THE SMOOTHED PÓLYA–VINOGRADOV INEQUALITY
Kamil Adamczewski
Department of Mathematics, Dartmouth College, Hanover, New Hampshire
[email protected]
Enrique Treviño
Department of Mathematics and Computer Science, Lake Forest College, Lake
Forest, Illinois 60045, USA
[email protected]
Received: , Revised: , Accepted: , Published:
Abstract
Let
χ
be
a
primitive
Dirichlet
character
to the modulus q. Let Sχ (M, N ) =
P
√
q log q.
M <n≤N χ(n). The Pólya-Vinogradov inequality states that |Sχ (M, N )| The smoothed Pólya–Vinogradov inequality, recently introduced by Levin, Pomerance and Soundararajan, is a numerically useful version of the Pólya–Vinogradov
inequality that saves a log q factor. The smoothed Pólya–Vinogradov inequality has
been used to settle a conjecture of Brizolis, namely that for every prime p > 3, there
is a primitive root g and an integer x ∈ [1, p − 1] such that g x ≡ x mod p. It has
also been used to improve the best known numerically explicit upper bound on the
least inert prime in a real quadratic field. In this paper we will prove a smoothed
Pólya–Vinogradov inequality which takes into account the arithmetic properties of
the modulus and we extend the inequality to imprimitive characters. We also find
a lower bound for the inequality.
1. Introduction
Let χ be a non-principal Dirichlet character to the modulus
q. It has been the
+N
MX
interest of mathematicians to study the sum χ(n). Pólya and Vinogradov
n=M +1
√
independently proved in 1918 that the sum is bounded above by O( q log q). Assuming the Riemann Hypothesis for L-functions (GRH), Montgomery and Vaughan
√
[3] showed that the sum is bounded by O( q log log q). This is best possible (up to a
constant), because in 1932 Paley [5] proved that there are infinitely many quadratic
characters χ such that there exists a constant c > 0 that satisfy for some N the
2
INTEGERS: 15 (2015)
N
X
√
inequality χ(n) > c q log log q.
n=1
Recently, in [2], Levin, Pomerance and Soundararajan considered a “smoothed”
version of the Pólya–Vinogradov inequality. Instead of considering the character
sum over an interval, they consider the following weighted sum
X
n − M
∗
Sχ (M, N ) :=
χ(n) 1 − − 1 .
N
M ≤n≤M +2N
The theorem they prove is the following:
Theorem A. Let χ be a primitive Dirichlet character to the modulus q > 1 and let
M, N be real numbers with 0 < N ≤ q. Then
X
∗
n − M
√
N
Sχ (M, N ) = χ(n) 1 − − 1 ≤ q − √ .
N
q
M ≤n≤M +2N
Levin, Pomerance and Soundararajan used the inequality to prove that for every
prime p > 3, there is a primitive root g and an integer x ∈ [1, p − 1] such that
g x ≡ x mod p, i.e., that the discrete logarithm base g has a fixed point. The
second author (see [6]) used the smoothed Pólya–Vinogradov inequality to improve
an upper bound for the least inert prime in a real quadratic field. The inequality is
not new, as it was used by Hua in [1] to improve a bound on the least primitive root
mod p. However, while Hua presented his paper as an introduction of a method
with numerous applications, we didn’t find other papers that used this technique.
Hopefully this paper will help bring this useful method to the spotlight it deserves.
In this paper we will prove several related results. In section 2 we will prove a
theorem that takes into account arithmetic information from the modulus q to give
a better upper bound for some ranges of N :
Theorem 1. Let χ be a primitive character to the modulus q > 1, let M, N be real
numbers with 0 < N ≤ q and let m be a divisor of q such that 1 ≤ m ≤ Nq . Then
X
n − M
φ(m) √
χ(n) 1 − − 1 ≤
q.
N
m
M ≤n≤M +2N
We also prove the following theorem which expands the range of N and will be
crucial to extend the inequality to imprimitive characters.
