An asymptotic estimate related to Selberg's sieve

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JOURNAL OF NUMBER THEORY lo, 83-94 (1978)
An Asymptotic Estimate Related to Selberg’s Sieve
S. GRAHAM
Department of Mathematics, University of Michigan, Ann Arbor Michigan 48109
Communicated by D. J. Lewis
Received June 20, 1977
Suppose 1 < z, < zs < N, and let A*(d) = r(d) max(log&/d), 0) for i = 1,2. We show that
We then use this to improve a result of Barban-Vehov which has applications to zero-density theorems.
Let A&) denote a real-valued arithmetic function with the property that )b(l) = 1, A&I) = 0 for d > z0 . If all prime divisors of a number n exceed z0 , then Cdl,, X,(d) = 1, so consequently
The quadratic form on the right is asymptotically minimized, subject to the constraints on A,, , by taking
(1)
But it is well known [5] that
c CGYd I - - log Q + O(1);
qgo (P(q) (2)
hence the choice (1) suggests that we might instead take
u4 = P(4oog(zo/4~l(log zox
Indeed, with either of these choices for A,, , it is not difficult to show that
83 0022-314X/78/0101-0083~2.00/0
Copyright 0 1978 by Academic Pray, Inc. AU rights of reproduction in any form reserved.

84 S. GRAHAM
uniformly in A4, and we thus obtain an upper bound for n(N + M) - v(M). This estimate becomes imprecise if z,, exceeds N1/2, and this seems to be in the nature of things, but when M = 0 we obtain an asymptotic expansion which is sharp for all values of z. .
We suppose throughout that 1 < z, < z, , and we let
44 = l-44 10&i/4 if d<zi, ZTZ 0 if d>z,. (3)
THEOREM. Let 1 < z1 < z2 < N. Then
lzGN ( gfl 4(4)( c A2(e)) = N 1% 21 + *(N). 8112
The supposition z2 < N occasions no real loss of generality, since the value of Cdl,, A@) is independent of z, when 1 < n < zi .
If z, < 2, , we define
Ad = V2(4 - 441/h&2/z1)
= A4 if 1 <d < z,,
= ~(d)oog(z2/d)/~os(z2/23> if z, < d < zg , (4)
ZZ 0 if d>z,.
Barban and Vehov [l] proved
(5)
Their proof was sketchy, but Motohashi [6] has supplied the details. We use the above theorem to prove the following
COROLLARY. Suppose 1 < z, < z, . If z2 ~2 N, then
,<-N cg Xd2 = log&z,) + * ( log2&l) ). 2
If z, < N -=c z, , then
The estimate (5) has been used by Motohashi [7] and Jutila [4] to prove zero-density theorems for L-functions which are sensitive near u = 1. In a later paper, we use the corollary to prove a quantitative form of Jutila’s

ASYMPTOTIC ESTIMATE 85
result, and thus obtain a new estimate for the constant in Linnik’s theorem on the least prime in an arithmetic progression.
We employ the usual notation for the classical arithmetic functions, and in addition we put
Note that without subscripts, A(n) and n(n) denote the arithmetic functions of Liouville and von MangoIdt, respectively.
2. PREPARATORY LEMMAS
At various points we appeal to weak quantitative forms of the prime number theorem. For convenience, we present them in advance.
For any A > 0,
c ~(d <A Q(log 2Q)-A, q<Q
c CL(~) 4-l <A (log 2Q)-A, q<Q
c p(4) b3 4 = -1 + O&x 2Q)-3, Q<Q 4
(6)
(7)
(8)
c log!’ = Q + O,(Q(log 2Q)-9 (9) PC:0
LEMMA 1. If a -=c 0 then
Bellman [2] and Halberstam [3] have proved results which are similar but much stronger. Here we give a simple proof which is sufficient for our purposes.
ProaS. The first inequality is trivial. For the second inequality, we have
qFQ 0,2(q) = qFQ dsq duea < Q c daW7 el” d,e < Q C [d, e]-l+a < Q c d2(4 ++a -& Q-
d.e n

