In this section we will exploit the flexibility afforded by our general os- cillation results to obtain refinements to previous results on the limitations to the equidistribution of primes. We stated in Theorem 1.5 (see Introduction) the result for primes in short intervals and we now state the analogous results for primes in arithmetic progressions.
Theorem 5.1. Letbe large and suppose thathas fewer than(log)1−ε prime divisors below log. Suppose that
(log)1+ε≤y≤exp(β
log/
2 log log)
for a certain absolute constant β > 0, and put x = y. Define for integers a coprime to
∆(x;, a) =
ϑ(x;, a)− x ϕ()
x ϕ().
There exist numbers x± in the interval (x, xyD/log(logy/log log)), and integers a± coprime to such that
∆(x+;, a+)≥y−δ(,y), and ∆(x−;, a−)≤ −y−δ(,y).
HereDis an absolute positive constant which depends only onε, andδ(ã,ã) is as in Theorem 1.5.
These bounds are 1 if y = (log)O(1), and y−τ(1+o(1)) if y = exp((log)τ) with 0 ≤ τ < 1/2. The corresponding result in [4, Th. A1],
gives the weaker bound y−(1+o(1))τ /(1−τ) (though our bound is obtained there assuming the Generalized Riemann Hypothesis). Our constraint on the small primes dividing is less restrictive than the corresponding condition there, though our localization of the x± values is worse (in [4] the x± values are localized in intervals (x/2,2x)).
Theorem 5.1 omits a thin set of moduli having very many small prime factors. We next give a weaker variant which includes all moduli.
Theorem 5.2. Let be large and suppose that (log)1+ε≤y≤exp(β
log/
log log)
for a certain absolute constant β > 0, and put x = y. There exist numbers x± in the interval (x, xyD/log(logy/log log)), and integersa± coprime to such that
∆(x+;, a+)≥ y−δ1(,y)
log log log and ∆(x−;, a−)≤ − y−δ1(,y) log log log, where δ1(x, y) = (log logy+O(1))/(log logx). Here D is an absolute positive constant which depends only on ε.
Theorem 5.2 should be compared with Theorem A2 of [4]. Our bound is y−τ(1+o(1)) ify= exp((log)τ) with 0< τ <1/2. The corresponding result in [4, Th. A2], gives a weaker bound y−(3+o(1))τ /(1+τ), though our localization of the x± values is again much worse.
To prove Theorems 1.5, 5.1 and 5.2 we require knowledge of the distribu- tion of primes in certain arithmetic progressions. We begin by describing such a result, which will be deduced as a consequence of a theorem of Gallagher [5].
For 1 ≤ j ≤ J := [logz/(2 log 2)], consider the dyadic intervals Ij = (z/2j, z/2j−1]. Let Pj denote a subset of the primes in Ij, and let πj denote the cardinality of Pj. We letQ denote the set of integers q with the following property: q = J
j=1qj and each qj is the product of exactly [πj/2] distinct primes inPj. It is clear that all the elements ofQare squarefree and lie below Q:=zjπj/2, and that|Q|=J
j=1
πj
[πj/2]
.
There is a constant c1 such that at most one primitive L-function with modulus q between √
T and T has a zero in the region σ >1−c1/logq, and
|t| ≤T. Further if this exceptional Siegel zero exists then it is real, simple and unique (see Chapter 14 in [2]). We call the modulus of such an exceptional character a Siegel modulus. Below Q there are log logQ Siegel moduli.
Denote these by ν1, ν2, . . . , ν, and for each select a prime divisor v1, . . . , v. Assume none of v1, . . . v belongs to ∪Jj=1Pj, which guarantees that there are no Siegel zeros for any of the moduli dwheredis a divisor of someq ∈ Q.
Proposition 5.3. Suppose that exp(√
logx) ≤ Q ≤ xb where b is a positive absolute constant, and letxexp(−√
logx)≤h≤x. Then 1
|Q|
q∈Q (a,q)=1max
ϑ(x+h;q, a)−ϑ(x;q, a)−h/ϕ(q) h/ϕ(q)
exp −α
√√logx logz
,
where α is a positive absolute constant.