Theorem 2. Let χ be a primitive character to the modulus q > 1 and let M, N be
real numbers with N > 0. Then,
X
n − M
q 3/2 N
N
χ(n) 1 − − 1 ≤
1−
.
(1)
N
N
q
q
M ≤n≤M +2N
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In particular, |Sχ∗ (M, N )| <
√
q.
Remark 1. The theorem
was
stated without proof as Corollary 3 in [2]. Also note
N
N
that if 0 < N < q, then
=
and therefore Theorem A follows from Theorem
q
q
2.
With Theorem 2, we are able to extend the smoothed Pólya–Vinogradov inequality for imprimitive characters, namely we prove
Theorem 3. Let χ be a non-principal Dirichlet character to the modulus q > 1 and
let M, N be real numbers with N > 0. Then,
X
n − M
4 √
χ(n) 1 − − 1 < √ q.
N
6
M ≤n≤M +2N
One of the remarkable things involving the smoothed Pólya–Vinogradov inequality is that it is not very hard to prove and it is a tight
since one can
inequality,
√
show that there exist M and N such that Sχ∗ (M, N ) > c q for some positive
constant
c and some character χ mod q. Indeed, in section 3 we will prove that
∗
Sχ (M, N ) > 22 √q. The proof was motivated by the proof of Theorem 9.23 in [4].
π
Finally, in the last section, we show a table computing Sχ∗ (M, N ) for many moduli.
2. Upper bound and corollaries
We begin by recreating the proof of Theorem A. We do so because the proofs of
Theorem 1 and Theorem 2 branch out from this proof.
Proof of Theorem A. We follow the proof in [2]. Let
H(t) = max{0, 1 − |t|}.
We wish to estimate |Sχ∗ (M, N )|.
Using the identity (see Corollary 9.8 in [4])
q
χ(n) =
1 X
χ̄(j)e(nj/q),
τ (χ̄) j=1
where e(x) := e2πix and τ (χ) is the Gauss sum
q
X
χ(a)e(a/q) , we can deduce
a=1
q
Sχ∗ (M, N ) =
X
1 X
χ̄(j)
e(nj/q)H
τ (χ̄) j=1
n∈Z
n−M
−1 .
N
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The Fourier transform (see Appendix D in [4]) of H is
Z ∞
1 − cos 2πs
b
b
when s 6= 0, H(0)
= 1.
H(s) =
H(t)e(−st)dt =
2π 2 s2
−∞
b
Therefore H(s)
is nonnegative for s real. In general, if
f (t) = e(αt)H(βt + γ),
(2)
with β > 0, then
1
s−α
b s−α .
fb(s) = e
γ H
(3)
β
β
β
Using α = j/q, β = 1/N and γ = −M/N − 1, then by Poisson summation (see
Appendix D in [4]) we get
q
X j
j
N X
∗
b
χ̄(j)
e −(M + N ) n −
H
s−
N . (4)
Sχ (M, N ) =
τ (χ̄) j=1
q
q
n∈Z
b is nonnegative and that |τ (χ̄)| = √q for primitive
Using that χ(q) = 0, that H
characters, we have
q−1
∗
N XX b
j
N X b kN
Sχ (M, N ) ≤ √
H
n−
N =√
H
.
q j=1
q
q
q
n∈Z
k∈Z/qZ
Therefore
!
X
kN
b
b
H
−
H(kN
)
q
k∈Z
k∈Z
!
NX b
√ X N b kN
H
H(kN )
= q
−
q
q
q
k∈Z
k∈Z
!
N b
√ X N b kN
H
− H(0) .
≤ q
q
q
q
∗
N
Sχ (M, N ) ≤ √
q
X
k∈Z
q
N
Using
in (2) and (3) yields that the Fourier transform of
α = γ = 0 and β =
qt
is
N
1
s−0
s−0
N b sN
b
e
· (0) H
= H
.
