Prescribing the binary digits of primes

0.19 MB·English

by  Jean Bourgain

PRESCRIBING THE BINARY DIGITS OF PRIMES
JEAN BOURGAIN
2
1
0
2 Abstract. We present a new result on counting primes p < N = 2n for

n
which r (arbitrarily placed) digits in the binary expansion of p are specified.
a
J Compared with earlier work of Harman and Katai, the restriction on r is

9 4/7
relaxed to r < c n . This condition results from the estimates of
logn

] Gallagher and Iw(cid:16)aniec(cid:17)on zero-free regions of L-functions with ‘powerful’
T
N conductor.
.
h
t
a (0). Summary
m
[
This work is motivated by the paper [H-K] on the problem described in the

2
v title. We prove the following
5
9
8
3
Theorem. Let N = 2n, n large enough, and A 1,...,n 1 such that
.
5 ⊂ { − }
0
4/7
n
1
r = A < c . (0.1)
1 | | logn
: (cid:18) (cid:19)
v
i Then, considering binary expansions x = x 2j (x = 1 and x = 0,1 for
X j<n j 0 j
1 j < n) and assignments α for j A, we have
r j P
a ≤ ∈
N
p < N; for j A, the j-digit of p equals x 2−r . (0.2)
j
|{ ∈ }| ∼ logN
A few comments. Statement (0.2) can also be formulated as an asymptotic
formula. Next, our result has an analogue for q-ary expansions (with essentially
the same proof) and we restricted ourselves to q = 2 only for simplicity.
This research was partially supported by NSF grants DMS-0808042and DMS-0835373.
1
2 JEAN BOURGAIN
The paper [H-K] establishes a similar result under the assumption
√n
A < c . (0.3)
| | logn
A key ingredient in [H-K] and also here is the use of zero-free regions for
L-functions L(s,χ). In particular, the restriction (0.1) follows from results
of Gallagher and Iwaniec ([G], [I]) that provide the zero-free region 1 σ <
−
c 1 , γ < T where ρ = σ+iγ, for special moduli q that are powers
(logqT.loglogqT)3/4 | |
of a fixed integer (here q = 2j). This region is larger than what’s available in
the general case. Note that we dismiss here a possible Siegel zero, which is not
a concern for q specified as above. Although the Gallagher-Iwaniec theorem is
one of the elements in the [H-K] argument, its full potential was unfortunately
not exploited. As in [H-K], we use the circle method but with a smaller minor
arcs region, leading to more saving for that contribution. As one expects, the
major arcs analysis is more involved.
Note that there are two multiplicative character sums entering the discussion
(after conversion of the exponential sums in the circle method)
Λ(n)χ(n) (0.4)
n<N
X
and
f(n)χ(n) (0.5)
n<N
X
where f = 1 or a suitably modified version.
[xj=αj forj∈A]
Themainadditionalideainthispapermayberoughlyexplainedasfollows. In
general,againdismissing Siegelzeros, L(s,χ)hasazero-freeregion1 σ < c ,
− logqT
γ < T. But, as implied by the usual density estimates, for ‘most’ characters,
| |
this zero-free region is much larger leading to better bounds on (0.4). For the
remaining ‘bad’ characters, which are few (including possible Siegel zeros), we
seek non-trivial estimates on the sum (0.5). This is clearly reasonable, in view
of the additive structure of f. Perhaps not surprisingly, it turns out that the
only situation that escapes this analysis (that in fact can be carried out as long
PRESCRIBING THE BINARY DIGITS OF PRIMES 3
r < n2−ε) are precisely moduli of the form 2j. Thus at the end, it is the size of