Proof. If (a, q) = 1 then using the orthogonality of characters, we have
ϑ(x+h;q, a)−ϑ(x;q, a) = 1 ϕ(q)
χ (modq)
χ(a)
x+h
x
χ(p) logp
= 1
ϕ(q)
x≤p≤x+h (p,q)=1
logp+O 1 ϕ(q)
χ=χ0
x+h
x
χ(p) logp .
By the prime number theorem the first term above is h/ϕ(q){1 +O(exp(−c
logx))},
which has an acceptable error term. We now focus on estimating the second term on average.
Below the superscript∗will indicate a restriction to primitive characters.
Observe that
χ=χ0
x+h
x
χ(p) logp=
d|q d>1
∗
χ (modd)
x+h
x
χ(p) logp+O
p|qd
logp
=
d|q d>1
∗
χ (modd)
x+h
x
χ(p) logp+O(d(q) logq).
Thus
q∈Q
χ=χ0
x+h
x
χ(p) logp=
1<d≤Q
q∈Q d|q
1 ∗
χ (modd)
x+h
x
χ(p) logp+O(|Q|xε).
(5.1)
Observe that if d is to have any multiples in Q then we must have d= J
j=1dj with each of thedj being composed of at most [πj/2] distinct primes from Pj. Therefore
q∈Q d|q
1≤ J j=1
πj−ω(dj) [πj/2]−ω(dj)
≤ J j=1
2−ω(dj) πj
[πj/2]
= 2−ω(d)|Q| ≤2−loglogdz|Q|,
where ω(n) denotes the number of prime factors of n, and the final estimate follows since all prime divisors of d are below z and so ω(d) ≥ logd/logz.
From these remarks we see that the right side of (5.1) is
≤ |Q|
1<d≤Q
2−loglogdz ∗
χ (modd)
x+h
x
χ(p) logp+O(|Q|xε), (5.2)
where the on the sum over d indicates that the sum is restricted to d as above; note that such dhave no Siegel zeros.
Define J0 := [1,exp(√
logx)], andJk := (exp(√
logx)zk−1,exp(√
logx)zk] for 1 ≤ k ≤ [(logQ−√
logx)/logz]. Theorem 7 of Gallagher [5] implies that the contribution of terms d ∈ Jk (for k ≥ 1) is (for a positive absolute constantα)
≤ |Q|hexp −α2 logx
√logx+ (k+ 1) logz −
√logx+klogz 2 logz
h|Q|exp −α
√√2 logx logz
, and also that the contribution of the terms d∈J0 is |Q|hexp(−α√
logx).
Summing over kwe deduce that the quantity in (5.2) is |Q|hexp(−α
logx/
logz) which proves the proposition.
Corollary 5.4. Suppose that exp(√
logx) ≤Q≤xb where b is a pos- itive absolute constant. Select z to be the largest integer such that
p≤zp ≤ Q2−o(1). Then there exists an integer q ∈[Q1−c/log logQ, Q], whose prime fac- tors all lie in [√
z, z] with
p|q1/p1−ξ/logz ≥ eξ/(10ξ) for 1 ≤ ξ ≤ 23logz, such that
ϑ(2x;q, a)−ϑ(x;q, a)− x ϕ(q)
x
ϕ(q)exp −β
√logx
√log logx
, (5.3)
for all (a, q) = 1, where β is a positive absolute constant.
Proof. Whenz is chosen as above we havez∼2 logQ so that√
logx z logx. If z/2j < √
z then let Pj = ∅. If z ≥ z/2j ≥ √
z then let Pj be the set of primes in Ij, omitting the prime divisors v1, . . . v of Siegel moduli;
in these cases πj =π(z/2j−1)−π(z/2j) +O(log logx). By Proposition 5.3 we can selectqsatisfying (5.3), with ∼z/(2j+1log(z/2j)) prime factors in eachIj
wherej≤[logz/log 4], so that
p|q1/p1−ξ/logz ≥eξ/(10ξ) for 1≤ξ≤ 23logz.