β
β
β
q
q
Therefore, by Poisson summation, we have
X ql ∗
N
N
N
√
√
Sχ (M, N ) ≤ √q
H
− √ = qH(0) − √ = q − √ .
(5)
N
q
q
q
l∈Z
ql q We used that q ≥ N which implies that for l 6= 0 and l ∈ Z, N
≥ N ≥ 1 which
ql
implies H N
= 0.
H
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Proof of Theorem 1. Following the proof of the previous theorem, we arrive at (4).
b is nonnegative, and that
From there, using that if (n, m) > 1 then χ(n) = 0, that H
√
|τ (χ̄)| = q for primitive characters, we have
q
∗
N X X b
j
N X b kN
Sχ (M, N ) ≤ √
H
H
n−
N =√
.
q j=1
q
q
q
n∈Z
(6)
k∈Z
(k,m)=1
(j,m)=1
Using inclusion exclusion we get
X kdN √ X µ(d) X dN kdN ∗
N X
b
b
Sχ (M, N ) ≤ √
µ(d)
H
= q
H
.
q
q
d
q
q
d|m
k∈Z
qt
Nd
Since the Fourier transform of H
X µ(d) X
∗
Sχ (M, N ) ≤ √q
H
d
d|m
k∈Z
d|m
is
l∈Z
dN b
q H
ql
Nd
=
sdN
q
, by Poisson summation
φ(m) √
√ X µ(d)
q
H(0) =
q.
d
m
d|m
We used that q ≥ mN which implies that for l 6= 0 and l ∈ Z, Nqld ≥ Nqm ≥ 1,
and hence H Nqld = 0.
Proof of Theorem 2. In the proof of Theorem A, we only used that N ≤ q in the
last inequality of (5). Therefore, from (5), we have
!
ql
N
√ X
∗
|Sχ (M, N )| ≤ q
H
−
.
N
q
l∈Z
To get the desired result we need only prove
X ql N
q N
N
H
+
≤
1−
.
N
q
N q
q
l∈Z
Note that H
ql
N
= 0 for |l| >
N
q .
Also H
ql
N
=H
−ql
N
. Using these two
facts together with H(0) = 1, we get
X ql X ql X
ql
H
=1+2
H
=1+2
1−
.
N
N
N
N
N
l∈Z
l≤
q
l≤
q
Therefore
X ql N
2q X
N
q
N
N
H
=1+2
−
l =1+2
−
+1 .
N
q
N N
q
N
q
q
l∈Z
l≤
q
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j k
Letting θ = Nq and using that Nq = Nq + θ, we get
X ql N
2N
q N2
H
+
=1+
− 2θ −
(1
−
2θ)
−
θ(1
−
θ)
N
q
N q2
q
l∈Z
=
2N
N
q
N
q
+ 1 − 2θ −
− (1 − 2θ) + θ(1 − θ) =
+ θ(1 − θ).
q
q
N
q
N
Therefore (1) is true. Once we have (1), we can conclude that
√
|Sχ∗ (M, N )| < q. Indeed, if N < q, then
q 3/2 N
N
N
√
√
Sχ∗ (M, N ) ≤
1−
= q − √ < q;
N
q
q
q
and if N ≥ q, we have
Sχ∗ (M, N )
q 3/2
≤
N
N
q
√
q √
N
q 3/2
1−
≤
≤
< q.
q
4N
4
We finish the section with the proof of Theorem 3 :
Proof of Theorem 3. In this proof, we follow the ideas used in [4] (page 307) to
extend the Pólya–Vinogradov inequality from primitive characters to general characters.
Let χ be induced by a primitive character χ∗ of modulus d > 1. This is possible
since χ is non-principal. In the case that χ is primitive, then χ∗ = χ. Letting χ0
be the principal character mod q, we have that χ = χ∗ χ0 . Therefore χ(n) = χ∗ (n)
for n an integer coprime to q, and χ(n) = 0 otherwise.
