Multiplicative functions of polynomial values in short intervals

Transcription

1 ACTA ARITHMETICA LXII Multilictive functions of olnomil vlues in short intervls b Mohn Nir Glsgow 1 Introduction Let dn denote the divisor function nd let P n be n irreducible olnomil of degree g with integer coefficients In 1952, Erdős [2] showed tht there exist constnts c 1 nd c 2, which m deend on P, such tht c 1 x log x n x d P n c 2 x log x for x 2 This result ws generlized b Delmer [1] who showed tht for n l N, c 1 xlog x s d l P n c 2 xlog x s n x with s = 2 l 1 nd where the c i m deend, in ddition, on l The lower bound in Erdős s result is firl strightforwrd but the uer bound is combintion of chrcteristicll ingenious ides In 1971, Wolke [6] clrified these ides nd together with severl imortnt contributions of his own, showed tht n suitble sum of the form n x f n cn be similrl bounded Here f is n non-negtive multilictive function with f l c 1 l c 2, c 1, c 2 constnts, nd { n } is sequence of nturl numbers with structure menble to the sieve method Aling his results to fn = dn nd n = P n, he recovered Erdős s result nd, indeed, gve severl other interesting lictions The ver generlit of Wolke s results ment tht the bounds obtined lcked uniformit with resect to n rticulr clss of sequences { n } In 1980, Shiu [5] obtined such uniformit for the clss of rithmetic rogressions, ie for liner olnomils nd refined the Erdős Wolke method, in this rticulr cse, to include lrger clss of multilictive functions s well s to obtin short-intervl result He considered the clss of non-negtive multilictive functions f which stisf the weker conditions f l A l 0, fn A 1 εn ε for n ε > 0 nd constnts A 0, A 1

2 258 M Nir He showed tht, n mod k fn ϕk 1 log x ex x, k f uniforml for k 1 β nd x α < x Here ϕk is Euler s function, denotes rime number, α, β R with 0 < α, β < 1 2 nd, k N with < k nd, k = 1 The imlied constnt in the nottion, lthough not exlicitl stted s such in [5], deends onl on α, β, A 0 nd the rticulr function A 1 In this er we extend these ides further to estimte the generl sum f P n while reserving sufficient uniformit with resect to the olnomil P n to obtin bound which imlies Shiu s theorem when lied to P m = km +,, k = 1 nd lso, in the generl cse, extends Erdős s result to olnomils hving distinct zeros not necessril irreducible nd to n ling in short intervl The uniformit which we obtin in our theorem hs the interesting imliction tht the rent generlistion to n ling in n rithmetic rogression is, in fct, corollr of the theorem 2 Definitions We define P to be the clss of olnomils P with integer coefficients, of degree g, with non-zero discriminnt D nd hving no fixed rime divisors Let ϱm denote, for m N, the number of solutions n mod m of the congruence P n 0 mod m The condition tht P hs no fixed rime divisor is equivlent to ϱ < for ll rimes It is well known see eg Ngell [4] tht i ϱb = ϱϱb if, b = 1, ii ϱ l g if D, iii ϱ g, nd iv if σ D with σ 1 then ϱ l = ϱ 2σ+1 if l > 2σ, nd b ϱ l l 1 ϱ l N We note tht ii nd iv iml tht ϱ l g 2σ nd hence tht v ϱm gd 2 ωm m N, where the discriminntl fctor D is defined b D = σ σ D, ϱ 0 We lso define the size of P, denoted b P, b P = mx i i where P t = 0 i g it i Note tht, n N, P n g + 1 P n g

3 Multilictive functions 259 The clss M of multilictive functions which we consider is the sme s tht in Shiu [5], ie M consists of non-negtive multilictive functions which stisf the following two conditions: i There exists ositive constnt A 0 such tht f l A l 0, rimes nd l N ii For ever ε > 0, there exists ositive constnt A 1 = A 1 ε such tht fn A 1 n ε, n N We note tht the function ϱn itself belongs to M with A 0 = gd 2 nd A 1 = A 1 g, D, ε For convenience, we lso extend the definition of f nd ut f0 = 0 3 Nottion t, x, nd z denote ositive rel numbers, nd q, with or without subscrits, denote rime numbers,, b, d, g, i, j, k, l, m, n, r nd s, with or without subscrits, denote nturl numbers, P + n nd P n denote, resectivel, the gretest nd the lest rime fctor of n 2 For technicl convenience, we dot the convention P 1 = nd P + 1 = 1, Ωn nd ωn denote, s usul, the number of rime fctors of n with nd without counting multilicit, resectivel, α, β, ϑ, ε nd δ denote ositive rel numbers less thn 1, c, c 1, c 2, denote ositive rel constnts nd Ψx, z = 1 n x, P + n z The deendence of the constnt imlicit in the nottion on the olnomil P, unless emhsised otherwise, will lws be t most on the degree g nd on the rime ower divisors of the discriminntl fctor D It m, however, deend on ll constnts ssocited with f nd with the intervl 4 Results Our min theorem is the following Theorem Let f M, P P nd let α, δ R with 0 < α, δ < 1 For n x, R with x, 2 nd x α x we hve f P n 1 ϱ fϱ ex x x rovided tht x c P δ, where the constnt c deends onl on g, α, δ nd the function A 1 ssocited with M

