TAIL OF A MOEBIUS SUM WITH COPRIMALITY CONDITIONS

Transcription

1 #A4 INTEGERS 8 (208) TAIL OF A MOEBIUS SUM WITH COPRIMALITY CONDITIONS Akhilesh P. IV Cross Road, CIT Camus,Taramani, Chennai, Tamil Nadu, India akhilesh.clt@gmail.com O. Ramaré 2 CNRS / Institut de Mathématiques de Marseille Aix Marseille Université, Camus de Luminy, Marseille, France olivier.ramare@univ-amu.fr Received: 2/4/7, Acceted: 2/20/7, Published: /6/8 Abstract We show that P (k,q)= µ/k2 = o(/) uniformly in q. A more recise bound is given, as well as an extension to similar sums. The recise rate of decay is however, unknown.. Introduction and Results Averages of multilicative functions have been a major subject of research for many years. The case of non-negative functions has been articularly well studied. When using convolution identities, corimality conditions naturally enter the scene, where the modulus to which the variable should be corime, say q, can belong to a very wide range. This means that one should use estimates that are uniform with resect to q, or at least where the deendence on this q is exlicitly described. The case of a non-negative function is easier; see for instance [2] or [6, Lemma 2.3]. The more di cult case of oscillating multilicative functions is the one that we address here. The examle P nalex,(n,q )= µ(n)/n has already been tackled by H. Davenort in [2, Lemma ], more exlicitely by A. Granville and the second author in [3, Lemma 0.2], and exanded by T. Tao in []. The more surrising but very interesting examle P nalex,(n,q )= µ(n)2!(n) /n has been investigated in [0, Lemma 4]. Note also that, when aiming at refined estimates for non-negative multilicative functions, we may have to handle averages of oscillating multilicative functions The first author has been artly suorted by the Marie Curie IRSES rogram Moduli, roject No The second author has been artly suorted by the Indo-French Centre for the Promotion of Advanced Research CEFIPRA, roject No 540-.

2 INTEGERS: 8 (208) 2 twisted by corimality conditions as is the case in [5]: though we were studying P dale µ2 (d)/'(d), we had to rove the following theorem. Theorem. For any real number (k,q )= 4 and any modulus q, we have µ 'k ale From another angle, one may simly say that with the case of non-negative multilicative functions being under control, the next ste is to remove this nonnegativity condition. The roof of Theorem uses some adhoc maniulations and does not disclose whether the sum under examination, when multilied by, goes to 0. Our first corollary is the following. Corollary. We have lim su max! q (k,q )= µ 'k = 0. The leading idea, which we borrow from [8], is to comare the function µ/('k) to a comletely multilicative one, namely /k 2 where we denote by (n) the Liouville function. Here (n) = ( ) (n) is the Liouville function, (n) denoting the total number of rime factors of the ositive integer n, counted with multilicity. We now resent our main theorem. Theorem 2. Let > and G be two real numbers. There exist two constants C = C(, G) and c 0 (, G) > 0 with the following roerty. For any arithmetic function g such that P d g(d) /d ale G and for all D, g(d) /d 2 ale G/D, for any modulus q and any (k,q )= (k,q )= d D 3, we have (? g) k 2 ale C ex c 0 log log /. Corollary is an easy consequence of this result, with = 2. We could similarly handle the summand k 7! µ/k 2. It is interesting to know how far one can go in the quality of the decreasing rate. The essential case is when? g is simly. It is not di cult to modify the roof of the above result to infer that, under the Riemann Hyothesis, one has, for any " > 0, k 2 ", (assuming the Riemann Hyothesis). () (log ) 3 "

3 INTEGERS: 8 (208) 3 On the other side, by considering the modulus q = Q ale, we readily check that lim su! max q (k,q )= k 2 (log ). (2) The surrising fact is that we have not been able to imrove on this estimate. Even simly getting a larger constant on the right-hand side seems di cult. This /( log ) may be the true order of decay, but we do not venture any conjecture. An extremely similar roof also roves the next theorem. Theorem 3. Let > and G be two real numbers. Let be a Dirichlet character. There exist two constants C = C(, G, ) and c 0 (, G, ) > 0 with the following roerty. For any arithmetic function g such that P d g(d) /d ale G and 8D, g(d) /d 2 ale G/D, for any modulus q and any (k,q )= d D 3, we have (? g) k 2 ale C ex c 0 log log /. The reader may wonder whether we are now able to imrove on Theorem, and the answer is no! When following the roof we roduce here, we have to rely on an exlicit version of Lemma 3. Reading [7] and [9], or [], it is not di cult to see that we can only hoe for an uer bound of the shae c/( log ), for a small enough constant, or c 0 /((log ) 2 ) for a rather large constant c 0. Continuing in this manner, we would reach a bound O(/( log log )) in Theorem 2 which is alas numerically of the same strength as O(/) for decent values of. 2. Lemmas Lemma. Let e 6 be a real number. Let q be a modulus all whose rime factors are below. We have kale, (k,q)= ale 7 '(q) q. This is an easy consequence of the Selberg sieve and is stated as above in [4, Theorem 3.5]. The following is the trivial consequence of the above that we shall use.