86 S. GRAHAM
LEMMA 2. For any integer r and any A > 0,
c e log (+) = & + OA(C1,,(r) log-A 2Q), (10) x0
m.r1-1
c s ~~!it (+) = ; + + OA(u-,,,(r) log-A 2Q), (11) ,r3&
Proof: Let
c A4 ,?z%
-+j <A w&) lwA 2Q.
.@={(d:p[d=>pIr).
We see easily that
dg Ma3 = ~(4 if (4 r) = 1, db = 0 if (n,r) > 1.
Hence
c ,9x2
+og(+) = c ; c %log(-$). de mg0/4
By (7) and (a), the above is
= d; d-l + 0 (d; d-l) + 0~ (dg d-l WA(y)). 00 d<Q
Here the fist term is =r/‘lcp(r). The first error term is
= Q-W n (1 + (pW - 1)-l)
wr
< (log 2QP cwdr). The last inequality follows from the fact that
1 + (JF - 1)-f < 1 + p-l/S

ASYMPTOTIC ESTIMATE 87
for all but Cnitely many p. In the second error term, we consider separately those d < Q1lz and Q112 < d < Q. In the first place,
while on the other hand,
z9 < Q-lj6 & d-2/3 <A (log 2QY u--1,2(r).
To prove (1 l), we use the identity
This implies
By (lo), this is
(d.rkl
= -& IJ (1 - P-Y + Tw’r
+ OA (u-1,2(r) 1 d-3/2 log-A (F)) a<0
K(I) 73 = p x + Odu-l12(r) lwA 2Q>,
and this proves (11).

88 S. GRAHAM
To prove (12), we note that for any c > 1,
- c i.@ log (%). ycII K(n)
(14)
We take c = 1 + log-A 2Q. By (11) (with A replaced by 2A) the difference between the first two sums on the right-hand side of (14) is
The third sum is
‘g (log c) c n-1 < log2 c. o<n<oo Since log c < log-A 2Q, this completes the proof of (12).
LEMMA 3. For any integer r,
c ,9%&
P’@) = $ & Q 4 O(Q”“u-l,,(r)).
Proof. Let 9 be as in (13). We see easily that
aTB 44 pYm> = p”(n) if h r) = 1, dnz==n zzz 0 if (n,r) > 1.
Hence the expression in the lemma is
= a2 wI~,a P2(d
=$QZ as.9
$) + 0 (Q1i2 aL cW2)
= f --& Q + 0 (Q112 n (1 + W2 - W')) PIP
=- z2 -& Q + O(Q"2~-m(r>).

ASYMPTOTIC ESTIMATE 89
3. PROOF OF THE THEOREM FOR z1 z2 <N
It is clearly immaterial to the result whether the range of summation is 1 < it < N or 1 < n < N; we work with the former range, since it allows a slight simplification in the proof.
If qz, < N, then
The error term is
(16)
For the main term, we have
= 1 48 n2(4 c +,) d.e de rj(d.e)
= c P2(‘) v(r)
TSZl r2 (1
m3lP * 1% ($))( & 9 1% ($)).
(m.r)=l (n,r)=l
By (lo), the above is
= v2(r) 94.) c l-2 ‘<*I 1% + 0 (~420 lo!T2 ($q”
= )-p&f+ o( c 2+log-2($L)) ‘<Zl
+0(X %%
~21:@) log-2 (+j).
The first error term is clearly majorized by the second, which is
Q zy’ C k-2 - 2k C k
e1,2m> z-‘2, < r<Z-~+lr,
(17)

90
and by Lemma 1, this is
S. GRAHAM
<zk-2<l. k
The theorem in the case z,z2 < N follows from (2) and (15)-(18).
(18)
4. THE CASE z1 z, > N
The estimate of the remainder term given in (16) does not allow one to use the proof of the previous section when N = o(zlze). Indeed, some well- known examples of Selberg suggest that one must use special properties of the integers if one is to extend the theorem to this case.
We will make essential use of the relation
C p(d) log (5) = A(n) if n > 1, dP
= logz if n=l.
It follows that for n > 1,
This implies
+ l~<NA(n) I & p 6) log (-&, + g /J (5) log kg WzlZ~ e<*lza
+ &( g I-1 GPog (*))I ; lJ (:)log($)l- (19) a<n/E, s<nps
By (9) and partial summation, the first sum on the right-hand side of (19) is =Nlog N + O(N). In the second sum, we may write 12 = pk.