4 260 M Nir The deendence of the constnt imlicit in the nottion on the olnomil P is onl on i the degree g nd on ii the discriminntl fctor D mentioned erlier It cn, of course, deend on α, δ nd the constnts ssocited with the clss M We mke the observtion tht, in generl, one would exect n uer bound for the sum considered in our theorem to deend on ϱ l for ll l 1 Since our bound deends, exlicitl, onl on ϱ nd it is well known tht ϱ l cn be lrge for rimes which divide the discriminnt of P, it is erhs to be exected tht there is some deendence on D in the imlicit constnt We now stte the corollr of the theorem which extends the result to rithmetic rogressions Corollr With the sme hotheses s in the Theorem, let, in ddition,, k N with k, k, P = 1 nd k 1 β, where β is n constnt with 0 < β < 1 Then f P n, n mod k k 1 ϱ/ k x 1 ϱ ex x, k fϱ rovided tht x c 1 P δ where the constnt c 1 deends onl on g, α, δ nd the function A 1 ssocited with M The deendence of the imlicit constnt on the olnomil P is s described in i nd ii in the sttement of the Theorem There is, in ddition to those mentioned there, deendence on β The condition k, P = 1 ensures tht the sequence P n, n mod k hs no fixed rime divisor The Corollr lied to P n = n, so tht D = 1, P = 1, ϱ = 1,, recovers Shiu s theorem in its entiret In fct, ll the imlictions of Shiu s theorem, s mentioned in [5], cn now be extended to the olnomil cse We give just one exmle: For r, l N, r 2, we hve, n mod k d l r P n k ϕk k r l 1 log x uniforml in, k nd with, k N, k, k, P = 1, x α x nd k 1 β rovided tht x 1 Here α nd β re n rel numbers with 0 < α, β < 1

5 Multilictive functions 261 The imlicit constnts in this exmle, nd onl in this exmle, m deend on P, M, r, l, α nd β The function d r n is, s usul, the number of ws of writing n s roduct of r fctors, tking ccount of ordering 5 Preliminr lemms We shll mke use of the following lemms Lemm 1 Let Gt Z[t] with degree g Let N nd let x,, z > 0 with z x Then 51 {n : x < n x, P Gn z, Gn, = 1} g 1 ϱ <z, where ϱ is the number of solutions m mod of the congruence Gm 0 mod P r o o f This is strightforwrd liction of Brun s sieve First note tht if 1 is n fixed rime divisor of G, then either 1 or 1 < z imlies tht the left-hnd side of 51 is zero nd the result is trivill true Suose therefore tht n such 1 stisfies 1 nd 1 z We follow the nottion of Hlberstm nd Richert [3] nd consider Brun s sieve with A = {Gn : x < n x}, B = { : < z,, rime}, { w = ϱ if < z,, 0 otherwise It is esil checked tht Ω 0 is stisfied with A 0 = g nd tht, 0 ω 1 1 g + 1 so tht Ω 1 holds with A 1 = g + 1 Lemm 22, 52 of [3] imlies tht Ω 2 k holds with k = A 0 = A 2 = g The condition R in Brun s sieve [3], 68 is trivill stisfied nd the result follows Lemm 2 For n t 2, we hve i 1 ϱ g s g 1 ϱ <t <t 1/s uniforml in s, rovided tht 1 s log t ii 1 ϱ g g ϕ <t 1/2, iii For n F M, F nϱn n n t ex t <t F ϱ 1 ϱ

6 262 M Nir where the imlicit constnt deends on g, D nd on the constnts A 0 nd A 1 ssocited with F P r o o f Let T be defined b T = t 1/s <t, g t 1/s <t 1 ϱ 1 Using the fct tht ϱ min{g, 1}, we deduce tht 0 log T = log 1 ϱ t 1/s <t = log 1 ϱ + log 1 ϱ g t 1/s <t, >g Thus T g s g nd i follows To show ii, note tht 1 ϱ <t 1/2, <t t 1/s <t, >g 1 + O g1 g log s + O g 1 g g<<t 1/2, <t 1/2 B i nd simle clcultion, this is g 1 ϱ s required For iii, we hve n t F nϱn n 1 ϱ 1 ϱ 1 1 g = g<<t 1/2, ϕ g 1 + F ϱ + t l=2 F ϱ ex + t t <t F l ϱ l l l 2 Since ϱ l gd 2 nd F l c l/3, we deduce tht F l ϱ l l 1 nd iii follows t l 2 1 ϱ 1 1 ϱ F l ϱ l l

