NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF 1 MODULO SHIFTED PRIMES

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1 NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF 1 MODULO SHIFTED PRIMES PAUL POLLACK Abstract An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multilicative function over a corime arithmetic rogression We rove a variant where the arithmetic rogression is relaced by a sifted set As an alication, we show that the normalized square roots of 1 mod m are equidistributed mod 1 as m runs through the shifted rimes q 1 1 Introduction Many roblems in elementary and analytic number theory require estimates for mean values of arithmetic functions One of our most useful tools for obtaining uer bound estimates is the following theorem of Peter Shiu [Shi80], which bounds the mean value of a nonnegative-valued multilicative function over a corime arithmetic rogression Let M be the collection of nonnegative-valued multilicative functions f satisfying the following two conditions: I There is a constant A 1 > 1 such that f k A k 1 for all rime owers k II For every ɛ > 0, there is a constant A ɛ > 0 such that fn A ɛn ɛ for all n 1 Theorem A Brun Titchmarsh for multilicative functions, [Shi80] Let f M, 0 < α, β < 1, and let a, k be integers satisfying 0 a < k and gcda, k = 1 Then for all sufficiently large x, we have that 11 fn y 1 ϕk log x ex f, x y<n x n a mod k whenever a, k, y satisfy k < y 1 α, x β < y x Here the imlied constant in 11, as well as the thresshhold for sufficiently large, deends only on α, β, the constant A 1 in I above, and the function A ɛ in II Theorem A has roved highly influential, with AMS Mathematical Reviews recording more than 70 citations to [Shi80] so far No doubt this is due to its imressive generality and the ease with which it can be lugged in as an auxiliary tool in number-theoretic investigations Shiu s work is based on methods described by Erdős [Erd5] and Wolke [Wol71]; work in a similar direction had also been carried out by Vinogradov and Linnik [VL57], Barban [Bar64a], and Barban and Vehov [BV69] In this aer, we ut forward a variant of Theorem A where the role of the corime rogression a mod k is taken by a sifted set 010 Mathematics Subject Classification 11N37 rimary, 11N36, 11J71 secondary 1 k

2 PAUL POLLACK Theorem 11 Brun s uer bound sieve for multilicative functions Let f M, let 0 < β < 1, and let k be a nonnegative integer For each rime x, let E be a union of ν nonzero residue classes modulo, where we suose that each ν k Let x y<n x n S S = In words, S is the set of all integers n not belonging to any E If x is sufficiently large, then 1 fn y log x ex f ν for all y satisfying x β < y x Here the imlied constant in 1, as well as the thresshhold for sufficiently large, deends only on β, k, the constant A 1 in I above, and the function A ɛ in II Remarks i Theorem 11, while obviously a close relative of Theorem A, does not obviously imly it nor vice versa ii It may initially seem strange that we require E to only contain nonzero residue classes This does not entail any loss of generality, since we can effectively remove n not corime to a given P by relacing the function fn with 1 gcdn,p fn iii Keeing the last remark in mind, one easily deduces from Theorem 11 that the number of n x that avoid any rescribed ν k residue classes modulo, for each rime x, is k x ex ν In this way, Theorem 11 generalizes Brun s uer bound sieve As a first examle of Theorem 11, if we remove a single nonzero residue class modulo for each rime x 1/, then the sum of the divisor function dn over the remaining n [1, x] is Ox, where the imlied constant is absolute Another immediate alication of Theorem 11 is an uer bound for the mean value of fn with n restricted to shifted twin rimes Corollary 1 Let f be a function belonging to M Let 0 < β < 1/ For all x 3 and y x β, x], x fq 1 A1,A ɛ,β log x ex f 1, x y<q x q, q rime Taking fn = t ωn and y = x in Corollary 1, we deduce that for each fixed t 0, and all x 3, t ωq 1 x t0 log x log xt 1, q x q, q rime uniformly for 0 < t t 0 One can now extract information on the distribution of ωq 1 by varying t For examle, mimicking the roof of Theorem 010 in [HT88] yields: Uniformly for E c

