ON THE DISTRIBUTION OF THE PARTIAL SUM OF EULER S TOTIENT FUNCTION IN RESIDUE CLASSES

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1 C O L L O Q U I U M M A T H E M A T I C U M VOL. * 0* NO. * ON THE DISTRIBUTION OF THE PARTIAL SUM OF EULER S TOTIENT FUNCTION IN RESIDUE CLASSES BY YOUNESS LAMZOURI, M. TIP PHAOVIBUL and ALEXANDRU ZAHARESCU Urbana, Il Abstract. We investigate the distribution of Φn = + P n i= ϕi which counts the number of Farey fractions of order n in residue classes. While numerical comutations suggest that Φn is equidistributed modulo q if q is odd, and is equidistributed modulo the odd residue classes modulo q when q is even, we rove that the set of integers n such that Φn lies in these residue classes has a ositive lower density when q = 3, 4. We also rovide a simle roof, based on the Selberg Delange method, of a result of T. Dence and C. Pomerance on the distribution of ϕn modulo 3.. Introduction. Let ϕ denote Euler s totient function, which counts the number of ositive integers less than n that are corime to n. Define n Φn := + ϕi. Then Φn is the number of Farey fractions of order n, which also corresonds to the number of lattice oints, y with 0 y n that are visible from the origin. C. Pomerance gave an outline in [8] see eercise 0 age 45 of the roof that there are infinitely many values of Φn in every residue class modulo 3. His idea is to eloit the fact that the Dirichlet series Ls := n= χ 3ϕn/n s has a ole at s =, where χ 3 is the unique non-rincial character modulo 3. This gives a motivation to study the distribution of Φn modulo 3, and more generally one can ask for an asymtotic formula for the number of ositive integers n such that Φn k mod q. In this aer, we investigate this question in the cases q = 3 and q = 4. Note that Φn is odd for all n. We define A k := {n : Φn k mod 3}, B j := {n : Φn j mod 4}. Moreover, if A N, we denote by A the number of ositive integers j with j A. 00 Mathematics Subject Classification: Primary N69; Secondary N64. Key words and hrases: Euler s totient function, distribution in residue classes. [] i=

2 Y. LAMZOURI ET AL. Table Table A 0 A A B B Tables and suggest that Φn is equidistributed among the three residue classes modulo 3, and among the classes and 3 modulo 4. One can also remark that the convergence seems to be very fast, and that the error term tends to be aroimately of the size of the square root of the main term. To further investigate the distribution of Φn in residue classes, we have erformed numerical comutations for all moduli 3 q 00 for u to 0 7 and noticed that a similar henomenon occurs. Indeed these numerical investigations may suggest that for all q 3, Φn is equidistributed among the residue classes in Cq where { {a mod q} if q is odd, Cq := {a mod q : a, = } if q is even. We define E q := ma j Cq {n : Φn j mod q} Cq. Table 3 below contains values of E q for 3 q 0, and u to 0 8. E 3 E 4 E 5 E Table 3 E 7 E 8 E 9 E In 909, E. Landau [5] roved that the number of integers n having all rime divisors in r residue classes modulo q with r <φq is asymtotic to. Cr, q log r/φq as,

3 PARTIAL SUM OF EULER S TOTIENT FUNCTION 3 where Cr, q is a ositive constant. Since the condition q ϕn imlies that n has no rime divisors in the residue class mod q, it follows from Landau s result that ϕn is divisible by q for almost all integers n. This shows that Φn stays constant modulo q for a large roortion of the time, then it changes recisely at those integers n such that q ϕn. If q, 6 =, W. Narkiewicz [7] showed that ϕn is equidistributed among the residue classes relatively rime to q. However, as we shall see later, the distribution of ϕn in residue classes modulo 3 and 4 has a different behavior. Indeed we shall rove that ϕn has more values that are congruent to mod 3 than to mod 3, and 4 ϕn imlies that ϕn mod 4 for all n 3. Eloiting these irregularities and using sieve theory, we show that the sets A k and B j have ositive lower densities for k = 0,, and j =, 3. We should also note that our idea would not work in general, since such irregularities do not eist when the modulus q is corime to 6, by the result of Narkiewicz. Theorem. For j =, 3 we have where δ = 9/896 > /00. lim inf B j δ, Remark. The oor value of δ is not only due to the use of sieve theory, but also to the difficulty of understanding the gas between consecutive rimes. Indeed, for n 5, Lemma. below shows that Φn Φn mod 4 if and only if n = k or n = k, where is a rime congruent to 3 modulo 4. In the case where q = 3 we rove Theorem. For k = 0,, we have for some δ > 0. lim inf A k δ Remark. An elicit comutation allows one to take δ The roof of Theorem relies on understanding the distribution of ϕn in the residue classes and modulo 3. For i =, let N i := {n : ϕn i mod 3}. In [], T. Dence and C. Pomerance roved an asymtotic formula for N i using a combinatorial argument along with Landau s result. and Wirsing s theorem see [0] on mean values of multilicative functions. Using a direct aroach based on the Selberg Delange method we rovide a simler

