Integers without large prime factors in short intervals: Conditional results

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1 Proc. Indian Acad. Sci. Math. Sci. Vol. 20, No. 5, November 200, pp Indian Academy of Sciences Integers without large prime factors in short intervals: Conditional results GOUTAM PAL and SATADAL GANGULY, 5/, Nandaram Sen, First Lane, Kolkata , India School of Mathematics, Tata Institute of Fundamental Research,, Homi Bhabha Road, Mumbai , India Corresponding author. MS received 5 June 200; revised 3 July 200 Abstract. Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ/2 + it is bounded by log t 2 +o, we show that for any given α>0the interval, + log /2+o ] contains an integer having no prime factor exceeding α for all sufficiently large. Keywords. Smooth numbers; Riemann zeta function.. Introduction Suppose P n denotes the largest prime factor of an integer n> and let us declare P =. Given a positive real number y, an integer n is called y-smooth if P n y. Smooth numbers are important in many branches of Number Theory as well as in Cryptography. We refer the reader to the articles by Granville [Gra00], Hildebrand and Tenenbaum [HT93] and Pomerance [Po94] for highly readable and informative discussions on these topics. This article is about distribution of smooth numbers in short intervals, namely intervals of type, + ]. See the next subsection for basic facts about distribution of smooth numbers. One expects that smooth numbers are uniformly distributed among intervals of moderate size. This means the following: Consider the function ψx,y = { n x: P n y} which counts the number of y-smooth numbers up to x. Then it is believed that the following asymptotic formula holds for wide ranges of the variables x and z, ψx + z, y ψx,y z x ψx,y. However, this is known to be true only under the restriction that x/zis very small compared to x see Hildebrand Tenenbaum [HT93] and Friedlander Granville [FG93]. The ranges of y and z in which such an asymptotic formula holds for almost all n x are also investigated in the two works cited above. A challenging problem in this subject see [Gra00] or [FG93] is to prove that ψx + x β,x α ψx,x α x β 55

2 56 Goutam Pal and Satadal Ganguly holds for all 0 <α,β<. In a fundamental work, Balog [Ba87] proved this for all β>/2 and α>0 and his method was refined by Harman [Har9] who obtained the same result but with a much better smoothness condition, namely with y = explog x 2/3+o in place of x α. So far no one has been able to prove even for β = /2 with α>0 arbitrary. However, breaking this /2 -barrier is crucial for application to Lenstra s elliptic curve factorization algorithm, though the smoothness required is even stronger. This algorithm finds a prime factor p of a large integer N in expected time if there are many exp log p log log p-smooth numbers in the interval p p, p + p. The expected time here is Oexp 2 + o log p log log p. See [Gra00], 2f for Lenstra s algorithm. Even the Riemann hypothesis does not solve the problem even though the zeros of the Riemann zeta function are intimately connected with distribution of smooth numbers as we shall see see the end of 2. for a striking manifestation of this connection. Indeed, the strongest result in literature in this direction under the Riemann hypothesis denoted RH henceforth, due to uan [u99], says ψx + xlog x +o,x α ψx,x α >0 for all α>0. We have improved this to ψx + xlog x 2 +o,x α ψx,x α >0 under RH and a further hypothesis that St log t /2+ε which is widely believed to be true. See below for the definition of St and the reasons for believing this conjecture. Under RH alone, we have improved uan s result also, albeit by a very small amount see Theorem 2. Here and henceforth ε denotes fixed but arbitrarily small positive number whether or not it is explicitly mentioned. Let T > 0 and let NT denote the number of zeros of the Riemann ζ function in the region 0 <σ <, 0 <t T.IfT is not the ordinate of a zero of ζ, let ST denote the value of π arg ζ/2+it obtained by continuous variation along the straight line joining 2, 2 + it,/2 + it, starting with the value 0. If T is the ordinate of a zero of ζ, let ST = ST + 0. Let LT = 2π + log 2π T log T T + 7 2π 8. 2 Then an application of the Stirling asymptotic formula for the gamma function yields see Chapter 5 of [Da00] the asymptotic formula NT = LT + ST + O 3 T as T. Surprisingly little is known about the function St. The best unconditional bound is St = Olog t op. cit. and it has not been improved upon for more than hundred years. Under RH, one can show that StOlog t/log log t. Montgomery has shown that see Theorem 2 of [Mo77] under RH, St = ± log t/log log t 2 4

