ON THE RESIDUE CLASSES OF (n) MODULO t

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1 #A79 INTEGERS 3 (03) ON THE RESIDUE CLASSES OF (n) MODULO t Ping Ngai Chung Deartment of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts briancn@mit.edu Shiyu Li Det of Mathematics, University of California - Berkeley, Berkeley, California jjl357@berkeley.edu Received: 8//3, Acceted: //3, Published: /3/3 Abstract The rime number theorem is one of the most fundamental theorems of analytic number theory, stating that the rime counting function, (), is asymtotic to / log. However, it says little about the arity of (n) as an arithmetic function. Using Selberg s sieve, we give, for any fied integers r and t, a ositive lower bound on the roortion of ositive integers n such that (n) is congruent to r modulo t. Moreover, we generalize this to the counting function of any set of rimes with ositive density.. Introduction and Statement of Results The rime counting function () has been a toic of interest for mathematicians throughout history. One of the first main results about this function is the rime number theorem, which states that () log. This asymtotic carries dee imlications for the distribution of rime numbers. However, there are many roblems about rimes which remain unsolved in site of this theorem. Two eamles are the Goldbach conjecture, stating that every even number greater than is the sum of two rimes, and the twin rime conjecture, which claims that there are infinitely many airs of rimes that di er by. In sectacular fashion, there has been a sate of recent advances concerning the gas between rime numbers. In 005, Goldston, Pintz and Yıldırım [4] roved that there eist infinitely many consecutive rimes which are much closer than average. Recent work by Zhang [] and the olymath roject [0] shows that there are The authors are grateful to the NSF s suort of the REU at Emory University.

2 INTEGERS: 3 (03) infinitely many airs of rimes di ering by at most an absolute constant. These advances deend critically on sifting techniques, articularly Selberg s sieve. In this aer, we rovide another alication of Selberg s sieve, one which does not seem to have been considered reviously in the literature. We treat (n) as an arithmetic function, and consider its distribution mod t as n varies. The rime number theorem alone says little about the arity of (n) or the roortion of ositive integers n for which (n) is even. Nevertheless, numerical evidence shows that this seems to be the case about half the time: #{n < (n) is even}/ Define T r,t () to be the number of ositive integers n < such that (n) r mod t. In this aer, we rove the following. Theorem.. For any 0 ale r < t, we have that lim inf! T r,t () 6t. If we assume the Hardy-Littlewood rime k-tule conjecture, we can relace 6 by above. Remark. Recall that the Hardy-Littlewood rime k-tule conjecture states that given a = (,..., k ), where,..., k are distinct integers which do not cover all residue classes to any rime modulus, the number (; a) of ositive integers m < for which m,..., m k are all rime satisfies (; a) B(log ) k, where B is a certain nonzero infinite roduct over rimes. With this estimate, it is ossible to imrove Theorem.3 by a factor of k k!. In our alication, k =, elaining the imrovement of the lower bound by a factor of 8. Remark. Theorem. was obtained indeendently and almost simultaneously by Alboiu []. Theorem. is a secial case of a more general theorem. We say that a set S of rimes has density if #{ < : S}! as!. #{ < : rime} Define T r,t (; S) to be the number of ositive integers n < such that S(n) r mod t, where S(n) denotes the number of rimes in S less than n. Theorem.. If we assume the notation above and that S has ositive density, then T r,t (; S) lim inf! 6t. If we assume the Hardy-Littlewood rime k-tule conjecture, we can relace 6 by above.

3 INTEGERS: 3 (03) 3 Eamles. Two eamles of sets of rimes with ositive density follow.. Let K/Q be a finite Galois etension with Galois grou G, and let C be a conjugacy class in G. Then the set of unramified rimes for which the Frobenius automorhism is in C has density #C/#G by the Chebotarëv density theorem.. Given a ositive irrational >, the set of rimes of the form b nc for some integer n has density by [8, Thm. ]. In articular, we may consider T r,t (; q, a) to be the number of ositive integers n < such that (n; q, a) r mod t, where (; q, a) is the counting function of the rimes congruent to a mod q. In this case, we have the following stronger result. Theorem.3. For any 0 ale r < t, 0 ale a < q, where gcd(a, q) =, we have that where lim inf! (q) is Euler s totient function. T r,t (; q, a) 6 (q)t, Remark. Results similar to those in Theorems.,., and.3 can be obtained by making use of Brun s sieve. However, the constants will be weaker. To rove our theorem, we will rely on a judicious alication of Selberg s sieve. In articular, using either Selberg s sieve or the rime k-tule conjecture, we bound the number of rimes which have a articular small ga, thus also bounding the number of rimes which have any small ga. This lets us say that not all gas occurring at rimes nt+r can be small, and thus T r,t () cannot be too small.. Selberg s Sieve We now introduce the main tool, Selberg s sieve. In this, as well as many other sieve methods (see [], [7, Ch. 6]), we are given a finite set A and a set of rimes P, and we wish to estimate the size of the set S(A, P) := {a A (a, ) = for all P}. To do this, we use the inclusion-eclusion rincile. For each squarefree integer d, let A d be the set of elements in A divisible by d. Then S(A, P) = d P µ(d) A d, where P is the roduct of all the rimes in P. Sieve methods are most useful when the elements of A d distribute aroimately evenly in A, in which case one