3 MULTIPLICATIVE FUNCTIONS ON SIFTED SETS 3 0 ψ log log x 1/6, the number of rime airs q, q in [1, x] with ωq 1 log log x > ψ log log x is Oxlog x ex ψ / This last statement is a strengthened form of a theorem of Barban [Bar64b] We now describe our original motivation for roving Theorem 11 The alication is a riff on two theorems of Hooley An infamous conjecture of Landau one of his four unattackable roblems redicts that there are infinitely many rimes of the form x + 1 This is still oen, but the analogous roblem for rimes of the form x + y + 1 was settled by Hooley in 1957 [Hoo57] Put rn = #{x, y Z : x + y = n} Hooley shows not only that rq 1 > 0 for infinitely many rimes q, but that rq 1 has a ositive average value Theorem B For a certain ositive constant K, we have rq 1 K x x log x q x q rime Actually, Hooley s work was conditional on GRH, but the discovery of the Bombieri Vinogradov theorem allowed for this deendence to be removed with minimal changes to Hooley s argument See [EH66] In the intervening years, Linnik [Lin60] gave an alternative roof of Theorem B The following variant of Hooley s result was shown by Kátai in 1968 [Kát68] see also [Kát70] Theorem B For a certain ositive constant K, we have rq 1 K x x log x q x q rime q 1 squarefree From elementary number theory, we know that for squarefree n, the number of square roots of 1 modulo n is recisely rn In articular, Theorem 4 B imlies that 1 is a square modulo q 1 for infinitely many rimes q We now bring in the second theorem of Hooley First, a definition: If P T is a olynomial with integer coefficients, and k is any ositive integer, then a normalized root of P, modulo k, is a rational number of the form ϖ/k, where fϖ 0 mod k, 0 ϖ < k In 1964, Hooley roved the following equidistribution theorem for normalized roots [Hoo64] Theorem C Let P T Z[T ] be an irreducible olynomial of degree at least For each ositive integer k, list the normalized roots of P modulo k in any order, and then concatenate the lists sequentially for k = 1,, 3, The resulting sequence is uniformly distributed in [0, 1 After this set-u, the reader can erhas guess where we are headed We rove that when P T = T + 1, the conclusion of Hooley s Theorem C holds with the moduli restricted to the shifted rimes q 1

4 4 PAUL POLLACK Theorem 13 For each rime q, list the normalized square roots of 1 modulo q 1, and then concatenate the lists successively for q =, 3, 5, 7, The resulting sequence is uniformly distributed in [0, 1 Remark By a deeer and very different method, Duke, Friedlander, and Iwaniec [DFI95] have shown that Theorem 13 holds with the moduli running over rimes q rather than shifted rimes q 1 In both Corollary 1 and Theorem 13, with some effort we could have alied the Nair Tenenbaum generalization of Shiu s theorem [NT98] in lace of Theorem 11; this is because of the simle structure of the sieved-out residue classes in these examles For each rime, the classes being sieved out can be described as the mod roots of a olynomial of small height Nevertheless, we think that Theorem 11 which, in full generality, does not follow trivially from [NT98], and the mentioned alications, are worthy of attention in themselves We are also hoeful that future researchers will find our descrition of the EBLSVVW 1 method useful; we have formulated Proosition 1 with this ambition in mind Notation Most of our notation is standard A ossible exetion is our use of P + n and P n for the largest and smallest rime factors of n resectively; we adot the convention that P + 1 = 1 while P 1 = The notation d n indicates that d is a unitary divisor of n; that is, d n and gcdd, n/d = 1 We let 1 denote the function that is identically 1, and we use 1 C for the characteristic function of a roerty or set C We write ex for e πix We reserve the letter for rime numbers Proof of Theorem 11 1 Generalities Let f M, and let A 1 and A ɛ be as in I and II Let x 3, and let θ 0, 1 For each integer n [1, x], we may write where n = 1 j j+1 J, 1 J, and where j is chosen as the largest index for which We let 1 j x θ and 1 j n d := 1 j, and we refer to d as the canonical unitary refix divisor of n We make the assumtion that n has no roer rime ower divisor in the interval x θ/, x θ ] and that Ωn/d θ 1 In our eventual alication, we will be able to handle the excluded values of n by a searate argument Since Ωn/d j+1 j+1 J n x, we see that 1 j+1 x 1/Ωn/d x θ/ 1 Erdős Barban Linnik Shiu Vehov Vinogradov Wolke