Corollary 5.5. Fix 1/2 ≥ ε > 0. Suppose there is a large inte- ger with fewer than (log)1−ε prime divisors below log. Suppose also that
exp(3(log)1−ε)≤Q≤b,and that J is a positive integer ≤exp(√
log). Se- lect z to be the largest integer such that
p≤zp ≤Q2−o(1). Then there exists an integer q ∈ [Q1−c/log logQ, Q], coprime to whose prime factors all lie in [√
z, z] with
p|q1/p1−ξ/logz ≥eξ/(10ξ) for 1≤ξ ≤ 23logz,for which J
j=1 (a,q)=1max
ϑ((j+ 1)/2;q, a)−ϑ(j/2;q, a)−/2ϕ(q) /2ϕ(q)
Jexp −β
√log
√log log
, where β is a positive absolute constant.
Proof. If j > ε(logz)/10 letPj =∅. If 1≤j≤ε(logz)/10 let Pj be the set of primes in Ij, omitting the prime divisors v1, . . . v of Siegel moduli and any prime divisors of. Now, for 1≤j≤J replacex by j/2 and hby /2 in Proposition 5.3, and sum. This yields
J j=1
1
|Q|
q∈Q (a,q)=1max
ϑ((j+ 1)/2;q, a)−ϑ(j/2;q, a)−/2ϕ(q) /2ϕ(q)
Jexp −α
√log
√logz
. Thus we may choose aq as described in the result, proceeding as in the proof of Corollary 5.4.
Corollary 5.6. Given a large integer,letQ=b for sufficiently small b >0, and let J be a positive integer≤exp(√
log). Select z= 10 log. Then there exists an integer q∈[Q1−c/log logQ, Q],coprime to whose prime factors all lie in [z/2, z] with
p|q1/p1−ξ/logz beξ/logz for 1 ≤ ξ ≤ 23logz, for which
J j=1
(a,q)=1max
ϑ((j+ 1)/2;q, a)−ϑ(j/2;q, a)−/2ϕ(q) /2ϕ(q)
Jexp −β
√log
√log log
, where β is a positive absolute constant.
Proof. LetPj =∅forj >1, and letP1 be a set ofπ1 = [2blog/log log] primes in I1 = (z/2, z], omitting the prime divisors v1, . . . v of Siegel moduli and any prime divisors of; this is possible sinceI1 contains∼5 log/log log primes, and we are forced to omit at most ∼ log/log log. From here we proceed as in the proof of Corollary 5.5.
Proof of Theorem 1.5. Take Q =xb in Corollary 5.4 to obtain q which satisfies the hypotheses of Corollary 3.5. Let u:= logy/logzand select v± as in Corollary 3.5. We consider the Maier matrices M± with
(M±)r,s=
log(rq+s) ifrq+sis prime,
0 otherwise,
where x/q < r <2x/q and v± ≤s≤ v±+y. Let M± denote the sum of the entries ofM±. Using (5.3) to sum the entries in columnswe see that
M±= x
ϕ(q) 1 +O exp −β
√logx
√log logx
v±≤s≤v±+y (s,q)=1
1.
Letr+ denote the row inM+ whose sum is largest, and letx+:=qr++v+∈ (x,2x). Since there arex/q+O(1) rows, we have
ϑ(x++y)−ϑ(x+)≥ q
x(1 +O(x−1/2))M+,
and then Theorem 1.5 follows from the bounds in Corollary 3.5. The analogous argument works for M−.