4 INTEGERS: 3 (03) 4 would eect A d = g(d) + R d, where is aroimately the size of A, g(d) is a multilicative function that is the density of A d in A, so that 0 < g(d) <, and R d is a small error term. Tyically, a sieve method gives an estimate for S(A, P) in the following form S(A, P) = Y P( g()) + error term. The main term is what one would eect if sifting by the rimes in P are airwise indeendent events, and the error term varies from di erent sieve methods. However, when the rimes in P are fairly large, this indeendence condition fails. Hence, most sieves can only obtain an uer bound for the sifted quantity in question. The key idea of Selberg s sieve is to relace the Möbius function in the formula of S(A, P) by an otimally chosen quadratic form so that the resulting estimates are minimal. Formally, it is stated in the following theorem. Theorem.. (Selberg s sieve) [7, Thm. 6.5] With notation as above, let h(d) be the multilicative function given by h() = g()( g()) and set H(D) = h(d) for any D >. If we assume that then yaleale d< D,d P R d ale g(d)d, () g(d)d if d P, () g() log log(/y) for all ale y ale, (3) S(A, P) ale H(D) + O D log D With further restrictions on g(d), one may obtain an estimate for H(D) which is frequently useful in ractice. Lemma.. [6, Lemmas 5.3, 5.4] Suose the conditions in Theorem. hold. Moreover, suose there eist ositive real numbers ale, A, A, L such that 0 ale g() ale ; (4) A L ale g() log ale log z w ale A for all ale w ale z. (5) walealez If P is the roduct of all rimes < D, then H(D) = C(log D) ale + O. L log D,

5 INTEGERS: 3 (03) 5 where C = (ale + ) Y ( g()) ale. The imlied constant deends only on A, A, and ale. In articular, it is indeendent of L. Remark. This is a combination of Lemmas 5.3 and 5.4 of Halberstam and Richert s book [6]. Note that eactly this formulation aears in [5]. In our work, we will find the following theorem, which is roved as a direct alication of Selberg s sieve, to be useful. Theorem.3. [7, Thm. 6.7] Let a = (,..., k ) be distinct integers which do not cover all residue classes to any rime modulus and i ale c log for some absolute constant c for all i. Then the number (; a) of ositive integers m ale for which m,..., m k are all rime satisfies log log (; a) ale k k!b(log ) k + O, log where B = Y () k and () is the number of roots of the olynomial f(m) = (m ) (m k ) modulo. The imlied constant deends only on k and c. Proof. We aly Selberg s sieve (Theorem.). Take A to be the set of olynomial values f(m) = (m ) (m k ) for ositive integers m ale. Take g() = () and =. Note that () ale k for all rimes and () = k for su ciently large ; thus, conditions () - (5) hold for ale = k, A = k +, A = 5k, L = (k ) log(ma i ) = O k,c (log log ). By Lemma., we have that H(D) = k! Y k k () log D L + O k log D L = k k!b(log D) k + O k, log D hence, by Theorem. with D = log (k ), we have (; a) ale S(A, P) + ma{l, log log } D ale k k!b(log ) k + O k. log The result follows.

6 INTEGERS: 3 (03) 6 Remark. The rime k-tule conjecture redicts that (; a) B(log ) k. Remark. For our urose, it is imortant that the imlied constant deends only on k and c but not the individual values of i since we will take i to infinity as! in the roof of Lemma Proof of Theorem. We are now ready to begin the roof of the main theorem. For each ositive integer k, let S k be the number of ositive integers n such that S(n) = k. We label the elements in S by q, q.... in increasing order, and we note that S k = q k+ q k. In other words, S k is just the ga between q k and q k+. Observe that for each rime q m, T r,t (q m ; S) = S k. kalem k r(mod t) By a simle alication of the rime number theorem, we see that, for all A >, T r,t (, S) = (q k+ q k ) + O t log A, (6) q k <, k r(mod t) where O t indicates that the imlied constant may deend on t. The roof of Theorem. relies on the following key lemma, which gives an uer bound on the number of small gas. Lemma 3.. Let f() tend to infinity as! and f() = O(log ). Then the number of airs of rimes ( i, j ) with i, j < such that is at most 0 < i j < f() 8 log f()( + o()). To rove this lemma, we start with an auiliary estimate, which follows from Theorem.3. Lemma 3.. Let a be a ositive integer such that a ale c log for some ositive constant c. Then the number of rime solutions i, j < to i j = a is at most 8 log log log B(a) + O log

7 INTEGERS: 3 (03) 7 where B(a) := Y a Y -a and the imlied constant deends only on c. Assuming the rime k-tule conjecture, the uer bound can be imroved by a factor of 8. Proof. In Theorem.3, take k =, a = (0, () = if - a. Then the result follows. a). Note that () = if a and We also need the following estimate. Lemma 3.3. Assuming the notation in Lemma 3., we have that B(a) = f()( + o()). aalef(), a Proof. where aalef(), a B(a) = Y aalef(), a a = Y 3 (a) := Y a, 3 Y -a aalef(), a +. (a), Note that (a) is a multilicative function, so L(s, ) = n= = Y = Y = (n) n s + () + () s s + ( ) Y s +... s s +... ( s )( ).