7 Lemm 3 For ll x 3, we hve for some bsolute constnt c Multilictive functions 263 Ψx, log x log log x c ex P r o o f This is Lemm 1 of Shiu [5] Lemm 4 Let F M For n t 2, we hve F n n c F 1 ex t 1/2 n t P + n t 1/s t 3 log x log log x 1/ s log s uniforml in s, 1 s log t/ log log t, where the constnt c 1 deends onl on the constnts A 0 nd A 1 ssocited with F P r o o f This is essentill Lemm 4 of Shiu [5] with the imlicit constnt exlicitl described nd lied with k = 1 6 Proof of the Theorem Let z = 1/2 nd for ech n in x, x], write where if we define P n = n b n P n = α 1 1 α j j α j+1 j+1 α l l, 1 < 2 < < l, with j chosen such tht n = α 1 1 α j j n z < n α j+1 j+1 If no such j exists, define n = 1 The ssocited b n is just defined b b n = P n / n Note tht n, b n = 1 with P b n > P + n We write P b n s q n nd divide the set of n in x, x] into four clsses: I: q n z 1/2, II: q n < z 1/2, n z 1/2, III: q n log x log log x, n > z 1/2 nd IV: log x log log x < q n < z 1/2, n > z 1/2 We estimte f P n for n belonging to ech clss in turn First we hve 1 61 f fb n I f P n z

8 264 M Nir where 1 indictes sum over those n in x, x] such tht P n, P b z 1/2 nd, b = 1 where b = P n / Now z 1/2 Ωb b = P n g + 1 P xg x g+1/δ so tht Ωb 1 nd hence fb A Ωb 0 1 This imlies tht the inner sum in 61 is We write this sum s r i mod 1 n r i mod where r i denote the ϱ solutions of the congruence P m 0 mod To investigte the inner sum in 62, write n = m+r i nd P r i = λ i, λ i Z Then P n = P m + r i = P r i + mqm, r i for some Qt, r i Z[t], of degree g 1 The conditions on the inner sum in 62 now reduce to x r i < m x r i, P λ i + mqm + r i z 1/2,, λ i + mqm + r i = 1 Since / z 1/2, we cn l Lemm 1 with Gt = λ i + tqt + r i to deduce tht the inner sum in 62 is 1 ϱ 1 <z 1/2, where ϱ 1 is the number of solutions of the congruence λ i + tqt, r i 0 mod Since P t + r i = λ i + tqt + r i, it is esil verified tht ϱ 1 = ϱ for, so tht 62 reduces to 1 ϱ r i Hence, from 61, n I f P n z <z 1/2, fϱ 1 <z 1/2, 1 ϱ

9 Using Lemm 2ii, we deduce the bound 1 ϱ <z Multilictive functions 265 z fϱ g ϕ We now note tht the function F = f/ϕ g M nd el to Lemm 2i, iii to deduce 1 ϱ fϱ ex <x s required We now turn to n II nd note tht to ech such n, there corresonds rime nd n exonent s such tht s P n, z 1/2 nd s > z 1/2 For ech z 1/2, let s denote the lest integer s with s > z 1/2 Hence s 2 nd s minz 1/2, 2 Thus z 1/2 1 s Hence we deduce tht 1 n II z 1/2 z 1/2 z 1/4 z 1/2 + s P n s 1 = + O1 x z 1/4 < z 1/2 2 z 1/4 ϱ s s z 1/2 + O1 z 1/4 + z1/2 7/8 We now show tht the n which belong to III re lso few in number For ech n III,, P n, z 1/2 < z nd P + < log x log log x Hence n III = z 1/2 < z P + <log x log log x z 1/2 < z P + <log x log log x Now if z cg, D then P n ϱ + O1 ϱ gd 2 ω gd 2 c log z/ log log z z 1/8 On the other hnd, if z < cg, D then, trivill, ϱ < cg, D z 1/2 < z P + <log x log log x ϱ