Proof of Theorem 5.1. Take Q = b in Corollary 5.5 to obtain q which satisfies the hypothesis of Corollary 3.3. Let u := logy/logz and select U in Corollary 3.3 so that U(1−A/logU) = u; then put S± = [zU±], so that S±∈[zu, zu+Cu/logu]. We consider the Maier matrices M± with
(M±)r,s=
log(rq+s) ifrq+s is prime,
0 otherwise,
where R < r < 2R with R := [/(2q)] and 1 ≤ s≤ S±. Let M± denote the sum of the entries of M±. Using Corollary 5.5 to sum the entries in columns we see that
M±=
2ϕ(q) 1 +O exp −β
√logx
√log logx s≤S
± (s,q)=1
1.
Now, the sum of the entries in rowr equals 0 if (r, )>1; and equalsϑ(S±+ qr;, qr)−ϑ(qr;, qr) = ϑ(S±+qr;, qr) if (r, ) = 1, sinceqr < and qr is not prime. The number of integers r ∈ [R,2R] with (r, ) = 1 is Rϕ()/+ O(τ()) = ϕ()/(2q) +O(ε). Therefore, denoting by r+ the row for which ϑ(x+;, a+), withx+=S++qr+ and a+=qr+, is maximized, we obtain
ϑ(x+;, a+)≥ 2q
ϕ()(1 +O(−1/2))M+,
and then Theorem 5.1 follows from the bounds in Corollary 3.3. The analogous argument works for M−.
Proof of Theorem 5.2. We proceed exactly as in the proof of Theorem 5.1 but replacing the use of Corollary 3.3 by Corollary 3.4, and the use of Corollary 5.5 by Corollary 5.6.
6. Further examples 6a. Reduced residues. Letq be square-free. Writing
n≤x (n,q)=1 n≡a (mod)
1 =
d|q
à(d)
n≤x d|nn≡a (mod)
1,
we see that this is
=
0 if (a, q, )>1
x
φ(q/(q,))
q/(q,) +O(τ(q)) if (a, q, ) = 1.
(6.1)
Corollary 6.1. Letq be a large square-free number,which satisfies(1.3), and defineα:= (log logq)−1
p|q(logp)/pwithη= min(1/100, α/3). Then for η(logq)η/2 ≥u ≥5/η2 there exist intervals I± ⊂[q/4,3q/4] of length at least (logq)u such that
n∈I+ (n,q)=1
1≥ φ(q)
q |I+| 1 + exp −u
η(1 + 25η) log(2u/η3)
,
and
n∈I− (n,q)=1
1≤ φ(q)
q |I−| 1−exp −u
η(1 + 25η) log(2u/η3)
.
Deduction of Corollary 1.2. This follows immediately upon noting that z≤(logq)η and replacing u/ηby u.
Proof of Corollary 6.1. Take a(n) = 1 if n ≤ q with (n, q) = 1 and a(n) = 0 otherwise. Recall that η = min(α/3,1/100). Since q has at most logq/log logq prime factors larger than logq we see that
p|q p>logq
logp p ≤1.
Therefore from our assumption that
p|qlogp/p=αlog logqwe may conclude that there exists (logq)η ≤z≤(logq)/3 such that
p|q z1−η≤p≤z
1 p ≥η2. (6.2)
Take to be the product of the primes in [z1−η, z] which divide q, so that is a divisor of q and ≤ ez(1+o(1)) ≤ q13. Given (logq)η/2 ≥ u ≥ 5/η2 we obtain by (6.2) and Corollary 3.2 (we check readily that η≥z−1/10 using η ≥20 log log logq/log logq) that there exist points u± ∈[u, u(1 + 22η)] such that, withy± = [zu±],
(6.3a)
n≤y+ (n,)=1
1≥ 1 + exp −u(1 + 25η) log 2u η2
φ() y+, and
(6.3b)
n≤y− (n,)=1
1≤ 1−exp −u(1 + 25η) log 2u η2
φ() y−.
Consider now the “Maier matrices”M±whose (r, s)thentry is (R+r)+s with 1 ≤ r ≤ R and 1 ≤ s ≤ y±, and R = [q/(4)]. As usual we sum a(n) as n ranges over the elements of this matrix. Using (6.1), note that the sth column contributes 0 unless (s, ) = 1 in which case it contributes Rφ(q/)/(q/) +O(τ(q)). Thus the contribution of the matrix is
Rφ(q/)
q/ +O(qε)
s≤y± (s,)=1
1.
