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1 Math-Net.Ru All Russian mathematical portal A. Perelli, A. Zaccagnini, On the sum of a prime a k-th power, Izv. RAN. Ser. Mat., 1995, Volume 59, Issue 1, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read agreed to these terms of use Download details: IP: December 23, 2018, 12:22:45

2 СЕРИЯ МАТЕМАТИЧЕСКАЯ Том 59, 1,1995 УДК 511 A. Perelli, A. Zaccagnini On the sum of a prime a fc-th power The distribution of Hardy-Littlewood numbers in short intervals is considered. Bibliography: 13 titles. 1. Introduction Let к ^ 2 be a fixed integer p denote a prime. We consider the problem of the representation of integers n as p -f m fc, m is an integer. Hence we may restrict our attention to those n for which the polynomial x k n is irreducible over Q. Denote by n-irr such n. For any n-irr let R k (n)= A(h), д(п,р) = Qk{n,p) {m(modp): m n = O(modp) } etw. n (i-«^)-no-^)/0-i) the infinite products being convergent by the prime ideal theorem. It is an easy consequence of Eisenstein's irreducibility criterion that x h n is irreducible unless n is powerfull. Hence, from the well-known asymptotic formula with remainder for the counting function of the powerfull integers, see e.g. Ivic [I], Theorem 14.4, we have that if N is sufficiently large H ^ N then with One expects that {n e[n,n + H], n-irr } = [N,N + H)\ R(N, Я, к) ^(^я^жя 1 / 2^^1/ 6. (i) R k (n) - г^'ь&^п) (2) as n -» oo over n-irr. Let L = log JV. In the case к = 2, Perelli-Pintz [P-P] showed that (2) holds for all n [N,N+H] but О (HL~ A ) exceptions, provided Я ^ iv 1 / 2 * 6, A > 0 is any constant. In this paper, we extend the method used in [P-P] to prove an analogous result for every к ^ 3. There are two difficulties in doing this. The first one lies in the treatment of the singular series, we use a recent result of Nair-Perelli [N-P], while the second lies in the treatment of the minor arcs, which requires the use of a variant of WeyFs inequality, see the Lemma below. A. PERELLI, A. ZACCAGNINI 1995

3 186 A. PERELLI, A. ZACCAGNINI In order to state our results, we need to introduce some further notation. Let Rt(n) = Rl(n,N,Y)= Л(Л), h-\-m k =n N-Y^h^N Y/2^m fe <3/2Y P fc» = P (n,n,y) = i Our main result is the following h-\-m=n N-Y^h^N Y/2<rn<3/2Y ml/fc_1 - THEOREM 1. Let к ^ 3, e,a > 0, iva+ ^Y ^N max (yw+^nrm ^Я^У. Tften 2 Л1Е(п) - P fc *(n)6 fc (n) 2 < е, А * ЯУ 2 / & ~ л. n-irr We note that the constant in the <C-symbol is ineffective, due to the use of the Siegel-Brauer theorem in the proof. We also note that a more natural lower bound for H in Theorem 1 would be Я > yi-i/fc+e^ Indeed, the weaker bound stated in Theorem 1 is due to the condition Я ^ jsji/2+e n Theorem 1 of Nair-Perelli [N-P]. More precisely, this condition comes from the use of a high-moment large sieve inequality in the first part of the proof of Theorem 1 of [N-P], which in this paper is used to approximate the partial sum of the singular series, arising from the major arcs, with the full singular series. From Theorem 1 we get the following COROLLARY. Let к ^ 3, e, A > 0, JVn +e ^Y ^N max fy 1_ i +,ivh^ ^ H ^ У. Then for all n [N,N + Я] but 0(HL~ A ) Rt(n) = P^(n)G k (n) + 0(Y l ' k L- A ). exceptions we have Moreover, if N& l -\)+e ^ # ^ N, then all n E [N,N + Я] but 0(НЬ~ А ) exceptions are a sum of a prime a k-th power. In a previous version of this paper we had the weaker bound Я ^ max (Ni2^1~'k^ 1 jy-i/2+e^ n ^e seconc i assertion of the Corollary. We wish to thank Dr. K. Kawada for pointing out that the condition Я ^ jv 1 / 2 * 6, coming from our treatment of the singular series, could be dropped by a nice combination of the arguments leading to (19) below those in sect. 4 of Plaksin [P]. We note that in the range М г ~ 1 / к + е < Я < TV, the proof of Theorem 1 also gives the asymptotic formula R k (n) = n^k6 k (n) + o(n 1 ' k L- A> )