8 INTEGERS: 3 (03) 8 Observe that L(s, ) = (s)a(s), where A(s) = Y + s ( ) 3 is absolutely convergent in Re(s) > 0. theorem [9, Cor. 8.8], aalef(), a (a) = aalef()/ = Y 3 Hence by the Wiener-Ikehara Tauberian (a) = A() f() ( + o()) f() + ( + o()). ( ) Therefore aalef(), a B(a) = Y 3 " = f()( + o()). # f() + ( + o()) ( ) Using Lemma 3. and 3.3, we can give an uer bound on the number of airs of rimes with rime ga in a given range. Proof of Lemma 3.. By Lemmas 3. and 3.3, the number of such airs ( i, j ) is bounded by 8 log log log B(a) + O log aalef() = 8 log log log + O B(a) + O(f()) log = 8 log f()( + o()). aalef(), a If we take f() = log with small, then we can give an uer bound on the number of consecutive rimes with an unusually small rime ga. From this regularity result, we will now deduce the main theorem.

9 INTEGERS: 3 (03) 9 Proof of Theorem.. Recall from (6) that T r,t (; S) = q k <, k r(mod t) (q k+ q k ) + O t log A Given a function f() which tends to as! and f() = O(log ), by Lemma 3., we know that #{ n < n+ n < f()} ale 8 log f()( + o()). (7). Note that #{q n < n r(mod t)} = () ( + o()) t ( + o()) t log by the rime number theorem and the estimate in [, Cor., P.69]. Hence 8 #{q n < q n+ q n f() and n r(mod t)} t log log f() (+o()) In articular, for f() ale T r,t (; S) = = = = q k <, k r(mod t) z= y=z log /(8t) z= t log Therefore, we have and so log /(8t), the main term is nonnegative. Hence (q k+ q k ) + O t log A = q k <, k r(mod t), q k+ q k =y t log log 8t ( + o()). 6t + O t log A y= q k <, k r(mod t), q k+ q k =y log /(8t) z= y z= q k <, k r(mod t), q k+ q k z 8 log z ( + o()) + O t log A! 8 log log ( + o()) 8t T r,t (; S) lim inf! T r,t (; S) ( + o()), 6t 6t. + O t log A + O t log A

10 INTEGERS: 3 (03) 0 If we assume the rime k-tule conjecture, the uer bound in Lemma 3. can be imroved by a factor of 8. Proceeding analogously in our above roof, we obtain lim inf! T r,t (; S) t. Proof of Theorem.3. Theorem.3 follows mutatis mutandis by considering only rime gas divisible by q and rimes congruent to a mod q in Lemma 3., in which case we obtain an etra factor of (q) in the denominator of the uer bound in (7). We also recall that (; q, a) ()/ (q) by Dirichlet s theorem on arithmetic rogressions. Acknowledgements The authors would like to thank Ken Ono for running the Emory REU in Number Theory and making valuable suggestions. We would also like to thank Robert Lemke Oliver for his guidance in both research and writing. References [] M. Alboiu, On the arity of the rime counting function and related roblems. Prerint. [] A. Cojocaru and M. Murty, An introduction to sieve methods and their alications. London Math. Soc. Stud. Tets, 006. [3] S. R. Finch, Mathematical Constants. Encycloedia Math. Al., vol. 94, Cambridge Univ. Press, 003. [4] D. Goldston, J. Pintz and C. Yıldırım, Primes in tules. I. Ann. of Math. () 70 (009), no., [5] D. Goldston, S. Graham, J. Pintz and C. Yıldırım, Small gas between roducts of two rimes. Proc. Lond. Math. Soc. (3) 98 (009), no. 3, [6] H. Halberstam and H.-E. Richert, Sieve methods. London Math. Soc. Monogr. Ser., 974. [7] H. Iwaniec and E. Kowalski, Analytic number theory. Amer. Math. Soc. Colloq. Publ., 004. [8] D. Leitmann and D. Wolke, Primzahlen der Gestalt [f(n)]. (German) Math. Z. 45 (975), no., 8-9. [9] H. Montgomery, R. Vaughan, Multilicative Number Theory I. Classical Theory. Cambridge Stud. Adv. Math., 007. [0] Polymath, Imroving the bounds for small gas between rimes. htt://michaelnielsen.org/olymath/inde.h?title=bounded gas between rimes. [] J. B. Rosser and L. Schoenfeld, Aroimate formulas for some functions of rime numbers. Illinois J. Math. Volume 6, Issue (96), [] Y. Zhang, Bounded gas between rimes. Ann. of Math., to aear.