Let r be the product of primes that divide q but not d. Then when (n, r) > 1,
we have χ(n) = 0. If (n, r) = 1, then χ(n) = χ∗ (n). Therefore
MX
+2N
X
n − M
n − M
∗
χ(n) 1 − − 1 =
χ (n) 1 − − 1
N
N
M ≤n≤M +2N
(n,r)=1
n=M
X
n − M
=
χ (n) 1 − − 1
µ(k)
N
M ≤n≤M +2N
k|(n,r)
X
X
n − M
− 1 .
=
µ(k)
χ∗ (n) 1 − N
X
k|r
∗
M ≤n≤M +2N
k|n
Now, writing n = km and using that χ∗ is totally multiplicative, we get
!
m − M
X
X
X
M N
∗
∗
k
µ(k)χ (k)
χ (m) 1 − N
− 1 =
µ(k)χ∗ (k)Sχ∗
,
.
k
k k
M +2N
M
k|r
k
≤m≤
k
k|r
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By Theorem 2, |Sχ∗ (M/k, N/k)| <
√
d. Hence, taking absolute value, we have
r
X√
√
q
∗
ω(r)
ω(r)
|Sχ (M, N )| <
.
(7)
d=2
d≤2
r
k|r
Since 2ω(r) is a multiplicative function, and for p ≥ 5, 2 <
√
p, we have
Y 2
2ω(r)
2
4
2
√ =
√ ≤√ ×√ =√ .
p
r
2
3
6
p|r
(8)
Combining (7) with (8) yields the desired result.
3. Lower bound
Theorem 4. Let χ be a primitive character to the modulus q > 1 and let M, N be
positive integers. Then
2
πN
+2N
MX
sin
q
n−M
1
S2 (N ) := max χ(n) 1 − − 1 ≥ √ 2
1≤M ≤q N q
N
π
n=M
sin q
Proof. Let
S3 (N ) :=
q
X
e
M =1
M
q
MX
+2N
n=M
n − M
− 1 ,
χ(n) 1 − N
and note that
q MX
+2N
X
n − M
|S3 (N )| ≤
− 1 χ(n) 1 − N
M =1 n=M
+2N
MX
n − M
− 1 = qS2 (N ).
≤ q max χ(n) 1 − 1≤M ≤q N
n=M
Therefore we can focus on S3 (N ).
q
X
MX
+2N
n − M
M
S3 (N ) =
e
χ(n) 1 − − 1
q
N
M =1
n=M
q
2N X
n
X
M
=
e
χ(n + M ) 1 − − 1 .
q
N
n=0
M =1
(9)
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Now we can do a change of variable, to go from M to L − n:
q
2N X
n
X
L−n
χ(L) 1 − − 1
S3 (N ) =
e
q
N
n=0 L=1
q
2N
n
X
X
n L
=
e −
1 − − 1
e
χ(L).
q
N
q
n=0
L=1
Therefore,
S3 (N ) = τ (χ)
2N
n
X
n 1 − − 1 = τ (χ)S4 (N ).
e −
q
N
n=0
Now it’s time to work on S4 (N ):
2N
N
2n
n
X
X
X
n n n
n n
S4 (N ) =
e −
1 − − 1 =
e −
+
e −
2−
.
q
N
q N
q
N
n=0
n=0
n=N +1
By making the change of variable m = 2N − n, we get
2N
N
N
−1 e
−
X
X
q
1
n
m
S4 (N ) =
e −
n +
e
m.
N n=0
q
N
q
m=0
Using the identity
N
X
nxn = x
n=0
N xN +1 − (N + 1)xN + 1
N xN +1 − (N + 1)xN + 1
=
,
2
(x − 1)
(x1/2 − x−1/2 )2
with x = e(α) and α = − 1q , we get
S4 (N ) =
1 N e ((N + 1)α) − (N + 1)e (N α) + 1
2
N
e α2 − e − α2
+
e (2N α) (N − 1)e (−N α) − N e (−(N − 1)α) + 1
.