4 ON THE SUM OF A PRIME AND A fc-th POWER 187 for all n e [N,N + H] but 0(HL~ A ) exceptions. Moreover, the N Y e in the above results may be replaced by a suitable power of L, at the cost of some complications in details. We also note that the cirguments used in the proof of Theorem 1 allow to treat the problem of the representation of nq as a sum of a prime a fc-th power, n [iv, N -f H] with exceptional set 0(HL~ A ), giving completely analogous results with a suitable uniformity in q. Let Ek (N) = {N ^ n ^ 2N: n is not the sum of a prime a fc-th power}. Our second result is a conditional bound for Ek{N) under the Generalized Riemann Hypothesis (GRH). The estimate E k {N) < М г ~^к\ with some 5(k) > 0, has been independently proved by A. I. Vinogradov [Vi] Brunner-Perelli-Pintz [B-P-P] for к = 2 by Zaccagnini [Z] for к ^ 3. Vinogradov states that, under GRH, E 2 {N) < N%+ E 3 (N) < N$+, although the method actually gives E 2 (N) < N l / 2 + e. THEOREM 2. Assume GRH let к ^ 2, e > 0. Then К = 2 k ~ x. E fc (iv)«fc, iv 1 -w+, We note that the N factor can be replaced by a suitable power of L. In the case к = 2, Mikawa [M] obtained, under GRH, the factor L 5, this can be reduced to L 3+e by refining the arguments. Theorem 2 is obtained using the method of Hardy-Littlewood [H-L] Weyl's inequality, by a suitable treatment of the singular series. The bound given in Theorem 2 is not the best that we can get for larger values of k. Indeed, using I. M. Vinogradov's results (see e.g. Theorem 5.3 of Vaughan [Va]) instead of WeyFs inequality, we can replace the exponent in Theorem 2 by 1 ^^ fc +, с > 0 is a suitable constant. However, we are unable to obtain the "expected" exponent 1 1/k -I- e. 2. Proof of Theorem 1 Throughout the paper we will denote by с a suitable positive constant, depending at most on fc, whose value will not necessarily be the same at each occurrence. Our method for the proof of Theorem 1 is based on the techniques in Perelli-Pintz [P-P] Nair-Perelli [N-P]. We will also use some arguments in Zaccagnini [Z]. Hence we wiu refer to these papers whenever possible, giving only a brief sketch of the needed changes. We may assume that e > 0 is sufficiently small, N is sufficiently large, JV, Y H are integers that H = y 1 " 1 /^6.

5 188 A PERELLI, A. ZACCAGNINI Outline of the method. Let В > 0 be a suitable constant which will be chosen later in terms of A k. Let P = L B, Q = HL~ B, Tl(q 1 a) be the Farey arc with centre at a/q of the Farey dissection of order Q, let aw(9,o) = {^ + 4,M<^}. Clearly, m'(q, а) С 9Я(<7> a). Let M = Y x l 2k set g Ш= U U 9Jl '(«' a )' m: I I Q' Q \SW, /(a) = J <p(q) \ q a < *y-, (a,q) = l,p<g< M; 0 otherwise, * means that (a, g) = 1. We have т = E e W N-Y<n<N fl (n)-p fc»6 fc (n) 2 «N^n^N-hH n-irr n-irr «E / S(a)F k (a)e(-na)da+ f I(a)F k (a)e{-na)da-p^n)& k {n) n-irr 1ГГ + E /(5(a)-/(a))F fe (a)e(-na)da = E OT + E m > < 3 ) N<n<N+H say, Hence S(a) = 5(a, W, У) = ]Г A(n) e(ne) iv-y<n<iv F fc (a) = F fc (a,y)= J] e{m k a). Y/2^.m k^3/2y = [ (S(0 - I(0)F k (0 I (S(a) - 1(a)) F k (a)k(a - t)dad (4) with iv^n^iv-f-я