5 Let t be the largest ositive integer for which MULTIPLICATIVE FUNCTIONS ON SIFTED SETS 5 j+1 = j+ = = j+t Then 1 j+t is a unitary divisor of n, so that the choice of j forces 1 j+t > x θ Thus, d = 1 j > x θ / t j+1 If t j+1 > x θ/, then t > 1, by 1; moreover, some ower of j+1 is then a roer rime ower divisor of n belonging to x θ/, x θ ], contradicting our assumtions on n So t j+1 x θ/, and, from the last dislayed equation, d > x θ/ From 1, there is a unique integer r satisfying x θ/r+1 < j x θ/r Since x Ωn/d j+1 x Ωn/d θ/r+1, we have Ωn/d r + 1 θ 1, so that fn = f 1 j f j+1 J f 1 j A Ωn/d 1 fda r+1θ 1 1 We collect the salient results in the following roosition Proosition 1 Let f M, and let θ 0, 1 Let x 3 Let n be an integer in [1, x], and assume n does not have a roer rime ower divisor in x θ/, x θ ] Let d be the canonical unitary refix divisor of n If Ωn/d /θ, then x θ/ < d x θ, x θ/r+1 < P + d x θ/r for some integer r, and fn exorfd Here the imlied constants deend only on θ, A 1, and A ɛ Comletion of the roof of Theorem 11 Fix θ = β/4 We ut those n S belonging to x y, x] into the following three ossibly overlaing categories For each category we then bound the corresonding contribution to n S x y,x] fn 1 Arithmetically atyical n: Here we include all n with a roer rime ower divisor in x θ/, x θ ] We also ut in this category all n which have a logx θ -smooth divisor in the interval x θ/, x θ ] Those n for which Ωn/d < /θ, where as above d is the canonical unitary refix divisor 3 All remaining n S We handle category 1 using the crude ointwise bound fn ɛ n ɛ The number of n x y, x] divisible by a roer rime ower k x θ/, x θ ] does not exceed y + 1 x θ 1 + y k m xθ + yx θ/4 yx θ/4 x θ/ < k x θ k m>x θ/ m squarefull

6 6 PAUL POLLACK Similarly, the number of n x y, x] ossessing a logx θ -smooth divisor e x θ/, x θ ] is at most x θ 1 + y e xθ + yx θ/ 1 x θ/ <e x θ P + e logx θ e x θ P + e logx θ x θ + yx θ/3 yx θ/3 To justify assing from the first to the second line, we use that the count log T -smooth numbers u to T is T o1, as T ; see, eg, [MV07, Corollary 79, 09] Since fn x θ/8 say, it follows that the contribution to n S x y,x] fn from arithmetically inconvenient n is yx θ/4 + yx θ/3 x θ/8 yx θ/8 This is negligible, since the our target uer bound the right-hand side of 1 is ylog x k 1 For n falling into category, we have that n = d j+1 J, where J j < θ 1 If all of j+1, j+,, J are bounded by x θ /, then n dx θ / /θ x θ Since fn x θ, these n contribute x 3θ yx θ, which is once again negligible So we may assume that j+t > x θ / for some ositive integer t J If t is chosen minimally, then utting we see that d is a unitary divisor of n, that that and that d = 1 j+t 1, d x θ, P n/d > x θ /, fn = fd fn/d Thus, these n make a contribution that is 3 d x θ fd fd A θ 1 1 fd x y/d <m x/d md S P m>x θ / We use Brun s sieve to bound the sum on m Let be a rime not exceeding x θ / The subscrited conditions imly that m avoids the residue class of 0 modulo and, if d, also an additional ν residue classes modulo Since y/d x β /x θ x θ /, Brun s sieve bounds the sum as y d θ / d y d log x 1 ν + 1 θ / θ / d 1 ν d 1 1 1