2
N
e −α − e α
2
2
Therefore, by taking common denominator and multiplying out, we get that
S4 (N ) equals
N e ((N + 1)α) − (N + 1)e (N α) + 1 + (N − 1)e(N α) − N e((N + 1)α) + e(2N α)
,
2
N e α2 − e − α2
which equals
2
2
e(2N α) − 2e(N α) + 1
e(N α) e N2α − e − N2α
e(N α) (sin N πα)
=
=
2 .
2
2
N
N
(sin πα)
N e α2 − e − α2
e α2 − e − α2
(10)
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From earlier we know, qS2 (N ) ≥ |S3 (N )| = |τ (χ)||S4 (N )|. Using |τ (χ)| =
that |e(x)| = 1 and (10) yields the theorem.
√
q,
Now we are ready to prove our main lower bound result.
Corollary 1. Let χ be a primitive character to the modulus q > 1 and let M, N be
positive integers. Then
M +2N
X
n − M
2√
χ(n) 1 − max − 1 ≥ 2 q.
M,N N
π
n=M
Proof. If q is even, let N = 2q . Therefore (9) becomes
2
πN
sin
q
1
2
1
2√
S2 (N ) ≥ √ 2 = √ 2 ≥ 2 q.
N q
q
q
π
sin πq
sin πq
The last inequality comes from sin1 x ≥ x1 .
If q is odd, let N = q−1
2 , then
2
2
πN
π
sin
cos
q
2q
2
1
S2 (N ) ≥ √ √ 2 =
2 .
N q
(q
−
1)
q
π
π
sin q
sin q
π
π
From this and sin πq = 2 sin 2q
cos 2q
, we get
S2 (N ) ≥
2
√ 4(q − 1) q
1
sin
π
2q
2 ≥
2 q √
2√
q > 2 q.
π2 q − 1
π
Remark 2. If we consider N = 3q for 3 | q, N = q−1
3 for q ≡ 1 (mod 3) and N =
q−2
for
q
≡
2
(mod
3),
then
we
can
improve
the
constant
from π22 ≈ 0.202642 to
3
√
5(5+ 5)
≈ 0.227973. With N around 2q
≈
5 the constant improves a bit more to
16π 2
0.229115. The optimal value for N under this technique is around N = .371q where
the constant is approximately 0.230651.
9
4π 2
4. Numerics
Let χ be a Dirichlet character mod q and let
∗
Sχ (M, N )
F (χ) = max
.
√
M,2N ∈Z
q
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INTEGERS: 15 (2015)
√
Note that F (χ) exists because |Sχ (M, N )|/ q is a bounded continuous function,
periodic in M and going to 0 as N → ∞ (by Theorem 2). In the previous sections
we gave upper and lower bounds for F (χ). Indeed
2
π 2 ≤ F (χ) < 1. Now, let
G(q) = max F (χ),
χ
and
H(q) = min F (χ),
χ
where the max and the min range over primitive characters mod q. By writing a
program in Java we created the following table of values for G(q) and H(q) which
show that there’s room for improvement in the upper and lower bounds, for example
it seems 25 < F (q) < 45 . The reason a program could be written to find G(q) and
H(q) even though M and N range through all integers is that the periodicity of
χ mod q allows us to restrict ourselves to 0 ≤ M < q and M ≤ 2N < M + q with
M, 2N ∈ N.
Acknowledgements
We would like to thank Carl Pomerance for his useful comments regarding this
paper and his encouragement.
References
[1] Loo-Keng Hua, On the least primitive root of a prime, Bull. Amer. Math. Soc.
48 (1942), 726–730. MR 0007399 (4,130e)
[2] M. Levin, C. Pomerance, and K. Soundararajan, Fixed points for discrete logarithms, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197,
2010, pp. 6–15.
[3] H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative
coefficients, Invent. Math. 43 (1977), no. 1, 69–82. MR 0457371 (56 #15579)
[4] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I.
Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655 (2009b:11001)
[5] R.E.A.C. Paley, A theorem on characters, J. Lond. Math. Soc. 7 (1932), 28–32.
