/c 3 δ 2 δ2 log/ϕ/ log log/ By taing δ satisfying we find that the last epression is 2 δ2 = log2/ϕ/ log log/, log log/ + 2 log log/ log2/ϕ + 2 log2a/c 3 log log/ This proves our proposition Our argument in Proposition 3 does not wor if the interval y, y + ] is replaced by the interval, ] In fact, the countereample occurs when y is quite large, namely, log y = log/ +δ The main idea of proof of our Theorem is the classification of positive integers n = p α α pα p p α l l = rw, say where is determined by Then n = r the former case, w p α pα2 2 pα + < pα pα2 2 pα+ w In each w, either p is large or p α can be estimated by sieve method When pα is a large prime power In is a large prime power, there are at most two possibilities for the eponent α and a trivial estimate for the w then suffices The sum r can be estimated by Hardy-Ramanuan type argument in [3] This yields the! in the denominator in Lemma 8

8 8 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG 2 Lemmas We need the following results to prove our Theorem Throughout, we let K denote a positive absolute constant which need not have the same value at each occurrence but K is effectively computable Lemma 4 For 2 y < z < X, we have p logx/p = log z log y log log log X logx/z logx/y + O y<pz In particular, for 2 z X, we have 9 pz p logx/p log z logx/z + log y logx/y log log z + K log X Proof This lemma can be proved by the partial summation that z p logx/p = z logx/t p dt pt y p t log 2 X/t pt y<pz and the well-nown estimate = log log t + c + O p log t pt y for t 2 and some constant c For any w >, we let P w := p<w p where the product is over all primes less than w Lemma 5 For any X, Y a, =, we have Y <nx+y n,p w= K > 0 and any integers a and > 0 such that X ϕ log w + w 2 ϕ log 2 w Proof This is Theorem 36 of [2] or Theorem 38 of [6]

9 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 9 Lemma 6 Let X, Y > 0 and let a, be coprime positive integers with < X We have Y <p α X+Y p α a mod KX ϕ logx/ Here the summation on the left hand side is over all prime powers p α Proof For > X/2, the result is trivial We may henceforth assume X/2 If X p >, then the sieve bound Lemma 5 yields our result as before For any prime p, let β < β 2 < be such that 0 Y < p βi X + Y, p βi a mod, i =, 2, Then for > i, p β p βi is a multiple of and so is p β βi since p, = a, = Also, X > p β p βi = p βi p β βi > 0 Hence p βi < X/ In other words, all but one of the prime powers p α in the sum of the lemma satisfy Y <p α X+Y p α a mod p X/ p α < X/ In particular, for each p there are logx/ + powers of p satisfying 0 Thus log X X + p X/ This completes the proof of our lemma For any positive integer m = p α pα2 2, where p pα < p 2 < < p are primes and α, α 2,, α we let and Define u = v = um := p α pα2 2 pα pα+, vm := p α pα2 2 pα p2

10 0 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG Lemma 7 Let ξ 2 Define P 0 ξ := For h, we have log ξ P h ξ := m=p α pα h h umξ m logξ/m log log ξ + Kh h! log ξ and Q h ξ := m=p α pα h h um>ξ vmξ m K h! log ξ log log ξ + Kh Proof First, P ξ = p α st p α+ ξ p α logξp α = p logξp + = p ξ log ξ log log ξ /2 log ξ /2 +O pξ /3 pξ /3 log log ξ/p [ log p ] α=2 log log ξ + K + O log ξ log log ξ + K log ξ log ξ/p [ log p ] α=2 log 2 logξ/2 α + log ξ p α pξ /3 p α logξp α p 2 log ξ + O log ξ by Lemma 4 We now use mathematical induction on h For each prime power p α, α, let Lp a := {mp α : p, m =, ωm = h, ump α ξ} For each n counted in the sum P h+ ξ, we have n = p α pα2 2 pα h+, un ξ Hence n belongs to Lp αi i for i =, 2,, h + and h + P h+ ξ = p α n Lp α n log ξ n h+

11 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM For n = mp α Lp α, the condition ump α ξ implies um ξ/p α Also Lp α is an empty set unless p α+ ξ Thus, the last double sum is h! p α st p α+ ξ p α+ ξ p α P h ξ/p α p α log logξ/pα + K h logξ/p α, by induction hypothesis on P h ξ/p α Then by P ξ, we find that as desired P h+ ξ h +! Net we prove the bound for Q ξ Q ξ = log log ξ + Kh p α+ ξ h +! log ξ log log ξ + Kh+, p α st p α+ >ξ p 2 ξ p ξ p α = p ξ p 2 ξ p = ξ α+> log ξ log p p ξ p α For h 2, using the above bound for Q ξ, we have as desired Q h ξ = = m=p α pα h h um>ξ vmξ n=p α pα h h unξ K m n=p α pα h h unξ = K P h ξ p α logξ/p α p + + K p log ξ n=p α pα h h unξ n Q ξn n log ξn For each integer n >, let n = p α pαt t n p α h h st p α h + h >ξn p 2 h ξn K h! log ξ log log ξ + Kh, p α h h denote its canonical factorization in which p < p 2 < < p t For any ξ >, define the following sets: H = H ξ := {n : ωn, p α+ > ξ}