7 MULTIPLICATIVE FUNCTIONS ON SIFTED SETS 7 We can extend the roduct over all rimes x without changing the order of magnitude We conclude that y 1 1 ν 1 ν 1 d log x where x y d <m x d md S P m>x θ / y d log x gd := d 1 d 1 ν gd, 1 min{k, 1} Inserting these bounds back into 3, we find that the remaining n in category contribute y 1 ν fd gd log x d d x θ y 1 ν 1 + fg + f g + log x Now fg/ = f/ + O1/, while k fk g k / k 1 It follows that the final dislayed roduct on is ex f/, leading to an uer bound for the entire exression of y 1 ν f ex, log x which in turn is y log x ex f ν This exression coincides with that on the right-hand side of 1, and so the contribution from the n in category is accetable Finally, we turn to category 3 We subdivide the n in that category according to the value of r for which holds Since we are assuming that n does not have a logx θ -smooth divisor in x θ/, x θ ], we only consider r for which x θ/r > logx θ, so that 4 r < logxθ log logx θ By Proosition 1, for each fixed r in the range 4, the corresonding n make a contribution that is exor fd 1 x θ/ <d x θ P + d x θ/r x y/d<m x/d P + m>x θ/r+1 md S

8 8 PAUL POLLACK In this case, Brun s sieve gives that the sum on m is y 1 ν rθ 1 k+1 d θ/r+1 d θ/r+1 d y log x 1 ν gd d, where g has the same meaning as above Since rθ 1 k+1 = exor, we see that these n contribute y 5 exor 1 ν fdgd log x d x θ/ <d x θ P + d x θ/r The sum on d is estimated by the following secial case of [Shi80, Lemma 4] Lemma Let F M Then, for all sufficiently large Z, F n n ex F 1, 10 R log R n Z 1/ P + n Z 1/R uniformly for R satisfying 1 R log Z Here the thresshold for sufficiently large, as log log Z well as the imlied constant, deends at most on the constant A 1 and the function A ɛ associated to F in the definition of M Let F = fg, Z = x θ, and R = r Using that f M, it is straightforward to check that F M ; moreover, the A 1 and A ɛ corresonding to f suffice to determine, together with k, choices for A 1 and A ɛ corresonding to F By 4, R log Z Alying Lemma log log Z, fdgd fg ex 1 d 10 r log r θ x θ/ <d x θ P + d x θ/r Z ex 110 r log r ex f We used once more than fg/ = f/ + O1/ We ut this estimate back into 5 and then sum on r in the range 4 Since exor ex 1 r log r 1, 10 r we conclude that the total contribution of n from category 3 is y 1 ν f ex, log x which is y log x ex f ν, as desired This comletes the roof of Theorem 11