3
the [G], [I] zero-free region that dictates the restriction on A .
| |
1. Minor Arcs Contribution
Let N = 2n. Write
1
¯
Λ(k)f(k) = S(α)S (α)dα (1.1)
f
k≤N Z0
X
denoting
S(α) = Λ(k)e(kα) (1.2)
and X
S (α) = f(k)e(kα). (1.3)
f
We assume f(k) = 0 for k even, sincXe obviously k 1 (mod2) is a necessary
≡
condition.
We fix a parameter B = B(n) which will be specified in 6. At this point, let
§
us just say that logB n4/7, up to logarithmic factors.
∼
The major arcs are defined by
a B
(q,a) = α < where q < B. (1.4)
M − q qN
(cid:20)(cid:12) (cid:12) (cid:21)
(cid:12) (cid:12)
Given α, there is q < N such t(cid:12)hat (cid:12)
B (cid:12) (cid:12)
a B 1
α < < .
− q qN q2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
From Vinogradov’s estimate (Theorem 13.6 in [I-K])
(cid:12) (cid:12)
S(α) < q21N21 +q−12N +N45 (logN)3
| |
(cid:16) (cid:17)
N N
+ +N4/5 (logN)3. (1.5)
≪ √B √q
(cid:18) (cid:19)
Hence if q B,
≥
N
S(α) (logN)3. (1.6)
| | ≪ √B
4 JEAN BOURGAIN
Thus the minor arcs contribution in (1.1) is at most
N
(logN)3 S . (1.7)
f 1
≪ √B k k
Write k = k 2j with k = 0,1. Given A 1,...,n 1 and α = 0 or
0≤j<n j j ⊂ { − } j
1 for j A, we define
P
∈ x
f(x) = h (1.8)
αj 2j+1
jY∈A (cid:16) (cid:17)
where 0 h 1 is a smooth 1-periodic function satisfying
0
≤ ≤
h (t) = 1 if 1 t 1 1
0 n ≤ ≤ 2 − n
h (t) = 0 if 1 t 1
 0 2 ≤ ≤
(h is defined similarly). Thus
1 
ˆ
h < Clogn (1.9)
α 1
k k
and, by approximation, we can assume that say
supphˆ [ n3,n3] (1.9′)
α
⊂ −
the additional error term (for suitably chosen h ) being at most e−n < 1 and
α N
hence may be ignored.
Note that in the formulation of the main theorem, we did not specify the
error term as we do not intend to put emphasis on this aspect. Thus the
function f introduced in (1.8) suffices for our purpose, while establishing an
asymptotic formula with specified error term would require an approximation
of the functions 1 and 1 which is better than given by h and h .
[0,1[ [1,1[ 0 1
2 2
In particular f satisfies
S < (Clogn)|A|. (1.10)
f 1
k k
Substitution in (1.7) gives then the bound
N
(Clogn)|A|n3. (1.11)
√B
PRESCRIBING THE BINARY DIGITS OF PRIMES 5
2. Major Arcs Analysis (I)
Next, we analyze the major arcs contributions (q < B)
S(α)S¯ (α)dα. (2.1)
f
(a,q)=1 Z
X |α−a|< B
q qN
Write α = a +β. Defining
q
q
τ(χ) = χ(m)e (m)
q
m=1
X
we have (see [D], p. 147)
1
S(α) = τ(χ¯)χ(a) χ(k)Λ(k)e(kβ) +O (logN)2 (2.2)
φ(q)
" #
χ k≤N
X X (cid:0) (cid:1)
where by partial summation
χ(k)Λ(k)e(kβ) (1+N β )max ψ(u,χ)
(cid:12) (cid:12) ≤ | | u<N | |
(cid:12)kX≤N (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) B
(cid:12) <(cid:12) max ψ(u,χ) . (2.3)
q u<N | |
Wesubdividethemultiplicative charactersχinclasses and (tobespecified),
G B
depending on the zero set of L(s,χ).
Thus
µ(q)
S(α) = Λ(k)e(kβ) +(2.4)+(2.5)
φ(q)
" #
k≤N
X
where the first term comes from the contribution of the principal character χ
0
and
1
(2.4) = τ(χ¯)χ(a) χ(k)Λ(k)e(kβ)
φ(q)
" #
χ(modq) k≤N
X X
χ∈G
and
1
(2.5) =
φ(q) ···
χ(modq)
X
χ∈B
6 JEAN BOURGAIN
Hence
B
(2.4) . max ψ(u,χ) (2.6)
| | √q  u<N | |
χ(modq),χ∈G
by (2.3). The contribution of (2.6)in (2.1) is at most 
(2.6). S (α) dα.
f
| |
Z
S M(q,a)
(a,q)=1
Summing over q < B and using the bound (1.10) gives the estimate
B(Clogn)|A| max ψ(u,χ) (2.7)
( u<N | |)
χ∈G
where in (2.7) the conductor of χ is at most B.