4 INTEGERS: 8 (208) 4 Lemma 2. Let 0 be a real number. Let q be a modulus. We have 0 kale 0, (k,q)= Y q, ale 02 where the imlied constant is indeendant of q and 0. This also means that we may restrict the rime to any subset of the divisors of q that are below 02. The right-hand side would at most only increase. Proof. The lemma is obvious when 0 is below e 6. Let us then assume that 0 is larger than this bound. We can relace q by q = Y and aly Lemma. Our q, ale 0 next task is to show that one can extend the roduct '(q ) = Y q q, ale to encomass the rimes u to 02. Such an extension may modify the '(q )/q by a factor that is between and Y, <ale 02 and this last quantity is asymtotically bounded below by /2 by Mertens Theorem. We continue with an easy consequence of the Prime Number Theorem. Lemma 3. There exists a ositive constant c with the following roerty. Let be a real number. We have k 2 ex c log(2). k> 3. Proof of Theorem 2, the Case of the Liouville Function The modulus q being fixed, we define q = Y (3) q, ale

5 INTEGERS: 8 (208) 5 as well as = Y q, ale. (4) We next fix the real number. The roof slits in two cases, according to whether is small or large. Here is our first bound. A summation by arts discloses the estimate (k,q )= Z k 2 = 2 <kalet, (k,q )= dt Z t 3 ale 2 kalet, (k,q )= which is O( /), by Lemma 2 (with 0 = t and restricting in this lemma the roduct over to the rimes not more than ). This first bound is e cient when is small enough. Let us roceed to get a second bound. We handle the corimality condition by emloying the Moebius function: (k,q )= k 2 = d q µ(d) = d q d 2 d k> `>/d k 2 (`) `2. Please note that, in the receding lines, we used in an essential manner the comlete multilicativity of the -function. We slit the above sum over d according to whether d is larger or smaller than some bound D < /2. We next emloy Lemma 3 for the summations over ` attached to d ale D, and further notice that daled d ale Y q, aled ale Y This gives rise to the first contribution: q, ale Y q, D<ale dt t 3 ale /. µ2 (d) ex c log(/d) d 2 ex (/d) d q c log(/d). (5) When now d is larger than D, we only use the fact that the artial sums P`>/d are bounded (in fact at least by 2 /6) and are left with finding a bound for d>d d 2. (`) `2

6 INTEGERS: 8 (208) 6 Let m = Y. aled, -q An integer d that divides q is rime to m, and thus d>d d 2 ale d>d, (d,m)= d 2. It is better to disclose at this level our choice for D, namely D = / log. (6) By emloying Lemma 2 again and a summation by arts as above, we show that d>d d 2 Combining (5) together with (7) we reach '(m) md Q aled ( ) Q D aled, q ( ) D log D. (7) (k,q )= k 2 ex c log log + log ex c log log. (8) We can ut our two bounds in a single exression as follows: k 2 min, ex c log log (k,q )= /. The theorem for the c 0 = c/2, i.e., -function is a straightfoward consequence of this bound, with (k,q )= k 2 ex c 0 log log. (9) 4. Proof of Theorem 2, Generic Case We now have (9) at our disosal, and want to aly the deformation g to write (? g) k 2 = g( ) 2 k 2., (k,q )= (,q )= k>/, (k,q )=. We

7 INTEGERS: 8 (208) 7 We searate the sum over according to whether ale + 2 or not. In the first case, we emloy (9) and get, (,q )=, dale + 2 g( ) 2 k>/, (k,q )= In the second case, we simly write, (,q )=, d> + 2 k 2 ale C, g( ) 2 k>/, (k,q )= dale + 2 k 2 g( ) ex c 0 qlog log. ale 2 + G/ 2. 6 The theorem follows readily. References [] Olivier Bordellès, Some exlicit estimates for the Möbius function, J. Integer Seq. 8 (205) Article 5.., 3. [2] H. Davenort, On some infinite series involving arithmetical functions, Quart. J. Math., Oxf. Ser. 8 (937), 8-3. [3] A. Granville and O. Ramaré, Exlicit bounds on exonential sums and the scarcity of squarefree binomial coe cients, Mathematika 43 (996), [4] H. Halberstam and H.-E. Richert, Sieve methods, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 974. [5] Akhilesh P. and O. Ramaré, Exlicit averages of non-negative multilicative functions: going beyond the main term, Colloquium Mathematicum 47 (207), [6] O. Ramaré with the collaboration of D. S. Ramana, Arithmetical asects of the large sieve inequality, volume of Harish-Chandra Research Institute Lecture Notes, Hindustan Book Agency, New Delhi, [7] O. Ramaré, From exlicit estimates for the rimes to exlicit estimates for the Moebius function, Acta Arith. 57 (203), [8] O. Ramaré, Exlicit estimates on the summatory functions of the Moebius function with corimality restrictions, Acta Arith. 65 (204), -0. [9] O. Ramaré, Exlicit estimates on several summatory functions involving the Moebius function, Math. Com. 84 (205), [0] Sigmund Selberg, Über die Summe P µ(n) nalex, 2. Skand. Mat.-ongr., Lund 953, (954), nd(n) [] Terence Tao, A remark on artial sums involving the Möbius function, Bull. Aust. Math. Soc. 8 (200), [2] J.E. van Lint and H.E. Richert, On rimes in arithmetic rogressions, Acta Arith. (965),