4 4 Y. LAMZOURI ET AL. roof of the result from []. Moreover, we can also ehibit lower order terms in the asymtotics of N i. We have Theorem 3. Let be large and K be a ositive integer. Then there eist elicit constants λ j, β j for j =,..., K such that K N = λ + λ j log j= log j+/ + O K log K+/, K N = β + β j log log j+/ + O K log K+/, where λ = 3/ 3 3/4 π j= / +, + and β is given by the same eression as for λ, ecet that the factor + + is relaced by +. Moreover λ and β The asymtotic eansion in Theorem 3 can also be obtained along the lines rovided by J. Kaczorowski in [4].. Preliminary lemmas. First we characterize the values of n for which Φn Φn mod q, when q = 3, 4. Lemma.. Let n 5 be a ositive integer. Then Φn Φn mod 4 if and only if n = k or n = k, where is a rime congruent to 3 modulo 4 and k is a ositive integer. Moreover Φn Φn mod 3 if and only if n = 3 a m where a = 0 or and m is divisible only by rimes that are congruent to modulo 3. Proof. Write n = α 0 α α α l l where the i are odd rimes and the α i are non-negative integers. Then ϕn = α 0 l i= i α i i. Hence, if α 0, i mod 4 for some i l, or l then ϕn 0 mod 4. Conversely if n = k or n = k where is a rime congruent to 3 modulo 4 then ϕn = k mod 4. Similarly writing n = 3 a m with 3 m one can see that 3 ϕn if and only if a or m is divisible by a rime mod 3. In order to rove Theorems and we shall need the following alications of Selberg s uer bound sieve. Lemma.. Let a be a fied non-negative integer and d be a ositive integer such that d is even if a = 0, and d is odd if a. Then as

5 PARTIAL SUM OF EULER S TOTIENT FUNCTION 5 we have, uniformly in d, { : 3 mod 4, a + d is rime} 4 + o f d log, > where fd is the multilicative function satisfying f k = if > and f k =, for all ositive integers k. Proof. This is a corollary of Theorem 3. of [3]. Lemma.3. Let be large and d log be a ositive integer. Then as we have {n : nn + d mod 3} + o w f d log, where w = if mod 3 and w = 0 otherwise. Moreover f d is the multilicative function defined by f k = if mod 3 and f k = otherwise, for any ositive integer k. Proof. This follows from Theorem 5. of [3] by taking κ = and L = log log there. Lastly we rove estimates for mean values of the multilicative functions f and f that arise in the sieve bounds of Lemmas. and.3. Lemma.4. Let P be a set of odd rime numbers, and d be a ositive integer divisible only by rimes / P. Let f be the multilicative function defined by f k if P, = if / P, for any ositive integer k. Then for any ɛ > 0 we have fn = d + O ɛ,d ɛ. n d n Proof. First note that P fn =, n P where the roduct equals if n is divisible only by rimes / P. If d n

6 6 Y. LAMZOURI ET AL. then writing n = dm we deduce that fn = fm. Hence we get. fn = fm. n d n m /d Let h be the multilicative function defined by h = f µ, where is the Dirichlet convolution and µ is the Möbius function. Then one can check that h = f = for P and h = 0 otherwise. Moreover for rime and k we have h k = f k f k = 0. Let y = /d. Then we obtain fm = hr = [ ] y. hr r m y m y r m r y = y hr + O hr. r r y r y The error term on the RHS of the above estimate is.3 + h + log y. y y Moreover, for any ɛ > 0, the series hr r ɛ = r= P r>y + ɛ is absolutely convergent. This shows that hr hr r y ɛ r ɛ ɛ y +ɛ, which imlies r y hr r = P r= + O ɛ y +ɛ. Thus, the result follows uon combining this estimate with Proof of Theorem. Let M be the set of ositive integers 5 n such that 4 ϕn. Then write M = {b,..., b m } with 5 b < < b m where m = M, and set b 0 = 5 and b m+ = []. Using Lemma. along with the rime number theorem for arithmetic rogressions we obtain 3. M = π; 4, 3 + π/; 4, 3 + O = 3 4 log + O log.