Corollary 6.1 follows immediately from (6.3 a,b).
Proof of Corollary 1.1. Let q =
p∈Pp; note that A, the set of in- tegers up to q without any prime factors from the set P, is a subset of [1, q] of density φ(P)/P = φ(q)/q, which is strictly < 1 by hypothesis. Let = √
z≤p≤z/3, p∈Pp so that ≤ q1−δ+o(1) for some fixed δ > 0, and apply the argument in our proof of Corollary 6.1 above. In place of Corollary 3.2 we appeal to Corollary 3.3 (within place ofq, since the hypothesis onP implies thatH(ξ)eξ/ξ), and see that for suitably largeu≤√
z there exist integers y±≥zu such that
n≤y+ (n,)=1
1≥ 1 + exp(−u(logu+ log logu+O(1)) φ()
y+,
and
n≤y− (n,)=1
1≤ 1−exp(−u(logu+ log logu+O(1)) φ()
y−. We conclude that for largeu≤√
zthere are intervalsI±⊂[q/4,3q/4] of length
≥zu such that
n∈I+ (n,q)=1
1≥ φ(q)
q |I+| 1 + exp(−u(logu+ log logu+O(1))
,
and
n∈I− (n,q)=1
1≤ φ(q)
q |I−| 1−exp(−u(logu+ log logu+O(1))
.
Finally, by Proposition 3.8 we find that we can takey±≤zu+2, provided that u≤(1−ε) log logz/log log logz.
From the “fundamental lemma” of sieve theory (see [7]), it follows that these estimates are essentially optimal.
Example 6. Take q =
p≤zp and A to be the integers less than q that are coprime to q. We show how to tweak A to obtain a set B ⊂ [1, q] such that the symmetric difference|(A\B)∪(B\A)|is small, but such thatBiswell distributed in short intervals.
Letk= [q/(logq)4] and divide [1, q] intokintervals [mh+ 1,(m+ 1)h] for 1≤m≤k, and h=q/k= (logq)4+O(1). For 1≤m≤k consider whether
mh+1≤n≤(m+1)h (n,q)=1
1−φ(q)
q h≤(logq)3, (6.4)
holds or does not hold. If (6.4) holds then take B ∩[mh+ 1,(m + 1)h] = A ∩[mh+ 1,(m+ 1)h]. Otherwise pick an arbitrary set of [φ(q)h/q] numbers in [mh+ 1,(m+ 1)h] and take that to beB ∩[mh+ 1,(m+ 1)h].
By construction we see that for any interval [x, x+y]⊂[1, q]
n∈B x≤n≤x+y
1 = φ(q)
q y+O φ(q)
q h+ y logq
. (6.5)
Thus Bis well distributed in short intervals of length y≥(logq)5, say.
Further note thatAand Bare quite close to each other. Indeed, from the theorem in Montgomery and Vaughan [13] we know that for integers r≥1,
m≤k
mh+1≤n≤(m+1)h (n,q)=1
1−φ(q) q h
2r
∼ (2r)!
2rr! q hφ(q) q
r . (6.6)
It follows that the number of values m for which (6.4) does not hold is r
q/(logq)2r. Therefore for anyr≥1
|(A\B)∪(B\A)| rq/(logq)r. (6.7)
By (6.1) and (6.7) we therefore see thatBis also well distributed in arithmetic progressionsa (mod) providedr (logq)r.
Now take=√
w≤p≤wp=e(1+o(1))wwithw≤zso that|q. Foru≤√ w but large, our usual Maier matrix argument then gives that one of the following statements holds:
(i) There exists y ∈ [q/4, q] and a (mod) such that, with δ((a, ) = 1) being 1 or 0 depending on whether (a, ) = 1 or not,
B(y;, a)−δ((a, ) = 1)φ(q/)
q y≥exp(−u(logu+ log logu+O(1)))φ(q/) q y.