6 ON THE SUM OF A PRIME AND A fc-th POWER 189 Now we proceed as in sect. 2 of [P-P]. Subdividing the interval [l/q, 1 -f 1/Q] into H adjacent intervals Ij of length 1/Я, from (4) the Parseval identity we get Y <HYL 2 max [ \F k (a)\ 2 da. (5) If a Ij П m a fflt(q, a) for some P < q ^ Q (a, q) = 1, then a a/g ^ 1/Я hence Ij П m С (f - ^ f + ; 0- Otherwise, since for Ф ^ we get Я Я ^ -K, we have Ij П m С 3H(g, a) \ 9Я'(д, a) for some q ^ P (a, g) = 1. Thus, by (3) (5), Theorem 1 is an immediate consequence of the following estimates У" < HY 2 > k L~ A, (6) max Р<Я<Я1- J-2/H\ 2 /Н \<l J {a,q)=l dt) < y2/fe-ll"^-2 (7) provided N 7 / 12+ < Y < N. max JLB/Y\ я^р JL B \Q J /Y d V < y2/fc-l L -A- 2> The major arcs. We first treat the integral over ffl in Еда- From theorems of [Va] we have that F k [-+V V k {q,o) f k (r]) + A k {rj,q,a) (9) Q (8) Л(") = E 1/fc_1 e(?), F fc ( 9,a) = e (m fe^) «g 1 " 1 Y/2<m<3Y/2 m=l ^ ^' Writing (10) (11) a=l ^ ^ ' using (9)-(ll) instead of (9)-(ll) of [P-P], arguing as in sect.3 of [P-P] we get / S(a)F k (a)e(~na)da = P (n) М-Н к { Ъ п) V9<P

7 190 A. PERELLI, A. ZACCAGNINI By lemmas of [Z] the Cauchy-Schwarz inequality, the first error term in (12) is < LY 1 ' 2 L- B ' 2 Y 1 / k ~- 1 ' 2 < yi/*l-(b-2)/ 2> (13) since Hence from (12) (13) we get / S{a)F k {a)e{-nol)dol = Р (п)в к (щр) + o(y 1 l k L~' A l 2 ) (14) provided В ^ сд (15) & k (n,x)= T JM-H k (q,n). The singular series. Arguing as at the beginning of sect. 4 of [P-P], by (11) the arguments used to obtain (13) we get JO P<q<^M q^q' + О (Y ^L k ~ cb ) + О (M 3 / 2+ L 3B / 2 ). (16) Hence by the choice M = У г 1 2к from (14) (16) we have that / S(a)Fk(a)e(~~na)da + / I(a)Fk{a)e(-na)da Jm Jo = P k *(n)g k (n,m) + 0(Y 1 / k L- A / 2 ), (17) provided (15) holds. Now we proceed to approximate 6fc(n, M) by &k(n) for n 6 [iv, iv + Я], n-irr, with a suitable exceptional set. In order to use results which already appear explicitly in the literature, we first approximate ^(n, M) by ^^ А/Г \ " / using a short interval version of the arguments in sect. 12 of [Z], then we approximate Щ (7?, M) by 6к (п) using the results in [N-P]. However, we remark that suitable changes in the method used for the proof of Theorem 1 of [N-P] would allow us to obtain directly the approximation of 6fc(n, M) by &k(n). Write = 9{M) = {q e N: fi(q) ф 0 p\q => p < M }.

8 ON THE SUM OF A PRIME AND A fc-th POWER 191 Then Щ(п,М)-6*(п,М) = ]T A fc (g,n), (18) A k (q,n) = р\я q>m ^l[( e (n,p)-l). Choosing V = exp(l(logl) 3 / 2 ), A = L' 1 arguing as in (12.10)-(12.12) of [Z] we get J2\A k (q,n)\<l- A / 2. (19) qe@ In order to treat the sum over q G (M, V) we replace Lemma 8.2 of [Va] by Lemma 1 (11) of [N-P] (with Q = 1), argue as in sect. 12 of [Z]. In this way we obtain that for Я^Л^/з+е Y, Yl M<l,n) < HL~ 2B. (20) n-irr M<q<V Hence, from (18)-(20) we obtain that for n-irr, n [N,N + H}\ Si, SiK#L- B, (21) we have & k (n,m) = Щ(п, M) + 0(L~ B ). (22) For n-irr we have that Щ(»,М) = 6*(п)ехр[ 2 ^ i ){ Ш } - (23) From Theorem 1 of [N-P] (with <2 = 1), by partial summation we have that if Я ^ N 1 ' 2 ** then 2X e(d,p)-i <#io g - 4W x. (24) p-i AT^n^iV+tf'prsX n-irr Choosing X = 2 J M, j = 0, 1,..., arguing as for (33)-(35) of [N-P], from (23) (24) we obtain that for n-irr, n [N, N + H] \ S 2 U k (n,m) = e k (n){l + 0{L- B )}, again S 2 < HL~ B. From (21), (22), (25) (26) we see that 6 k (n,m) = 6 fc (n) + 0{L~ B (1 + 6 fc (n) )) (25) (26) (27)