[6] Enrique Treviño, The least inert prime in a real quadratic field, Math. Comp.
81 (2012), no. 279, 1777–1797.
11
INTEGERS: 15 (2015)
q
3
4
5
7
8
9
11
12
13
15
16
17
19
20
21
23
24
25
27
28
29
31
32
33
35
36
37
39
40
41
43
44
45
47
48
49
51
52
53
55
56
57
59
60
61
G(q)
0.577
0.500
0.596
0.567
0.471
0.533
0.603
0.462
0.666
0.516
0.452
0.610
0.622
0.461
0.635
0.615
0.467
0.628
0.615
0.481
0.640
0.654
0.476
0.604
0.653
0.470
0.701
0.604
0.484
0.652
0.655
0.482
0.603
0.669
0.473
0.675
0.588
0.479
0.639
0.644
0.478
0.626
0.672
0.471
0.694
H(q)
0.577
0.500
0.500
0.500
0.471
0.509
0.484
0.462
0.474
0.506
0.452
0.493
0.489
0.447
0.495
0.480
0.467
0.493
0.473
0.460
0.484
0.485
0.472
0.476
0.486
0.459
0.473
0.490
0.459
0.479
0.487
0.455
0.479
0.474
0.473
0.474
0.481
0.457
0.466
0.487
0.467
0.482
0.477
0.463
0.486
q
63
64
65
67
68
69
71
72
73
75
76
77
79
80
81
83
84
85
87
88
89
91
92
93
95
96
97
99
100
101
103
104
105
107
108
109
111
112
113
115
116
117
119
120
121
G(q)
0.610
0.481
0.624
0.678
0.480
0.638
0.681
0.466
0.720
0.607
0.487
0.676
0.683
0.481
0.634
0.684
0.470
0.701
0.611
0.487
0.689
0.656
0.483
0.621
0.669
0.474
0.718
0.630
0.484
0.685
0.688
0.485
0.619
0.696
0.473
0.716
0.630
0.482
0.697
0.688
0.486
0.624
0.692
0.476
0.690
H(q)
0.481
0.449
0.478
0.470
0.448
0.474
0.472
0.463
0.475
0.465
0.450
0.483
0.475
0.463
0.470
0.469
0.461
0.479
0.480
0.451
0.470
0.479
0.448
0.465
0.482
0.460
0.468
0.480
0.452
0.480
0.466
0.452
0.478
0.476
0.458
0.473
0.472
0.456
0.474
0.470
0.449
0.478
0.471
0.464
0.475
q
123
124
125
127
128
129
131
132
133
135
136
137
139
140
141
143
144
145
147
148
149
151
152
153
155
156
157
159
160
161
163
164
165
167
168
169
171
172
173
175
176
177
179
180
181
G(q)
0.627
0.488
0.697
0.709
0.484
0.647
0.708
0.470
0.694
0.615
0.488
0.711
0.704
0.480
0.645
0.706
0.474
0.747
0.620
0.488
0.690
0.717
0.485
0.629
0.701
0.483
0.703
0.636
0.478
0.715
0.724
0.483
0.623
0.698
0.468
0.715
0.636
0.487
0.711
0.691
0.484
0.636
0.721
0.472
0.714
H(q)
0.473
0.448
0.471
0.464
0.453
0.485
0.474
0.455
0.465
0.471
0.456
0.471
0.478
0.458
0.472
0.472
0.455
0.472
0.466
0.451
0.471
0.474
0.451
0.469
0.468
0.454
0.464
0.466
0.451
0.476
0.465
0.448
0.479
0.475
0.462
0.466
0.472
0.449
0.460
0.466
0.452
0.466
0.466
0.455
0.466
Table 1: A table showing the max and min of G(q) and H(q) for all moduli q ≤
181 that have primitive characters. It is worth noting that the reason the moduli
divisible by 4 has such a small G(q) is Theorem 1.