9 MULTIPLICATIVE FUNCTIONS ON SIFTED SETS 9 3 Equidistribution of square roots of 1 modulo shifted rimes: Proof of Theorem 13 We follow the original arguments of Hooley [Hoo64] as closely as ossible For each ositive integer k, let ϱk denote the number of square roots of 1 modulo k From elementary number theory, ϱ is multilicative and ϱ s for all rime owers s, so that ϱk ωk Moreover, as noted in the introduction, ϱk = 1 rk for squarefree values 4 of k Thus, letting Theorem B gives that 31 K = {q 1 : q rime}, ϱk k K For each air of integers h, k with k > 0, let Sh, k = x log x ϖ mod k ϖ 1 mod k ehϖ/k Trivially, Sh, k ϱk By Weyl s criterion and the lower bound 31, to rove Theorem 13 it will suffice to show that Sh, k = ox/ log x x k K for each fixed h 0 Cf the discussion on 48 of [Hoo64] In what follows, we let X = x 1/ log log x For each ositive integer k x, we write k = k 1 k, where k 1 is X-smooth and k is X-rough meaning that P k > X We decomose Sh, k = Sh, k + Sh, k = +, 1 k K k K k 1 x 1/3 k K k 1 >x 1/3 say Concerning, Cauchy Schwarz gives 1/ 3 1 k 1 >x 1/3 Sh, k k 1 >x 1/3 1/ Since Sh, k ϱk ωk, we have that Sh, k ωk xlog x 3 ; k 1 >x 1/3 the final estimate here follows, for instance, from Shiu s Theorem A or the much more elementary Theorem 01 in [HT88] On the other hand, by the estimate for Θx, y, z

10 10 PAUL POLLACK aearing at the bottom of 9 of [HT88], 1 x ex k 1 >x 1/ o1 log log x log log log x, as x A less recise estimate, still sufficient here, follows from [BV64] Putting these estimates back into 3, we see that = Ox/log xa for each fixed A Thus, it will suffice to show that 1 = ox/ log x, as x As in [Hoo64] see that aer s Lemma 3, we have Sh, k = Sh k, k 1 Sh k 1, k, where k 1 denotes the inverse of k 1 modulo k, and k denotes the inverse of k modulo k 1 Now using k 1 and k for generic X-smooth and X-rough numbers, 1 = Sh k, k 1 Sh k1, k k 1 k x k 1 k K, k 1 x 1/3 ϱk Sh k, k 1 = Θx/k 1, k 1, k 1 k x k 1 x 1/3 k 1 k K, k 1 x 1/3 where, for y x /3 and k 1 x 1/3, we set Note that by Cauchy Schwarz, 33 Θy, k 1 Θy, k 1 = k 1 k +1 rime k 1 k K ϱk ϱk Sh k, k 1 k 1 k +1 rime Sh k, k 1 The first arenthesized sum in 33 can be handled by Theorem 11 With P the roduct of the rimes not exceeding X, we have that ϱk 1 gcdn,p =1 ωn X k 1 k 1 k +1 rime n y k 1 n+1 rime To roceed, we observe that for k 1 n + 1 to be rime, either n y 1/, or n avoids the class of k1 1 mod for all rimes X not dividing k 1 The former case accounts for Oy 1/ values of n By Theorem 11, the latter case contributes y log y ex y 4 1 log y 1 ylog log x6 ex log y log X log X log x X< y k 1 such values We used here that log y log x, that log X = log x/ log log x, and that ex 1 k k 1 1 ϕk 1 log log 3k 1 log log x The contribution of y 1/ is negligible