(ii) There exists an interval [x, x+y]⊂[1, q] withy≥wu such that
x≤n≤x+y n∈B
1−φ(q)
q y≥exp(−u(logu+ log logu+O(1)))y.
But from (6.5) we see that case (ii) cannot hold if wu ≥ (logq)5 and if e−u(logu+log logu+O(1)) 1/logq. We conclude therefore that the distribu- tion of Bin arithmetic progressions is compromised, and that case (i) holds in this situation. In particular the expected asymptotic formula for B(y;, a) is false for some εexp((logq)ε).
Our argument also places restrictions on the uniformity with which Montgomery and Vaughan’s estimate (6.6) can hold. Given h and q with hφ(q)/q large, define η by q/φ(q) = (hφ(q)/q)η. We now show that if (6.6) holds thenr (log(hφ(q)/q))2+4η+o(1).
Fixε >0. Choose Lso that hφ(q)/q=L2+ε; then let u= (1−ε) logL/log logL and w= (logL)3+4η+4ε.
We follow the same argument as in Example 6, but replace “(logq)3” in (6.4) by “(hφ(q)/q)/L”, from which it follows that we replace “y/logq” in (6.5) by “y/L”, and that the upper bound in (6.7) is qh(2r/eLε)r. Case (ii) cannot hold in our range by (6.5). By the combinatorial sieve we know that |A(y;, a)−δ((a, ) = 1)φ(q/)y/q| 2π(z) so that, since case (i) holds,
|(A\B)∪(B\A)| φ(q)/L in our range. By (6.7) we deduce thatew+o(w)= L(q/φ(q))h(eLε/2r)rso thatr w/logL= (logL)2+4η+4ε, which implies the result.
6b. ‘Wirsing sequences’. LetP be a set of primes of logarithmic density α for a fixed number α∈(0,1); that is
p≤x p∈P
logp
p = (α+o(1)) logx,
as x → ∞. Let A be the set of integers not divisible by any prime in P and leta(n) = 1 ifn∈ Aand a(n) = 0 otherwise. Wirsing proved (see page 417 of [18]) that
A(x)∼ eγα
Γ(1−α)x
p≤x p∈P
1− 1 p
. (6.8)
Leth be the multiplicative function defined byh(p) = 0 ifp∈ P and h(p) = 1 if p /∈ P and take fq(a) = h((a, q)) and γq =
p|q,p∈P(1−1/p). Naturally we may expect that A(x;q, a)∼ fqγq(a)q A(x) and our work places restrictions on this asymptotic.
Letu≥max(e2/α, e100) be fixed. Then for largexwe see that the hypothe- ses of Theorem 2.4 are met with z = (logx)/3 and η = 1/logu. Combining Theorem 2.4 with (6.8) which shows that A(x)/x varies slowly (and there- fore equals (1 +o(1))A(y)/y for any y ∈ (x/4, x)), we attain the following conclusion.
Corollary 6.2. For fixed u ≥max(e2/α, e100) and large x there exists y ∈ (x/4, x) and an arithmetic progression a (mod) with ≤ x(3/logx)u such that
A(y;, a)−f(a)
γ A(y)exp(−u(logu+O(log logu)))A(y) φ().
Similarly using Theorem 2.5 with η= 1/logu and z= (13logx)M1 we ob- tain the following “uncertainty principle” showing that either the distribution ofA in arithmetic progressions with small moduli, or the distribution in short intervals must be compromised.
Corollary 6.3. Let u ≥ max(e2/α, e100) be fixed and write u = M N with bothM andN at least1. Then for each largexat least one of the following two statements is true.
(i) There exists y ∈(x/4, x) and an arithmetic progression a (modq) with q ≤exp((logx)M1) such that
A(y;q, a)−fq(a)
qγq A(y)exp(−u(logu+O(log logu)))A(y) φ(q). (ii) There exists y > (13logx)N and an interval (v, v+y) ⊂ (x/4, x) such
thatA(v+y)− A(v)−yA(v) v
exp(−u(logu+O(log logu)))yA(v) v . 6c. Sums of two squares and generalizations.