3 Integers without large prime factors in short intervals 57 and further he conjectures that St log t/log log t 2. 5 Farmer, Gonek and Hughes [FGH07] have given arguments from random matrix theory that suggests lim sup t St log t log log t = π 2. Here we assume RH and a bound on Stwhich is weaker than either of the two conjectures mentioned above, namely St log t /2+ε, 6 for ε arbitrarily small but positive. Our main result is Theorem. Under RH and the conjectural bound 6, we have for any given α > 0 and ε>0, a positive number 0 = 0 ε, α, such that whenever > 0, and Y log /2+ε, the interval, + Y ] contains an integer having no prime factor exceeding α. We proceed along the lines of uan [u99]. However, the conjecture on St allows us to obtain a good bound on the growth of logarithmic derivative of ζs on a vertical line sufficiently close to the critical line and this results in a better error term. We also choose M and M 2 a little differently which gives us a little extra saving. If we assume only RH, then in the proof of Lemma 5, we can use the bound St = Olog t/log log t and that will lead to the following result which gives a minute improvement over uan s result. Theorem 2. Under RH, we have for any given α>0 and ε>0, a positive number 0 = 0 ε, α, such that whenever > 0, and Y log log log +ε, the interval, + Y ] contains an integer having no prime factor exceeding α. We give proof only of the first theorem since the proof of the second will be identical except that the bound for St will be different. Remark. Recently Soundararajan [So0] has improved the result substantially on RH alone. He proves, on RH, that there are α -smooth numbers in intervals of length cα. Remark 2. Our proof shows that the number of α -smooth numbers in the intervals in question is actually /2 o. Notations and conventions. ε will denote positive real numbers which can be arbitrarily small and it need not be the same in different occurrences. s will denote a complex variable and its real and imaginary part will be denoted by σ and t respectively.

4 58 Goutam Pal and Satadal Ganguly 2. Preliminary steps 2. Distribution of smooth numbers It is of interest to know how many y-smooth numbers are there between and. Let x,y denote the number of y-smooth positive integers x. Dickman [Di30] was the first to prove an asymptotic formula of the kind x,y ρux as x with u = log x fixed. 7 log y The function ρ is monotonically decreasing, continuous and satisfies the following differential difference equation: uρ u = ρu u >, 8 with the initial condition ρu = 0 u. 9 This function is known as the Dickman function or the Dickman de Bruijn function. Note that it is constant for all sufficiently large x if y is a constant power of x. de Bruijn [Br5] showed that { } logu + x,y = xρu + O 0 log y holds uniformly in the range y 2, u log y 3/5 o ; that is, for y>explog x 5/8+o. In 986, Hildebrand [Hil86] improved the range to y 2, u exp{log y 3/5 o }; that is, for y>explog log x 5/3+o. 2 It is natural to ask in what range we can expect this asymptotic to be valid. Hildebrand [Hil84] showed that the above asymptotic formula holds uniformly for u y /2 o ; that is, for y log x 2+o, 3 if and only if the Riemann hypothesis is true. 2.2 A mean value result for Dirichlet polynomials We record the following well-known bound see, for example, Chap. 6 of [Mo7] or Theorem 9. of [IK04] on the mean value of Dirichlet polynomials which will be required later. Theorem 3. For any sequence {b n } of complex numbers and any positive real number R, we have R 2 b 0 n n it dt R + N b n 2. n N n N

5 Integers without large prime factors in short intervals The technique of counting smooth numbers The basic technique of counting smooth numbers used here, which goes back to Balog [Ba87], is the following. Let α be any fixed positive real number. Define a sequence {a m } where {, if p m p α, a m = 0, otherwise. Let M = 2 2 α 2 and M 2 = f where f<log 2 +ε. Define a Dirichlet polynomial Ms = a m m M m M s, 2 and define, for any positive integer n, A n ={m,m 2 : M <m,m 2 M 2,m m 2 n} and d n = a m a m2 r, n=m m 2 r, m,m 2 A n where is the von Mangoldt function, defined by { log p, if n = p t for some integer t, n = 0, otherwise, and the associated Dirichlet series can be written as n n n= s = ζ s ζs. If we can show that d n > 0, <n +Y with Y<, then there must be some integer n = m m 2 r between and + Y, with m and m 2 smooth and therefore n itself is smooth, because, r = n/m m 2 + Y/ M 2 = α. ζ s ζs 4. A bound on ζ s We shall have an occasion to use a bound of ζ s ζs inside the critical strip. So in this section we obtain a conditional bound assuming the conjecture 6. Theorem 4. Under RH and the assumption St log t /2+ε, ζ s ζs log t ε uniformly in 2 + log t σ = Re s σ <. To prove the above theorem we need the following lemma.

6 520 Goutam Pal and Satadal Ganguly Lemma 5. If St log t 2 +ε then we have N for every ε>0. T + NT = Olog T /2+ε, 4 log T Proof. Let ε>0 be fixed and let T>0. From 3 we have N T + log T = 2π NT { T + log T + log 2π 2π 2π T log + log T /2+ε, as T. The lemma follows. log T + log T log T + S T log T T + log T + log T /2+ε Now we shall prove the above theorem. Proof. Let R = [ log t ] log log t and 2 + log t σ σ <. Then by the formula see eq of [Ti86] ζ σ + it ζσ + it = t γ </ log log t + Olog t, s ρ where ρ = 2 + iγ varies over the zeros of ζ,wehave ζ σ + it ζσ + it t<γ<t+ log log t R k=0 R k=0 R k=0 t+ log k k+ t <γ t+ log t t+ log k k+ t <γ t+ log t log t 3 2 +ε + k 2 } T log T ST + Olog t + γ t log t Olog t + γ t log t 2 2 log t + Olog t + k 2 + Olog t, by Lemma 5