10 266 M Nir Thus 63 reduces, b Lemm 3, to Thus z 1/8 g,d Ψz, log x log log x + z1/2 z 1/2 7/ /8 n II Now note tht for n ε 1 > 0, n III f P n A 1 ε 1 P n ε 1 g + 1 ε 1 A 1 ε 1 P ε 1 x gε 1 Using x c P δ with ε 1 sufficientl smll, we deduce tht f P n 1/16 Hence f P n + f P n 15/16 log x g n II n III 1 ϱ fϱ ex <x s required It remins to consider n belonging to clss IV: f P n f n IV z 1/2 < z 2 fb where 2 indictes sum over those n in x, x] such tht P n,, b = 1 nd log x log log x < P b z 1/2 where b = P n / We divide the intervl for P b into subintervls z 1/s+1, z 1/s ] where Note tht s 0 log z/ log log z 2 s s 0 := [log z/ loglog x log log x] Consider now those n in 2 for which z 1/s+1 < P b z 1/s For such n, we hve P + z 1/s nd lso x z Ωb/s+1 P b Ωb b g + 1 P x g x g+1/δ so tht Ωb s Hence fb A s for some A > 0 Thus f P n n IV s 0 A s s=2 z 1/2 < z, P + z 1/s f 3 where 3 indictes sum over those n in x, x] such tht P n, 1

11 Multilictive functions 267, b = 1 nd z 1/s+1 < P b z 1/s where b = P n / So denoting b r i the ϱ solutions of P m 0 mod, we hve f P n n IV s 0 A s s=2 z 1/2 < z P + z 1/s f r i mod 3 n r i mod Writing n = m+r i, we deduce, s with the n in clss I, tht since z 1/s+1 / the inner sum is 1 ϱ nd hence f P n n IV s 0 A s s=2 s 0 A s s=2 z 1/2 < z P + z 1/s <z 1/s+1, z 1/2 < z P + z 1/s fϱ <z 1/s+1 fϱ 1 ϱ <z 1/s ϱ 1 ϱ 1 We now el to Lemm 2i, twice, together with simle clcultion to deduce the bound 1 ϱ s0 g A s s + 1 g fϱ ϕ <x s=2 z 1/2 < z P + z 1/s Lemm 4 with F = fϱ/ϕ g now imlies tht the inner sum bove is F ex 1 10 s log s fϱ ex 1 10 s log s z so tht we finll deduce tht f P n n IV <x <x 1 ϱ ex x 1 ϱ ex x x fϱ s + 1 g A s ex 110 s log s s 2 fϱ s required This comletes the roof of the Theorem

12 268 M Nir 7 Proof of the Corollr We denote b J the intervl x /k, x /k] nd write f P n = f P km + m J n mod k Put Gm = P km + We shll l our theorem to Gm First note tht deg G = deg P = g Recll tht if P t = g t g + + 0, g 0, is n olnomil with zeros α 1,, α g in C, then its discriminnt D is given b D = 2g 2 g α i α j 2 Hence Gt hs zeros α i /k, 1 i g, nd discriminnt 71 D 1 = g k g 2g 2 αi α 2 j = k gg 1 D k k i<j i<j Further, if ϱ 1 d denotes the number of solutions mod d of the congruence Gm 0 mod d, it is esil verified tht 72 ϱ 1 = 0 if k since k, P = 1 nd ϱ 1 α = ϱ α if k We deduce from 71 nd 72 tht D 1 = σ = σ = σ D 1 ϱ 1 0 σ D 1, k ϱ 1 0 σ D, k ϱ 1 0 σ = σ D, k ϱ 0 Hence D 1 D so tht the rime ower divisors of D 1 re mongst those of D Also note tht 72 imlies tht Gm hs no fixed rime divisors It remins to check the other hotheses of the Theorem The condition α1 x k k is stisfied for suitbl smll α 1 > 0 since k 1 β nd x α Further, since G k g P nd x c 1 P δ, we hve x c G δ 1 for suitbl smll δ 1 > 0 The Theorem now ields tht f P n n mod k k <x /k 1 ϱ 1 ex x /k σ fϱ 1

13 Using 72 we simlif this to 1 ϱ k k <x /k k <x /k k 1 ϱ/ k 1 ϱ/ <x k Multilictive functions 269 ex <x /k <x /k x /k k 1 ϱ fϱ ex 1 ϱ ex x k x k fϱ fϱ Since x /k x β, we cn el to Lemm 2i to finll obtin the bound k 1 ϱ fϱ ex 1 ϱ/ <x <x x k k s required k k1 ϱ/ <x 1 ϱ ex x k fϱ References [1] F Delmer, Sur l somme de diviseurs k x {d[fk]}s, C R Acd Sci Pris Sér A-B , A849 A852 x [2] P Erdős, On the sum dfk, J London Mth Soc , 7 15 k=1 [3] H Hlberstm nd H- E Richert, Sieve Methods, Acdemic Press, London 1974 [4] T Ngell, Introduction to Number Theor, 2nd ed, Chelse, New York 1964 [5] P S h i u, A Brun Titchmrsh theorem for multilictive functions, J Reine Angew Mth , [6] D W o l k e, Multiliktive Funktionen uf schnell wchsenden Folgen, ibid , DEPARTMENT OF MATHEMATICS UNIVERSITY OF GLASGOW GLASGOW G12 8QW SCOTLAND, UK Received on nd in revised form on