3. Major Arcs Analysis (II)
Next we treat χ = χ and χ .
0
∈ B
Assume χ is induced by χ which is primitive (modq ), q q. Then from [D],
1 1 1
|
p. 67
q q
τ(χ¯) = µ χ¯ τ (χ¯ ) (3.1)
1 1
q q
(cid:18) 1(cid:19) (cid:18) 1(cid:19)
which vanishes, unless q = q is square free with (q ,q ) = 1.
2 q1 1 2
The contribution of χ in (2.1) equals
q
τ(χ¯)
χ(k)Λ(k)e(kβ) f(k) χ(a)e ( ak) e( kβ) dβ.
q
φ(q) − −
" #" ! #
Z k≤N k<N a=1
|β|< B X X X
qN
(3.2)
We have
q
e (ak)χ(a) = e (ak)χ (a)
q q 1
a=1 (a,q)=1
X X
= e (a k)χ (a ) e (a k)
q1 1 1 1 q2 2
" #" #
(a1X,q1)=1 (a2X,q2)=1
PRESCRIBING THE BINARY DIGITS OF PRIMES 7
= χ (k)τ(χ )c (k) (3.3)
1 1 q2
(cf. [D], p. 65) and
µ q2 φ(q )
c (k) = (q2,k) 2 . (3.4)
q2 (cid:16)φ q(cid:17)2
(q2,k)
(cf. [D], p. 149). (cid:16) (cid:17)
From (3.1), (3.3), (3.4)
q
τ(χ¯) τ(χ ) 2 χ¯ (q )
1 1 2
χ(a)e ( ak) = | | µ (q ,k) χ (k)
q 2 1
φ(q) " − # φ(q1) φ q2
Xa=1 (q2,k) (cid:0) (cid:1)
(cid:16) (cid:17)
q χ¯ (q )
1 1 2
= µ (q ,k) χ (k) (3.5)
φ(q ) φ( q2 ) 2 1
1 (q2,k)
(cid:0) (cid:1)
(cf. [D], p. 66).
Returning to (3.2), rather than integrating in β over the interval β < B ,
| | qN
we introduce a weight function
qN
w β
B
(cid:16) (cid:17)
where 0 w 1 is a smooth bumpfunction on R such that w = 1 on [ 1,1],
≤ ≤ −
supp w [ 2,2] and
⊂ −
wˆ(y) < Ce−|y|1/2.
| |
Note that this operation creates in (2.1) an error term that is captured by the
minor arcs contribution (1.7) .
(cid:0)
Hence, substituting (3.5), ((cid:1)3.2) becomes
q B B u (q ,k )
1 ¯ (q ) wˆ (k k ) (k )Λ(k )f(k ) 2 2 (k )
φ(q )X1 2 qN qN 1 − 2 X 1 1 2 φ q2 X1 2
1 k1X,k2<N (cid:16) (cid:17) (cid:0) (q2,k2) (cid:1)
(3.6)
(cid:0) (cid:1)
and we observe that by our assumption on wˆ the k ,k summation in (3.6) is
1 2
restricted to k k < qNn3, up to a negligible error.
| 1 − 2| B
We first examine the contribution of the principal characters.
8 JEAN BOURGAIN
For = ,q = 1 and (3.6) becomes
0 1
X X
B B µ (q,k )
2
wˆ (k k ) Λ(k )f(k ) . (3.7)
qN qN 1 − 2 1 2 φ q
k1X,k2<N (cid:16) (cid:17) (cid:0) (q,k2) (cid:1)
(cid:0) (cid:1)
Recall the distributional property of the primes ([I-K], Corollary 8.30)
Λ(k) = x+O xexp c(logx)3/5(loglogx)−15 (3.8)
−
k≤x
X (cid:0) (cid:0) (cid:1)(cid:1)
and also our assumption logB n4/7. Performing the k -summation in (3.7),
1
≪
partial summation together with (3.8) give
B B µ (q,k )
qN wˆ qN(k1 −k2) f(k2) φ q 2 +O N exp −c(logN)3/5(loglogN)−15
kX1,k2 (cid:16) (cid:17) (cid:0) (q,k2) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
µ (q,k) (cid:0) (cid:1)
= f(k) +O N exp c(logN)3/5(loglogN)−15
φ q −
k (cid:0) (q,k) (cid:1)
X (cid:0) (cid:0) (cid:1)(cid:1) (3.9)
(cid:0) (cid:1)
Let κ(q) be a function satisfying the following
Assumption A. Let q < B be odd and square free. Then
0
N N
f(k) = E[f] +O κ(q ) 2−r (3.10)
0
q q
0 0
q0|Xk,k<N (cid:16) (cid:17)
where E[f] denotes the normalized average.
Remark. Since the function κ(q) at this stage remains unspecified and its
properties will be established later (in 4), we thought this presentation more
§
desirable than making a more technical statement here (cf. (4.26)).
PRESCRIBING THE BINARY DIGITS OF PRIMES 9
Assuming q square-free (sf) and odd, the first term of (3.9) equals
µ(q′)
f(k) =
φ q)  
q′|q q′ (q,k)=q′
X X
(cid:0)  
φ(q′)
µ(q′) µ(q′′) f(k)
φ(q)
Xq′|q qX′′|qq′ qX′q′′|k 
k<N
 