7 Integers without large prime factors in short intervals 52 log t 3 2 +ε R u=0 du + Olog t + u 2 log t 3 2 +ε log R log t 3 2 +ε log log t. Hence the theorem follows. 5. The proof Now, for any x [, + Y ], by the Perron formula, d n = 2+iT ζ s 2πi ζs M2 s x + Ys x s ds s x<n x+y + O 2 it T + O 00, where T is some positive real number for the moment, but later we shall choose T 4. We integrate this with respect to x, getting +Y d n dx = 2+iT ζ s 2πi 2 it ζs M2 sasds where x<n x+y + O Y T + OY 00, As = + 2Ys+ 2 + Y s+ + s+. ss + Now, to show that there is a smooth number between and + 2Y, it is enough to show that the left-hand side is positive for all large enough, which is shown in the next section. This integration results in saving one log factor. Our goal now is to show that +Y x<n x+y d n dx>0for all sufficiently large, and Y = f /2, f satisfying the conditions of the theorem. We move the contour to Re s = η = 2 + log +Y x<n x+y, and apply the residue theorem of Cauchy, getting d n dx = Y 2 M 2 + η it + 2πi 2 it 2πi η it + 2+iT Y +O 2πi η+it T since Res s= ζ s ζs =, and A = Y 2. Now, by 0, M = a m m = M m M 2 M2 M2 M t d a m m t M t η+it + O Y00 5 ρ/αdt log. 6

8 522 Goutam Pal and Satadal Ganguly So the first term, Y 2 M 2 Y 2 log 2, and we shall show that this term dominates all other terms. We have the bound + Y s s } min {Y σ, σ, s t where s = σ + it is as usual. This implies, As min {Y 2 σ σ + }, t 2. 7 The horizontal integrals have T in the denominator and will be shown to be very small by trivial estimation. Namely, using the second bound for As, and the bound ζ σ +it ζσ+it log T 2, which we can ensure by choosing T suitably, avoiding the zeros of ζs see Chap. 7 of [Da00], 2πi η it 2 it η 2 ζ s ζs M2 sasds ζ σ + it ζσ + it M2 σ + it Aσ + it dσ 3+ 2 log T 2 T 2 4, by choosing T 4. And similarly we get the same bound for the other integral 2+iT η+it. Now for estimating the vertical integral from η it to η + it, we break up the interval [0,T] into [0,/Y] and [/Y, T ]. I = 2π /Y 0 ζ η + it ζη + it M2 η + itaη + itdt Y 2 η /Y + M 2 M 2η 2η log ε by Theorem 3, Theorem 4 and the bound 7. Hence, I Y log 2 /Y α 2 2η 3 log log ε Y 2 log ε. f Recall that we have taken M 2 = /f and Y = f /2. The second integral is estimated by integration by parts, and we get I 2 = T ζ η + it 2π ζη + it /Y +η log ε 2 +η log ε 2 Y 2 log ε 2 f. T /Y M 2 η + itaη + itdt M 2 η + it t 2 dt /Y 2 /Y + M 2M 2η 2η

9 Integers without large prime factors in short intervals 523 by again using the same bounds. Finally, +Y d n dx = Y 2 M 2 + O Y 2 log ε + O 4 f x<n x+y Y + O + OY 00. T Since M 2 log 2 by 6 and T 4,ifwetakef>log 2 +ε then we can conclude that +Y d n dx >0 x<n x+y for all sufficiently large, which is what we wanted to prove. Acknowledgements This article grew out of the first-named author s Ph.D. thesis. He wishes to thank his advisor Prof. G Misra for constant encouragement and Prof. R Balasubramanian for suggesting the problem and also for many enlightening conversations and encouragements. He thanks Dr A Mukhopadhyay for many useful discussions. He also wishes to thank Indian Statistical Institute, Banglore and Calcutta centres and the Institute of Mathematical Sciences, Chennai where parts of the work for his thesis were carried out. The second-named author thanks Prof. R Balasubramanian and Dr A Mukhopadhyay for many interesting discussions he had regarding the problem considered here. He thanks the Institute of Mathematical Sciences, Chennai for warm hospitality and great working conditions and the Tata Institute of Fundamental Research, Mumbai for providing excellent environment where part of the work was done. References [Ba87] [Br5] [Da00] [Di30] [FG93] [FGH07] [Gra00] Balog A, On the distribution of integers having no large prime factors, Astérisque de Bruijn N G, On the number of positive integers x and free of prime factors >y, Nederl. Akad. Wetensch. Proc. A Davenport H, Multiplicative number theory, Third edition, Revised and with a preface by Hugh L Montgomery, Graduate Texts in Mathematics 74 New York: Springer-Verlag 2000 Dickman K, On frequency of numbers containing prime factors of a certain relative magnitude, Ark. Mat. Astr. Fys Friedlander J B and Granville A, Smoothing smooth numbers, Philos. Trans. Soc. London A Farmer D W, Gonek S M and Hughes C P, The maximum size of L-functions, J. Reine Angew. Math Granville A, Smooth Numbers: Computational Number Theory and Beyond, Proc. MSRI Conf. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography eds J Buhler and P Stevenhagen Berkeley: Cambridge University Press 2000 pp. 56

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