11 comared to this, and so MULTIPLICATIVE FUNCTIONS ON SIFTED SETS 11 k 1 k +1 rime ϱk ylog log x6 log x Turning to the second arenthesized sum in 33, we have that Sh k, k 1 = k 1 k +1 rime 0 a<k 1 Sah, k 1 k ā mod k 1 k 1 k +1 rime Writing a 0 for the least nonnegative residue of ā modulo k 1, we see that the values of k aearing in the second sum have the form a 0 + k 1 t for some t y/k 1 Moreover, either t y/k 1 1/, or both both a 0 + k 1 t and k 1 a 0 + k 1 t + 1 have no rime factors exceeding X Brun s sieve now imlies that the sum on k is and thus 0 a<k 1 Sah, k 1 y k 1 log X ex k ā mod k 1 k 1 k +1 rime k 1 1 ylog log x4 k 1 log x ylog log x4 k 1 log x, 1 0 a<k 1 Sah, k 1 Exactly as in [Hoo64] see Lemma 1 there, the sum on a is at most ϱk 1 k 1 gcdh, k 1, and now collecting estimates yields k 1 k +1 rime Sh k, k 1 ylog log x4 log x ϱk 1 where here and for the remainder of the argument, the imlied constants may deend on h Referring back to 33, Θy, k 1 y log x log log x5 ϱk 1 1/, so that 1 x log x k 1 x 1/3 ϱk 1 1/ k 1 Θx/k 1, k 1 log log x5 k 1 x 1/3 Bounding the sum on k 1 by an Euler roduct, as in [Hoo64] cf the dislay immediately receding that aer s eq 1, we find that k 1 x 1/3 ϱk 1 1/ k 1 log x 1/ We conclude that x log log x 5 1 log x 1 Since 1 > 1, this imlies that 1 = ox/ log x, as desired This comletes the roof of Theorem 13

12 1 PAUL POLLACK Acknowledgements The author is suorted by NSF award DMS He would like to exress his gratitude to Terence Tao for a helful blog ost [Tao11] exlaining Erdős s roof in [Erd5] References [Bar64a] M B Barban, Multilicative functions of Σ R equidistributed sequences, Izv Akad Nauk UzSSR Ser Fiz Mat Nauk , no 6, Russian [Bar64b], On the number of divisors of translations of the rime number-twins, Acta Math Acad Sci Hungar , Russian [BV64] M B Barban and A I Vinogradov, On the number-theoretic basis of robabilistic number theory, Dokl Akad Nauk SSSR , Russian [BV69] M B Barban and P P Vehov, Summation of multilicative functions of olynomials, Mat Zametki , Russian [DFI95] W Duke, J B Friedlander, and H Iwaniec, Equidistribution of roots of a quadratic congruence to rime moduli, Ann of Math , [EH66] P D T A Elliott and H Halberstam, Some alications of Bombieri s theorem, Mathematika , [Erd5] P Erdős, On the sum x k=1 dfk, J London Math Soc 7 195, 7 15 [Hoo57] C Hooley, On the reresentation of a number as the sum of two squares and a rime, Acta Math , [Hoo64], On the distribution of the roots of olynomial congruences, Mathematika , [HT88] R R Hall and G Tenenbaum, Divisors, Cambridge Tracts in Mathematics, vol 90, Cambridge University Press, Cambridge, 1988 [Kát68] I Kátai, A note on a sieve method, Publ Math Debrecen , [Kát70], On an alication of the large sieve: Shifted rime numbers, which have no rime divisors from a given arithmetical rogression, Acta Math Acad Sci Hungar , [Lin60] Yu V Linnik, An asymtotic formula in the Hardy Littlewood additive roblem, Izv Akad Nauk SSSR, Ser Mat , Russian [MV07] H L Montgomery and R C Vaughan, Multilicative number theory I Classical theory, Cambridge Studies in Advanced Mathematics, vol 97, Cambridge University Press, Cambridge, 007 [NT98] M Nair and G Tenenbaum, Short sums of certain arithmetic functions, Acta Math , [Shi80] P Shiu, A Brun-Titchmarsh theorem for multilicative functions, J Reine Angew Math , [Tao11] T Tao, Erdős s divisor bound, 011, blog ost ublished July 3, 011 at htt://terrytao wordresscom/011/07/3/erdos-divisor-bound/ To aear in the forthcoming book Sending symmetry [VL57] A I Vinogradov and Yu V Linnik, Estimate of the sum of the number of divisors in a short segment of an arithmetic rogression, Usehi Mat Nauk NS , no 476, Russian [Wol71] D Wolke, Multilikative Funktionen auf schnell wachsenden Folgen, J Reine Angew Math , Deartment of Mathematics, University of Georgia, Athens, GA address: ollack@ugaedu