Kummer’s Conjecture for Cubic Gauss Sums
D.R. Heath-Brown
Magdalen College, Oxford
1 Introduction
Kummer [5] computed the sums
S
p
=
p
X
n=1
exp(2πin
3
/p)
for primes p ≡ 1 (mod 3) up to 500, and found that S
p
/(2
√
p) lay in the intervals
[−1, −
1
2
], (−
1
2
,
1
2
), [
1
2
, 1] with frequencies approximately in the ratio 1 : 2 : 3.
He conjectured, somewhat hesitantly, that this might be true asymptotically.
Kummer’s conjecture was disproved by Heath-Brown and Patterson [4].
In order to state their result we must introduce a little notation. Let ω =
exp(2πi/3) and let (∗/∗)
3
be the cubic residue symbol for ZZ[ω]. For each
c ∈ ZZ[ω] such that c ≡ 1 (mod 3) the cubic Gauss sum is
g(c) =
X
d(mod c)
(
d
c
)
3
e(
d
c
),
where we have defined
e(z) = exp(2πi(z + z)) (1)
for all complex z. One then has
g(c)
3
= µ(c)c
2
c, (2)
where µ(∗) is the M¨obius function for ZZ[ω], see Hasse [2; pp. 443-445] for
example. It is therefore natural to normalize g(c) by writing
˜g(c) =
g(c)
|c|
.
One then finds that if p = N(π), where π ≡ 1 (mod 3) is a prime of ZZ[ω], then
S
p
2
√
p
= <(˜g(π)).
1
Heath-Brown and Patterson showed that the numbers ˜g(π) are uniformly dis-
tributed around the unit circle, thereby disproving Kummer’s conjecture.
To establish uniform distribution the natural route is to use the Weyl crite-
rion, which requires the sums
X
N(π)≤X
π≡ 1(mod 3)
˜g(π)
k
to be o(X/ log X) for each fixed non-zero integer k. The formula (2) shows that
˜g(π)
3
= −π/|π|. Thus if k is a multiple of 3, the required bound is a standard
consequence of the zero-free region for L-functions with a Gr¨ossencharakter.
When k = 3l + 1 we need to examine
X
N(π)≤X
π ≡1(mod 3)
˜g(π)(
π
|π|
)
l
.
Similarly, when k = 3l −1 we may restrict attention to sums of the above shape,
via the observation that g(c) = g(c). The principal result of Heath-Brown and
Patterson is then the estimate
X
N(c)≤X
c≡1(mod 3)
˜g(c)Λ(c)(
c
|c|
)
l
¿
ε
X
30/31+ε
+ |l|X
29/31+ε
, (3)
valid for any l ∈ ZZ and any ε > 0. Here Λ(c) is the von Mangoldt function,
defined as log N(π) if c is a power of the prime π, and 0 otherwise.
This type of bound probably does not express the whole truth, for Heath-
Brown and Patterson [4] (following Patterson [8], who presents a heuristic jus-
tification) have made the following conjecture.
Conjecture For any ε > 0 we have
X
N(c)≤X
c≡1(mod 3)
˜g(c)Λ(c)(
c
|c|
)
l
=
½
bX
5/6
+ O
ε
(X
1/2+ε
), (l = 0),
O
ε
(X
1/2+ε
), (l 6= 0),
(4)
where
b =
2
5
(2π)
2/3
Γ(
2
3
).
This is supported both by a heuristic argument and by the available numerical
evidence. It expresses a bias towards the ˜g(π) having positive real part, thereby
explaining the non-uniformity found by Kummer.
2
Unfortunately present methods appear to be inadequate for a resolution of
Patterson’s conjecture. The goal of the present paper is however to establish
an improved version of the result (3) of Heath-Brown and Patterson, which
only just fails to achieve the required degree of precision. Specifically we shall
establish the following bound.
Theorem 1 For any ε > 0 we have
X
N(c)≤X
c≡1(mod 3)
˜g(c)Λ(c)(
c
|c|
)
l
¿
ε
X
5/6+ε
+ |l|X
3/4+ε
,
for every l ∈ ZZ.
Possible improvements of (3) were investigated by Coleman in his thesis [1;
Chapter 2]. Coleman proved unconditionally that
X
N(c)≤X
c≡1(mod 3)
˜g(c)Λ(c)(
c
|c|
)
l
¿
ε
(X
29/32
+ |l|X
41/64
)(X(|l | + 1))
ε
,
and, subject to a ‘Large Values Conjecture’, (which is statement rather stronger
than that of our Theorem 2), that
X
N(c)≤X
c≡1(mod 3)
˜g(c)Λ(c)(
c
|c|
)
l
¿
ε
(X
5/6
+ |l|X
7/12
)(X(|l| + 1))
ε
.
Coleman explains that his factor (|l| + 1)
ε
can probably be dispensed with, so
that his second term is then better than that occuring in Theorem 1. This is due
to savings in the ‘T -aspect’ in Coleman’s analysis, related to the formulation of
his Large Values Conjecture.