9 192 A. PERELLI, A. ZACCAGNINI for n-irr, n E [N, N 4- H] \ S 3 From (17) (27) we have that v «Y^L-^ Y: is*wi 2 '9Л n-irr \S S \<HL~ B. (28) + F2/& E (l 6 *( n > M )! 2 + l 6^( n )l 2 ) + HY 2 ' k L~ A, (29) n-irr, ness provided (15) holds. From lemmas of [Z] we obtain that 6 fc (n,m)<l fc. (30) In order to treat the sums of б&(п) 2 in (29) we observe that &k(n) is closely connected with the residue Гк(п) of the Dedekind zeta-function (к п (з) at s = 1, K n = Q(a n ) a n is any root of a; fc n = 0. By a theorem of Dedekind we have that p \ D(n) implies g{n,p) = FK n (p), D(n) is the absolute value of the discriminant of K n Рк п (p) is the number of prime ideals ф of K n with N*p = p, N^ being the norm of ф. Since r fc (n) = 11 ( ( l - ^ *'-'1(п(^)) / l-i the infinite product being convergent by the prime ideal theorem, for n-irr we have that.( )"- П (i-j^) П П ('-J ЫпГ*= (1-^r) I1--II/I1-- V deg^j>l JV<P=p x п (.-ly'-'/l! п.д,(по-й/(-^)^ <*» deg ф denotes the degree of *P lli is a suitable absolutely convergent product. Hence &k(n) = C k (n)r k (n)~ 1 (32) Cfc(n) is defined by (31). Since D(n) < n* 5 " 1, the number of ф with ]V*p = p is at most k g{n,p) ^ fc, we see that Cfc(n) satisfies ir^cfcfaxl / (33)

10 ON THE SUM OF A PRIME AND A fc-th POWER 193 for n [N, N + H] n-irr. By Lemma 3 of [N-P] there exists a constant с such that the inequality /?>1 holds for at most one simple real zero /3 among the real zeros of the functions (к п (s) with n [JV, N + Я]^ n-irr. In view of (32), denote by n-good any n G [N,N + H], n-irr, such that (к п (/?) ^ 0 5 by n-bad otherwise. From a well-known extension of Hecke^s theorem to algebraic number fields, we have that r k (n) > 6(n)(l - P) + 1-5(n) (34) S(n) { 1 for n-bad 0 for n-good. From (32)-(34) lemmas 4, 7 8 of [N-P] we obtain that 6 fe (n) < I 2 for n»good, (35) &k(n) < N /s for n-bad (36) ]T 1 < N 1 ' 2^^. (37) n-irr, n-bad Hence from (29), (30) (35)-(37), for Я ^ N l l 2+e we finally obtain V «Y 2 ' k HL- A, (38) provided (15) holds. Tie minor arcs. We first prove (8). Writing m = # + Z, by the binomial theorem we get F k ( -+17 = e(l4-+4 e(/*,/w), (39) (Y/2-l) 1 / k /q^t^(sy/2-l) 1 / k /q / M ( ) = т? E (5)(^) j^" J '- Hence аду 1-1^ < j/^( ) < wyi-ia i=i l/ib,zl ^ h s " lce ^ ^ dj e Thus (39) lemmas of Titchmarsh [T] give fl/(qq) 1 / fl \ dt7<y 2 / fc - 1 L- B. (40) T Серия математическая, N*l