Example 3, revisited. We return to Balog and Wooley’s Example 3, the numbers that are sums of two squares. It is known that (see Lemma 2.1 of [1])
A(x;q, a) = fq(a)
qγq A(x) 1 +O
log 2q logx
1
5
. (6.9)
Take q to be the product of primes between √
logx and logx/log logx. Using the Maier matrix method, (6.9) and our Corollary 3.3 we obtain that for fixed u and large x there exist y± ≥ (logx)u and intervals [v±, v±+y±]⊂ [x/4, x]
such that
v+≤n≤v++y+
a(n)≥(1 + exp(−u(logu+ log logu+O(1))))y+A(x) x ,
and
v−≤n≤v−+y−
a(n)≤(1−exp(−u(logu+ log logu+O(1))))y−A(x) x . These are of essentially the same strength as the results in [1].
Further we also obtain that (for fixed u) there exists y∈(x/4, x) and an arithmetic progression a (mod) with≤x/(logx)u such that
A(y;, a)− f(a)
γ A(y)≥exp(−u(logu+ log logu+O(1)))A(x) φ(). Example 7. Let K be a number field with [K :Q]>1 and let R be its ring of integers. Let C1, . . . , Ch be the ideal classes of R, and define A(i) to be the set of integers which are the norms of integral ideals belonging to Ci. From the work of R. W. K. Odoni [14] we know that
A(i)(x)∼ci x (logx)1−E(K)
whereci >0 is a constant and E(K) denotes the density of the set of rational primes admitting inK at least one prime ideal divisor of residual degree 1. It is well known that E(K) ≥1/[K :Q] and also we know that E(K) ≤1−1/
[K:Q] (see the charming article of J-P. Serre [16]).
We now describe what the natural associated multiplicative functions h andfq should be. Defineδ(n) = 1 whennis the norm of some integral ideal in K andδ(n) = 0 otherwise. Clearly n∈ A(i) for some iif and only ifδ(n) = 1.
Naturally we would expect that
n≤x pk|n
δ(n)≈
j≥kδ(pj)/pj ∞
j=0δ(pj)/pj
n≤x
δ(n),
and so the natural definition of h is h(pk)
pk =
j≥kδ(pj)/pj ∞
j=0δ(pj)/pj.
Note thath(p) = 1 ifδ(p) = 1 (which happens ifphas a prime ideal divisor in K of residual degree 1, and so occurs for a set of primes with density E(K)) and thath(p)≤1/p+O(1/p2) ifδ(p) = 0 (and this happens for a set of primes
of density 1−E(K)>0). With the corresponding definition of fq(a) we may expect that for (q,S) = 1 (for a finite set of bad primes S including all prime factors of the discriminant ofK)
A(i)(x;q, a)∼ fq(a)
qγq A(i)(x).
By appealing to standard facts on the zeros of zeta andL-functions over num- ber fields one can prove such an asymptotic for small values ofq (for example, if q is fixed). Our work shows that that this asymptotic fails if q is of size x/(logx)u for any fixed u. We expect that one can understand the asymp- totics ofA(i)(x;q, a) for appropriate smallq in order also to conclude that the distribution ofA(i) in short intervals (of length (logx)u) is compromised. We also expect that similar results hold with R replaced with any order inK.
Example8. Letkbe a fixed integer, and chooserreduced residue classes a1,. . .,ar (modk) where 1≤r < φ(k). TakeA to be the set of integers not divisible by any prime≡ai (modk) and takeSto be the set of primes dividing k. Here h is completely multiplicative with h(p) = 0 if p ≡ aj (modk) for some j and h(p) = 1 otherwise, and fq(a) is defined appropriately. This is a special case of a Wirsing sequence, and so (6.8) and Corollary 6.2 apply (we see easily that in Corollary 6.2 may be chosen coprime tok). Note also that Example 3 essentially corresponds to the case k = 4 and a1 = 3. This also covers Example 7 in the case whenK is an abelian extension.