7 PARTIAL SUM OF EULER S TOTIENT FUNCTION 7 Put r = [ M /] and let L log be a ositive real number to be chosen later. Furthermore, define T d = {0 i r : b i+ b i = d} for all ositive integers d. Hence, we infer from 3. that 3. r B 3 i+ b i = i=0b d 3 L 8 log T d dt d L r T d L + O log. What remains is to obtain a good uer bound for d<l T d. Let us define K = { a : a = or a = where 3 mod 4} and let K d = {k, k K : k < k and k k = d}. Then 3.3 K d + OL. T d This reduces to finding an uer bound for K d. Note that K d = 0 when d mod 4. This leaves us with the following cases: Case : d 0 mod 4. There are two ossible ways for this to occur, namely when k = and k = q or k = and k = q, where and q are rimes congruent to 3 modulo 4. Therefore, Lemma. gives 3.4 K d 6 + o > f d log. Case : d mod. This can occur when k = and k = q or k = and k = q. In this case we deduce from Lemma. that 3.5 K d 4 + o f d log. > Hence, using 3.4 and 3.5, and aealing to Lemma.4 with P being the set of rimes >, we get K d + K d K d = = 6 d 0 mod o d 0 mod 4 f d + 4 L log. d mod d mod f d > log + o Thus, by combining the last estimate with equations 3. and 3.3 we

8 8 Y. LAMZOURI ET AL. obtain B L 7 log + o We choose L = 3 56 log to finally deduce B 3 δ + o, L L log + O log. where δ = The corresonding lower bound for B can be obtained along the same lines. 4. Proof of Theorem. Let N be the set of ositive integers n such that 3 ϕn, and write N = {a,..., a k } with a < < a k, and k = N. Put a 0 = and a k+ = []. Since the number of ositive integers i k such that a i N or a i+ N is at most N, using Theorem 3 we deduce that 4. { i k : a i, a i+ N } N N = N N δ + o, log where δ = λ β Let L log be a ositive real number to be chosen later, and suose that for some ositive integer i k we have a i, a i+ N that is, ϕa i ϕa i+ mod 3 and mina i a i, a i+ a i, a i+ a i+ L. Then for j = 0,,, there are at least [L] integers n [a i, a i+ ] such that Φn j mod 3. Let RL be the set of such integers i, and define S d = {b, b N : b < b and b b = d} for all ositive integers d. Since the number of integers i k such that mina i a i, a i+ a i, a i+ a i+ < L is bounded by 3 d<l S d, we infer from 4. that 4. RL δ + o 3 S d. log On the other hand there are at least [ RL /3] ositive integers i RL such that the intervals [a i, a i+ ] are disjoint. Hence, for j = 0,, we have δ L 4.3 A j 3 + o L S d + O. log log In order to obtain an uer bound for S d, we use sieve theory. Indeed by Lemma. we know that all n N are not divisible by any rime mod 3. Therefore, Lemma.3 gives S d C 0 + o f d log, mod 3

9 PARTIAL SUM OF EULER S TOTIENT FUNCTION 9 where C 0 = lim y = 3 lim y mod 3 y mod 3 y y , y using a comutation of A. Languasco and A. Zaccagnini [6]. Furthermore, using Lemma.4 with P being the set of rimes mod 3, we get S d C 0 + ol log. Combining this with 4.3 we obtain δ L L A j + o C 0 + O 3 log log log for j = 0,,. Thus, choosing L = α 0 log with α0 = δ/c 0 we deduce that A j δ + o, where δ = δ /7C , comleting the roof. 5. The distribution of Euler s function modulo 3: Proof of Theorem 3. For i =, let M i be the set of ositive integers n such that 3 ϕn and ϕn i mod 3. Then one can easily check that n N if and only if n M or n = 3d with d M /3. This imlies 5. N = M + M /3, and similarly we get 5. N = M + M /3. Hence it suffices to estimate M and M. Let n be a ositive integer such that 3 n and 3 ϕn, and write n = k j= a j j. Then ϕn = k j= a j j j and therefore j mod 3 for all j k. Moreover one has 5.3 ϕn P k j= a j k Ωn ωn mod 3, where Ωn resectively ωn is the number of distinct rime factors of n counted with resectively without multilicity. Let f and g be the arith-