11 194 A PERELLI, A ZACCAGNINI Now we turn to the proof of (7). By Gallagher's lemma (see [G]) we have, writing / = [x,x + %\, that m k l dx. (41) Writing d = m nwe get E \m k ei e m m = E«((- fc - n fc >f) E>>. m k /n k / О^^КЯ/У 1 " 1 /* (42) k 1 P(n,d)= E O'd*-'. From (41) (42) we have that E,W= E e(p { n, d )^ n k,(n+d) k el J-2/H\ \Я J dr]<h -2 у \у2 UF 2^- 1 l^d^h/y 1-1 / 1 * (43) E d = E e(^^)^fo.f-ec) nj H У х = (^ + f ) 1/fc, F 2 = (2F - f ) 1/fc. Hence by Abel's inequality we get E e(p(n, rf )^) < Я max <C H max ^ d Vl (44) We will need the following Lemma. The estimate provided by the Lemma allows us to save only a power of L for each P < q ^ Q. However, by a better treatment of the divisor function which arises in the proof, it is possible to obtain a version of the Lemma which allows to save a power of N when N ^ q ^ Q 1 " 6. LEMMA. Let F(x,y) = x 9 y + ]C?=o bj(y)x J g ^ 2 is a fixed integer Ь 3 (у) are real-valued functions. q ^ N we have E l<d<t J2 n<r e(af{n,d)) Let D > 0 is any constant К = 2 9 *. a- ^\ ^ -\ (a,q) = 1. Then for T, R, 1/K «gtd TR ( i ^ + L-2D +9 '+2g-lJ L(D + l)/k

12 ON THE SUM OF A PRIME AND A fc-th POWER 195 PROOF. We follow the ideas in the proof of Weyl's inequality. By Holder's inequality we get E n<r e(af{n,d)) < г 1 -*/*^ 1/K (45) E x = E E e KM) к By Lemma 2.3 of [Va] we have that E,«R K-g Yl 1E1 '" Yl Щ2 l^d^t h x \hikr g-l \h g -i\^r nelg-l e ( h l'-- h g-l p g-l( n '> h l>'--i h g--l> d )) P g -i(x; hi,..., V-ь rf ) = g\ adx + 3^L(hi + + /^-i) + (# - 1)! ab g -i(d) J^_i С [1, Д]. Since the terms with dh\ h g -\ = 0 contribute <C TR 9 " 1, we get ~~ (46) ^ h=l 'n / 9 -i Given any Z? > 0, the contribution to Si of the /i's with T 9 +I(/I) > L is g\tr g - ^RK- g +i L -D J2 T g+1 (h) 2 <TR K L- D +^+^. (47) b=l The contribution of the /i's with r^+i(/i) ^ L D is, by Lemma 2.2 of [Va], f rp ng 1 <**-.*» ' g mi (Si,in,-.) <TB* (i +1 + jij) -', (48) the Lemma follows from (45)-(48). Now we apply the Lemma with g = k 1, a = fca/g, F(x,y) = P(x 1 y) 1 T = H/Y 1 ~ 1 / k 1 R = у > = Б + fc 2-1. Observing that jfea/g = a'/q' for some g' > q (a ;, <?') = 1, from (43), (44) the Lemma we finally obtain that J-2/H\ \Я J dn<y 2 ' k - 1 L- B ' 2h -\ (49) Choosing В = ca, we see that (15) is satisfied, Theorem 1 follows from (38), (40) (49). 7*

13 196 A. PERELLI, A. ZACCAGNINI 3. Proof of the Corollary Since A is an arbitrary constant, the first assertion of the Corollary is an easy consequence of Theorem 1 (1). The second assertion of the Corollary is not, strictly speaking, a consequence of Theorem 1. However, writing Y = JV 7 / 12+, M = Y 1^2^ Г к( П ) = ]C l0 P ' N-Y^p^N Y/2^m fc <3/2Y by the same techniques we used in sect. 2 to prove (17) the estimate m «HY 2 < k L~ A, we obtain run) = PUn)&k(n 1 M) + 0(Y 1 / k L" A ) for all n [N, N + H] but 0(HL~ A ) exceptions, now H > yi-i/m-e. Hence in order to prove the Corollary it suffices to show that 6*(n,M)»L- c (50) under the same conditions on n. In contrast to sect. 2 we approximate &k{n,m) with Щ(га,Д), R = ехр(1 г / 2 ). Since *(п,м)-щ(п,д) = ^ Afc(g,n) fc fa,n) = Z 2 ' R<q^M q>m q$@(r) qe@(r) (51) say, arguing as for (19) with A = L x / 2 we get ^«expj-cl 1^}. (52) From the well-known formula g(n,p)-l= 53 XW, X(modp) x fc =xo XT^XQ see e. g. sect. 4.2 of [Va], for square-free q we obtain that Ц{д(щр)-г) = 53 x(n) р\я xec(q)