 
and substituting (3.10) we obtain
µ(q′) µ(q′′)
N E[f] (3.11)
φ(q/q′) q′q′′
Xq′|q qX′′|qq′
φ(q′) κ(q′q′′)
+O N 2−r . (3.12)
 φ(q) q′q′′ 
Xq′|q qX′′|qq′
 
Next
µ(q′) 1 1
(3.11) = N E[f] 1
φ(q/q′) q′ − p
Xq′|q Yp|qq′ (cid:18) (cid:19)
µ(q′) 1
= N E[f] φ(q/q′)
φ(q/q′) q
q′|q
X
N E[f] if q = 1
= (3.13)

0 otherwise.

Summing (3.12) over q < B sf and odd, we have the estimate (setting q =
 1
q′q′′)
κ(q ) 1
2−rN 1  
q φ(q′′)φ(q )
qqX11<sBf 1 q′′|qq12,X<(qB1,qs2f)=1 2 
 
10 JEAN BOURGAIN
κ(q)
< 2−rN(logB)2 . (3.14)
q
q<B
 X 
qsf,odd
 
For q sf and even, set q = 2q and notethat the firstterm of (3.9) equals
1
µ(q′)
f(k)
φ q /q′)  
Xq′|q1 1 (q1X,k)=q′
(cid:0)  
and we proceed similarly as above with the same conclusion and q replaced by
q . From the preceding, the contribution of the principal characters equals
1
κ(q)
2E[f]N +CN2−r(logN)2 +BN exp c(logN)3/5(loglogN)−51 .
q −
q<B
Xqsf  (cid:0) (cid:1)
 
  (3.15)
Consider next χ ,χ primitive.
1 1
∈ B
Returning to (3.6), estimate by
q B B µ (q ,k)
1 2
wˆ (k k) f(k) (k)
φ(q ) qN qN 1 − φ q2 X2
1 kX1<N(cid:12)Xk (cid:16) (cid:17) (cid:0) (q2,k) (cid:1) (cid:12)
(cid:12) (cid:12)
q B φ(cid:12)(q′) B (cid:0) (cid:1) (cid:12)
1 2 max wˆ (k k) f(k) (k) (3.16)
1 1
≤ φ(q1) q qX2′|q2 φ(q2) k1 (cid:12)k,(kX,q2)=q2′ (cid:16)qN − (cid:17) X (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We introduce another function α(q ,q ) satisfying
1 0
Assumption B.
Given q ,q < B,(q ,q ) = 1 with q sf and odd, χ (modq ) primitive,
0 1 0 1 0 1 1
J
f(k)χ (k) < 2−rα(q ,q ) | | (3.17)
1 1 0
(cid:12) (cid:12) q
(cid:12)(cid:12)k∈XJ,q0|k (cid:12)(cid:12) 0
(cid:12) (cid:12)
holds, whenever J (cid:12)[1,N] is an interva(cid:12)l of size N.
⊂(cid:12) (cid:12) ∼ B