10 0 Y. LAMZOURI ET AL. metic functions defined by { + Ωn ωn fn = if n imlies mod 3, 0 otherwise, { Ωn ωn gn = if n imlies mod 3, 0 otherwise. Then we deduce from 5.3 that 5.4 M = fn and M = gn. n n The Dirichlet series of f and g are defined by fn gn L f s := n s and L g s := n s, n= resectively, and are absolutely convergent for Res >. Our idea is to eress L f s resectively L g s as a ower of the Riemann zeta function ζs times a function H s resectively H s which is analytic in the half lane Res, and then use the Selberg Delange method to estimate n fn resectively n gn. There are two characters modulo 3, the rincial character χ 0 and the real character χ 3 defined by χ 3 n = n 3. We rove where n= Proosition 5.. Let s C with Re s >. Then L f s = ζs / H s and L g s = ζs / H s, H s := H s := 3 s / + Ls, χ 3 3 s / Ls, χ 3 s s s s + /, s s + /. Moreover both H s and H s can be analytically continued in a region Res c 0 / + log Ims + for some constant c 0 > 0. Proof. We shall only rove the statement for L f s, since the argument for L g s is similar. Since the function Ωn ωn is multilicative, we

11 PARTIAL SUM OF EULER S TOTIENT FUNCTION get 5.5 L f s = = = s + s + s + for Res >. On the other hand we have 5.6 k k ks = k χ 0 k ks k = log Ls, χ 0 log Ls, χ a a= as + s + s s + χ 3 k ks Now if mod 3 or k is odd then χ 3 k χ 3 = 0. This yields k χ 3 k χ 3 k ks = m Combining this with 5.6 we obtain 5.7 s = Ls, χ 0 / Ls, χ 3 / = m ms = 3 s / ζs / Ls, χ 3 / k χ k χ 3 k ks. log / s s. / s. The result then follows from 5.5, along with the fact that Ls, χ 3 is entire and does not vanish in a region Res c 0 / + log Ims +, for some constant c 0 > 0. Proof of Theorem 3. Using the Selberg Delange method more recisely Theorem 3 in Chater II.5 of [9] we infer from Proosition 5. that 5.8 fn = α + log n K k= α k log k+/ + O K log K+/,

12 Y. LAMZOURI ET AL. where α = H Γ / = 3/4 π / +, + since Γ / = π and L, χ 3 = π/3 3/, which follows from the Dirichlet class number formula see Chater 6 of []. Moreover the constants α k are defined by α k := Γ / k l+j=k l! Hl s j, and s j are the coefficients of the Laurent series of s s ζs / around the oint s =. Analogously to 5.8 we obtain a similar asymtotics for n gn with different constants α and α k where α is given by the same eression as for α ecet that the factor + + is relaced by +. Finally the result follows uon combining these asymtotic formulas with 5., 5. and 5.4. One can also arrive at the conclusion of Theorem 3 by alying the same method to the Dirichlet series χ 0 ϕn χ 3 ϕn n s and n s, uon noting that n= n ϕn i mod 3 with ɛ = and ɛ =. n= = χ 0 ϕn + ɛ i n n χ 3 ϕn, Acknowledgments. The authors are grateful to the referee for many useful comments and suggestions. The first author is suorted by a ostdoctoral fellowshi from the Natural Sciences and Engineering Research Council of Canada. The second author was suorted from the NSF grant DMS EMSWMCTP: Research Eerience for Graduate Students. Research of the third author is suorted by the NSF grant DMS REFERENCES [] H. Davenort, Multilicative Number Teory, 3rd ed., Grad. Tets in Math. 74, Sringer, New York, 000. [] T. Dence and C. Pomerance, Euler s function in residue classes, in: Paul Erdős , Ramanujan J. 998, 7 0.

13 PARTIAL SUM OF EULER S TOTIENT FUNCTION 3 [3] H. Halberstam and H.-E. Richert, Sieve Methods, London Math. Soc. Monogr. 4, Academic Press, London, 974. [4] J. Kaczorowski, Some remarks on factorization in algebraic number fields, Acta Arith , [5] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leizig, 909; nd ed., Chelsea, New York, 953. [6] A. Languasco and A. Zaccagnini, On the constant in the Mertens roduct for arithmetic rogressions. II. Numerical values, Math. Com , [7] W. Narkiewicz, On distribution of values of multilicative functions in residue classes, Acta Arith. 966/967, [8] P. Pollack, Not Always Buried Dee: A Second Course in Elementary Number Theory, Amer. Math. Soc., Providence, RI, 009. [9] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Stud. Adv. Math., 46, Cambridge Univ. Press, Cambridge, 995. [0] E. Wirsing, Über die Zahlen, deren Primteiler einer gegebenen Menge angehören, Arch. Math , Youness Lamzouri, M. Ti Phaovibul, Aleandru Zaharescu Deartment of Mathematics University of Illinois at Urbana-Chamaign 409 W. Green Street Urbana, Illinois 680, U.S.A. lamzouri@math.uiuc.edu haovib@illinois.edu Zaharescu@math.uiuc.edu Received November 00; revised 5 January