14 ON THE SUM OF A PRIME AND A k-тш POWER 197 C(q) = {x(modg), x primitive x k = Xo} \C(q)\ ^ (к 1) ш ( д \ Hence Е Нг-~=Ч И- = E E ^Ш E E E»*<»> Since X1X2 7^ Xo whenever gi ^ g2 5 by the Polya-Vinogradov inequality we get N<n<N+H <** E S c <«>i 2+L (f(q) ( E ^va^)i 2 R<q^M KR<q^M 4>{q) < ^ + м]х с <Яехр(-сЬ 1 / 2 ) 0 (53) Since Щ(п, R) > L~~ c, (50) follows from (51), (52) (53). 4 0 Proof of Theorem 2 Choose Q = cn 1 " 1^, P = М г / к Ь~~ с set m= (J y*a%,a), q^pa=l m: 1 ^ 1 \m f(g,<0 = {^+т7 ЮГ(д,а)} Write S(a) = ]P H n ) e ( na ) n^2iv an d Fk(a) = ^ e(m fc a), m k^2n hence, with suitable adjustments in the range of summation, S[^+v)=^T(v) + R(v,q,a) Fk{U^ = Yh^h{n)+^k{wY

15 198 A. PERELLI, A. ZACCAGNINI The Hardy-Littlewood method. Clearly, if n ^ 2N we have Rk(n) =([ I) S{o)F k {a)e{-na) da + E ^ E * e ( - % ) /, r( 4 )A fc ( 4,,,e)e(-»i,)d, + / 5(a)F fc (a)e(-na) da = V + V 4- V +*«, (54) The contribution of Ei can be evaluated individually, arguing as for (13) we get ^ =n 1 ' k 6 k (n,p) + 0(iPQ)^2N l^^2l c y (55) By the Bessel inequality Theorem 4.1 of [Va] we have that I I 2 i q * fi/gq E E 2 «J2^E / 1п\ТШн(^а)\^<МР^. (56) From Bessel's Weyl's inequalities we obtain that N I 2 ^ * ri/sq Г E «E E / R{r}, q, a)f k j d?7 Е( Л '<?) = Е / i2( 4,g,o) 2 d 4. a=1./-1/9q By (8) of Kaczorowski-Perelli-Pintz [K-P-P] we have ^(i?,g)<ivl 2 /Q, hence дт-1+2/fc+e «11 pl-2/k (57) Again from the Bessel Weyl inequalities we get ]T / m 2 < sup F fe (a) 2 / \S(a) \ 2 da<n 1+ k-tnc+ e. (58) a m N<n<2N J 0

16 ON THE SUM OF A PRIME AND A fc-th POWER 199 From (54)-(58) we deduce that R k (n) = n^k6 k (n,p) + 0(V/ fc L~ c ) (59) for all n G [N,2N] but 0(N 1 ~'^K + ) constant. exceptions, с > 0 is a suitably large The singular series. As in sect. 2, we approximate &k(n, P) by Щ(п, P), but this time we need a better bound for the size of the exceptional set. Using the notation of sect. 2, with Sf S>(P), from (18) we have б (п,р)-щ(п,р) = J2 4(n,g)+ j4(n,g) = 5i + S 2 ( 6 ) P^q^V q>v qe@ qe@ say. By Lemma 8.2 of [Va] arguing as in sect. 12 of [Z] we obtain that ]T 5i < N 1+ P-^ < N 1 '^ +e n-irr hence 5i < L~ c (61) for all n-irr, n G [N, 2iV], but 0(N 1 ~2k +e ) exceptions. On the other h, arguing as for (19) we get S 2 < L~ c (62) for all n-irr, n G [N,2N]. From (60)-(62) we have that в к (щр) = Щ(п,Р) + 0{L- C ), (63) с > 0 is a suitably large constant, for all n-irr, n G [N, 2iV], but 0(М г ~2к + ) exceptions. From Lemma 14.2 of [Z] we get n fc (n,p)>l" fc, (64) Theorem 2 follows from (1), (59), (63) (64).

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