We may apply standard techniques of analytic number theory to study A(x) and A(x;q, a). Consider the generating function A(s) =∞
n=1a(n)n−s which converges absolutely in Re(s) > 1 and satisfies the Euler product
p≡ai (modk)(1−p−s)−1. Further using the orthogonality relations of charac- ters ψ (modk) we see that
A(s) =
ψ (modk)
L(s, ψ)φ(k)1
b≡ai (modk)ψ(b)
B(s),
where B is absolutely convergent in Re(s) > 1/2. Further for a character χ (modq) with (q, k) = 1 we get that
A(s, χ) = ∞ n=1
a(n)χ(n)
ns =
ψ (modk)
L(s, ψχ)φ(k)1
b≡ai (modk)ψ(b)
B(s, χ), with B(s, χ) absolutely convergent in Re(s) > 1/2. For large x, if q ≤ exp(√
logx) with (q, k) = 1 is such that for all characters χ (modqk) (prim- itive or not) L(s, χ) has no zeros in σ ≥ 1−c/log(qk(1 +|t|) then we may conclude by standard arguments that
A(x;q, a) = fq(a)
qγq A(x) +O(xexp(−C
logx)), (6.10)
for some constant 1 > C > 0. Since k is fixed we may suppose that no divisor of it is a Siegel modulus. Let ν1, . . ., νt denote the Siegel moduli below exp(√
logx) (see §5 for details, and note thatt log logx), and select a prime factor vi for each νi. Choose q to be the product of primes between
√w and wwith w= (C/10)√
logx, taking care to omit the primes v1,. . .,vt
which fall in this range. Then (6.10) applies to this modulus q (which is of size exp((C/10 +o(1))√
logx). Now, applying the Maier matrix method (and our Corollary 3.3) we deduce that for large u ≤ √
w there exists an interval [v, v+y] in [x/4,3x/4] withy ≥(logx)u such that
A(v+y)− A(v)−yA(x) x
exp(−2u(logu+ log logu+O(1)))yA(x) x . (6.11)
Arguing more carefully, using a zero density estimate as in Section 5, it may be possible to improve the right side of (6.11) to exp(−u(logu+ log logu+ O(1)))yA(x)x .
6d. The multiplicative function zΩ(n) for z ∈ (0,1). Take a(n) = zΩ(n) where Ω(n) denotes the number of prime factors ofncounted with multiplicity and z is a fixed number between 0 and 1. We take S = ∅ and h(n) = zΩ(n) and fq(a) = zΩ((a,q)). For large x we know from a result of A. Selberg (see Tenenbaum [17]) that
A(x)∼x(logx)z−1 Γ(z) .
From Theorem 2.4 and the above, we deduce that for fixedu≥max(e2/(1−z), e100) and largex there existsy ∈(x/4, x) and an arithmetic progression a (mod) with≤x(3/logx)u such that
A(y;, a)−f(a)
γ A(y)exp(−u(logu+O(log logu)))A(y) φ(). Supposeq≤exp(√
logx) is such that for every characterχ (mod q) (prim- itive or not)L(s, χ) has no zeros inσ ≥1−c/log(q(|t|+ 2)) for some constant c >0. Then following Selberg’s method we may see that for some 1> C >0
A(x;q, a) = fq(a)
qγq A(x) +O(xexp(−C
logx)).
(6.12)
Let ν1,. . .,νr be the Siegel moduli below e√logx (see §5 for details; and note that r log logx), and select one prime factor vi for each νi. Choose q to be the product of primes between √
w and w for w = (C/10)√
logx, taking care to omit the primes v1, . . ., vr should they happen to lie in this interval.
Then (6.12) applies to this modulus q, and using the Maier matrix method and appealing to Corollary 3.3 we find that for large u≤√
w there exists an