ASYMPTOTIC ESTIMATE 91
For such it, we find that
2 ~($)log($--) = --log (5) if zi <A 1
d<nlzi = 0 a if p < zi .
Thus the second sum on the right-hand side of (19) is
= - zl;<N w%P) (1% (5)) - z z<N (logp) (log (5,) 2 .
+ O(N112 log 2N);
here the contribution of those n = pk with k > 1 has been absorbed into the error term. By (9) and partial summation, the above is
=N log(z,z,/N2) + O(N).
Thus to complete the proof of the theorem, it suffices to show that
= Nh (+) + O(N).
Let (4 e) = q, d = qr, e = qt (here (r, t) = l), and n = mqrt. The left- hand side of (20) is
We apply Lemma 3 and partial summation to the innermost sum. If w = max(zJt, zz/r) and y = min(zJt, z,/r), this sum is
(Here and below e represents Napier’s constant.)
641/10/x-7

92 s. CmAHAhf
Thus the expression in (21) may be written as
= NZ., + q&z + z&c + O(W12&),
say. Using (11) and (12) for the innermost sum in CA , we have
Using (7), (8), Lemma 1, and partial summation, we obtain
2” = Q<;,z2 4-l + 0 6<;,. 4-l lw2 ($)) = log (+) + O(1). (23) x a
To estimate & , we note that (11) and (12) imply
zBgcI c q@N/% ?J%<+@N/a~,
P(r) 1
+ Q<;,s2 ze,s <;N,Qz I 1 a-1~2w10g-2 (-+I.
For the first sum we use (6); for the second sum we use Lemma 1 and partial summation. We get

ASYMPTOTIC ESTIMATE 93
A similar procedure shows that
To estimate ZD , we first note that Lemma 1 implies
n;Q n-1/2a-l,3(n) log (+) < Q-lj2 c k2k/2 c ~-1/3W k Qrk<n<Q2-“+’
< Q112 c k2-kla < Q112. k
It follows that
zD << N’~2(N/z,~,)‘~a c q-3/2 < N112, Q
(26)
since N < zIz, . To prove (20), we combine (21) through (26). This completes the proof of
the theorem.
5. PROOF OF THE COROLLARY
BY (413
+ ,xI, ( c nncq2 eln
es22
- 2 XN ( zn A@))( c 44). , An
dS% es3
The right-hand side of the above is unchanged if min(N, z2) is substituted for z2 . This substitution, together with three applications of the theorem, proves the corollary.
ACKNOWLEDGMENTS
The author gratefully acknowledge the considerable assistance of Professor H. L. Montgomery with the content and presentation of this paper. He also wishes to thank Professor D. J. Lewis for his suggestions concerning style.

94 S. GRAHAM
REFERENCES
1. M. B. BARBAN AND P. P. VEHOV, On an extremal problem, Trudy Moscov. Ob.% 18 (1968), 83-90. See also: Trans. Moscow Math. Sot. 18 (1968), 91-99.
2. R. BELLMAN, Ramanujan sums and the average value of arithmetic functions, Duke Math. J. 17 (1950). 15%168.
3. H. HALBERSTAM, Four asymptotic formulae in the theory of numbers, J. London Math. Sot. 24 (1949), 13-21.
4. M. JUTILA, On Liik’s density theorem, to appear. 5. J. H. VAN LINT AND H. E. RICHERT, uber die Summe C p*(n)/&), NederL Akud.
Wetensch. Proc. Ser. A 67 (1964) (In&g. Math. 26 (1964), 582-587. 6. Y. Moromwn, On a problem in the theory of sieve methods (in Japanese), Res. Inst.
Moth. Sci. Kyoto Univ. Ktikytiroko 222 (1974), 9-50. 7. Y. MOTOIIASHI, On a density theorem of Linnik, Proc. Japan Acud., 51(1975), 815-817.