The proof of Theorem 1 follows the line of attack established by Heath-
Brown and Patterson, as will be explained in §2, but injects two new ideas into
the argument. The first of these replaces a pointwise b ound for a Dirichlet
series, which Heath-Brown and Patterson used in an application of Perron’s
formula, by a mean-value bound. This will be described in §3. Coleman’s
analysis includes a saving of the same type, although his argument is distinctly
more complicated. The second innovation is a far more sophisticated estimate
for the ‘Type II sum’, which is Σ
3
(X, u) in the notation of [4]. In order to do
this we shall establish the following.
Theorem 2 Let c
n
be an arbitrary sequence of complex numbers, where n runs
over ZZ[ω]. Then
X
N(m)≤M
∗
|
X
N(n)≤N
∗
c
n
(
n
m
)
3
|
2
¿
ε
(M + N + (M N)
2/3
)(MN)
ε
X
n
∗
|c
n
|
2
,
3
for any ε > 0, where Σ
∗
denotes summation over square-free elements of ZZ[ω]
congruent to 1 modulo 3.
It seems possible that the term (M N)
2/3
could be removed with further
effort, and the bound would then be essentially best possible. However the
above suffices for our purposes. It should be noted that if the variables are not
restricted to be square-free a result as sharp as Theorem 2 would be impossible.
The proof of Theorem 2 is modelled on the corresponding argument for sums
(over ZZ) containing the quadratic residue symbol, due to the author [3]. The
latter is distinctly unpleasant, but fortunately some of the diffculties may be
reduced in our situation by the introduction of the term (M N)
2/3
in Theorem 2.
None the less the proof of this result will form a substantial part of the present
paper.
2 Preliminary Arguments
We shall begin by introducing a litle more notation, following Heath-Brown and
Patterson [4]. We write
˜g
l
(c) = ˜g(c)(
c
|c|
)
l
,
H
l
(X) =
X
N(c)≤X
c≡1(mod 3)
˜g
l
(c)Λ(c),
and
F
l
(X, α) =
X
N(c)≤X
c≡1(mod 3), α|c
˜g
l
(c),
for any α ∈ ZZ[ω].
As in [4;§3] we apply Vaughan’s identity, though now with a minor modifi-
cation. We write
Σ
j
(X, u) =
X
a,b,c
Λ(a)µ(b)˜g
l
(abc),
where a, b, c run over square-free elements of ZZ[ω], subject to the conditions
a, b, c ≡ 1 (mod 3) and X < N(abc) ≤ 2X and
N(bc) ≤ u, j = 0,
N(b) ≤ u, j = 1,
N(ab) ≤ u, j = 2
0
,
N(a), N(b) ≤ u < N(ab), j = 2
00
,
N(b) ≤ u < N(a), N(bc), j = 3,
N(a), N(bc) ≤ u, j = 4.
4
Then
Σ
0
(X, u) + Σ
2
0
(X, u) + Σ
2
00
(X, u) + Σ
3
(X, u) = Σ
1
(X, u) + Σ
4
(X, u).
The reader will note that we have split Σ
2
(X, u) (as given in [4;§3]) into Σ
2
0
(X, u)
and Σ
2
00
(X, u). We shall suppose that 1 ≤ u ≤ X
1/3
. Then, just as in [4;§3],
we have
Σ
0
(X, u) = H
l
(2X) − H
l
(X),
|Σ
1
(X, u)| ≤ 3 log(2X)
X
N(α)≤u
max
X<z≤2X
|F
l
(z, α)|, (5)
|Σ
2
0
(X, u)| ≤ 2 log u
X
N(α)≤u
max
X<z≤2X
|F
l
(z, α)|, (6)
and
Σ
4
(X, u) = 0.
We shall bound Σ
1
(X, u) and Σ
2
0
(X, u) by means of the following estimate.
Lemma 1 For any ε > 0 and any α ∈ ZZ[ω] with N(α) ≤ X
1/3
we have
F
l
(X, α) ¿
ε
δ
l
X
5/6
N(α)
−1
+ X
2/3+ε
N(α)
−1/2
+ |l|X
1/2+ε
N(α)
−1/4
,
where δ
l
is 1 for l = 0 and is 0 otherwise.
This may be compared with Theorem 4 of [4] which is the bound
F
l
(X, α) ¿
ε
δ
l
X
5/6
N(α)
−1
+ X
3/4+ε
N(α)
−5/8
+ |l|X
1/2+ε
N(α)
−1/4
.
We take u = X
1/3
, so that (5) and (6) yield
Σ
1
(X, u), Σ
2
0
(X, u) ¿
ε
X
5/6+ε
+ |l|X
3/4+ε
,
on re-defining ε.
For the proof of Theorem 1 it therefore remains to obtain similar bounds for
Σ
2
00
(X, u) and Σ
3
(X, u), which will be achieved with the aid of Theorem 2. We
begin by recalling that
˜g
l
(vw) = ˜g
l
(v)˜g
l
(w)(
w
v
)
3
,
see Hasse [2; pp.443-445], for example. Thus, if we write
A(v) =
X
ab=v
N(a),N(b)≤u
Λ(a)µ(b)˜g
l
(ab)
5