12 2 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG H 2 = H 2 ξ := {n : ωn 2, p α+ ξ, p α pα2+ 2 > ξ} H 3 = H 3 ξ := {n : ωn 3, p α pα2+ 2 ξ, p α pα2 2 pα3+ 3 > ξ} and in general, for, H = H ξ = {n : ωn, p α pα 2 pα + ξ, p α pα pα+ > ξ} Lemma 8 Let, y > 0 For any coprime positive integers a, with 2 and ξ = /, we have y<n+y n H K ϕ log/ log log/ + K! Proof We split the set H into H = {n : p α pα 2 pα + ξ, p α pα p2 > ξ} =: H H 2, {n : p α pα p2 ξ, p α pα pα+ > ξ} say For each n H, write n = rw where r = p α pα, w = n r Note w >, and r = if = Furthermore, for n amod, we have r, = since a, = If n H, then the smallest prime factor in w is p, which is bigger than ξ/r Hence y<n+y n H r K r yr <w+yr w ra mod w,p r = rϕ log, r by applying the sieve bound in Lemma 5

13 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 3 The summation r is eactly that in P / Thus, y<n+y n H K ϕ r K ϕ log/ r log K = r ϕ P / log log/ + K! by Lemma 7 Before we move on, we tae note of the following simple fact: if y < n < n + y and n a n mod, then n n is divisible by as well as gcd n, n Furthermore since n, = = n,, we can conclude that n n is a positive multiple of n, n In particular, 2 n, n / Now suppose n = rw and n = rw are two integers in H 2 with the same r Then by 2, we have 3 w, w r If both w and w have the same smallest prime factor p, and p α w, pα w say, with α α, then it follows from 3 that pα At the same time, we r also have the condition on n in H 2 that r p α+ > This means that α is uniquely determined by the inequalities 4 p α r, pα+ > r We can now conclude that, for all n H 2, n = rw with a given r, either i w = p β for one value of β, which is determined uniquely by p ; or ii w = p β, where β taes eactly two values, one of which is α in 4 and another one larger than this Both of these two values are determined uniquely by p

14 4 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG We now return to the sum Those n in case i contribute Σ r p y<n+y n H 2 yrp α <m+yrp α m rp α a mod Note that α in the inner summation is uniquely determined by r and p The innermost sum is clearly rp α + Hence Σ r Q rp α p + r pξ/r + r, p in view of the conditions in H 2 number theorem, we have 5 Σ and the definition of Q Hence, by the prime Q Q + 2 r log r r + 2 P Klog log/ + K log/! The contribution of those n in case ii above is Σ2 r r 4 r p p p yrp α <m+yrp α rp α m rp α a mod rp α K Q, rp α + α α yrp <m+yrp α m rp a mod

15 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 5 as in the case of Σ The case that w has ust one prime power can be verified separately by using Lemma 6 Combining this with 5, we prove Lemma 8 3 Proof of the Theorem Again, when > /2, the result is trivial Therefore we may assume /2 Let s be given and set ξ = / > 2 easy to see that if n = p α pα2 2 pα p α pα2 2 pα pα+ ξ; and hence n ξ Thus y<n+y ωns By the definition of H ξ, it is does not belong to H H 2 H, then y<n+y n H Hs + nξ ωns Applying Lemma 8 to the first sum and Hardy-Ramanuan s inequality 3 to the second sum, we get the bound K s log log/ + K l ϕ log/ l! for the above right hand side This proves 6 in our Theorem For s log log/, we let λ = s/l where L = log log/ + K If λ <, then the summation at the right hand side of 6 is s l! log log/ + Kl = log log/ + Ks s! s log log/ + Ks s s! log log/ + Ks s! s s 2 l + log log/ + K s l λ s l λ l = λ log log/ + K s s! and if λ, then the summation at the right hand side of 6 is s l! log log/ + Kl = = log log/ + Ks s! log log/ + Ks s! s s s s 2 l + log log/ + K s l 2 s l L L L log log/ + Ks s e L e 2 L e s l L s!

16 6 TSZ HO CHAN, STEPHEN KWOK-KWONG CHOI, AND KAI MAN TSANG The last summation over l is s = e ll+/2l 0l L L + + L e t2 /2L dt L + L e l 2 /2L L<ls e t2 dt L This completes the proof of our Theorem Acnowledgment The authors wish to epress their gratitude to the referee for the thoughtful and gracious comments and suggestions References [] H Davenport, Multiplicative Number Theory, 3rd Edition, Graduate Tets in Math 74 Berlin-Heidelberg-New Yor 2000 [2] H Halberstam and HE Richert, Sieve Method, Academic Press 974, London [3] G H Hardy and S Ramanuan, The normal number of prime factors of a number n, Quarterly J Math 48 97, [4] A Hildebrand and G Tenenbaum, On the number of prime factors of an integer, Due Math J 56, No 3, 988, [5] E Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol Leipzig, 909 [6] H L Montgomery and R C Vaughan, Multiplicative Number Theory I Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press 2007, New Yor [7] M Orazov, Analogue of the Brun-Titchmarsh inequality Russian Izv Aad Nau Turmen SSR Ser Fiz-Tehn Khim Geol Nau 982, no 2, 90 9 [8] L G Sathe, On a problem of Hardy and Ramanuan on the distribution of integers having a given number of prime factors, J Indian Math Soc 7 953, 63-4; 8 954, 27-8 [9] A Selberg, Note on a paper by L G Sathe, J Indian Math Soc 8 954, [0] C A Spiro, Etensions of some formulae of A Selberg, Internat J Math Math Sci 8 985, no 2, [] C Tudesq, Étude de la loi locale de ωn dans de petits intervalles French English summary [Study of the local law of ωn in small intervals] Ramanuan J , no 3, [2] R Warlimont and D Wole, Über quadratfreie Zahlen mit vorgeschriebener Primteileranzahl German Math Z , no, 79 82

17 AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM 7 Department of Mathematical Sciences, University of Memphis, Memphis, TN 3852, USA address: tchan@memphisedu S6 Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A address: choi@mathsfuca Department of Mathematics, The University of Hong Kong, Hong Kong, SAR, China address: mtsang@mathshuh