Kloosterman sums and primitive elements in Galois ﬁelds by

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XCIV.2 (2000)

Kloosterman sums and primitive elements in Galois fields

by

Stephen D. Cohen (Glasgow)

1. Introduction. Let F

_q

denote the finite (Galois) field of order q, a power of a prime p. The multiplicative group F

^∗_q

of F

q

is cyclic of order q −1:

a generator is known as a primitive element of F

_q

. Hence F

_q

contains φ(q−1) primitive elements, where φ is Euler’s function. Generally, primitivity is a fragile property that may be destroyed when the element in question is modified through multiplication or addition. Nevertheless, if ξ is a primitive element, then so is 1/ξ.

When q = 2, H. Niederreiter [Ni] has expressed the number of irreducible polynomials of degree n (≥ 3) over the binary field F

₂

, having the coefficients of x

ⁿ⁻¹

and x both equal to 1, as a formula involving Kloosterman sums over F

₂n

. Thereby, this number is shown to be positive, except when n = 3.

An alternative formulation of this conclusion is that, except when n = 3, F

2ⁿ

contains an element ξ such that F

2ⁿ

= F

2

(ξ), and both ξ and 1/ξ have (F

₂n

, F

₂

)-trace equal to 1. In this paper we consider extensions F

_qn

of a general finite field F

_q

. The aim is to show that Kloosterman sums are adequate for the stiffer task of generalising the above result (when n ≥ 5) to yield the existence of a primitive element ξ of F

_qⁿ

such that T

_n

(ξ) = a and T

_n

(1/ξ) = b, where a and b are (arbitrary) given elements of F

_q

and T

n

(ξ) := ξ + ξ

^q

+ . . . + ξ

^qⁿ⁻¹

denotes the (F

qⁿ

, F

q

)-trace of ξ. The result to be proved is as follows.

Theorem 1.1. Let q be a prime power and n (≥ 5) be an integer. Suppose that arbitrary elements a and b of F

_q

are given. Then there exists a primitive element ξ of F

qⁿ

such that T

n

(ξ) = a and T

n

(1/ξ) = b, except when a = b = 0 and (q, n) = (4, 5), (2, 6), or (3, 6).

Theorem 1.1 is consistent with the pattern that, as n increases, one can expect to guarantee the existence of a primitive element satisfying additional

2000 Mathematics Subject Classification: 11L05, 11T24, 11T30.

[173]

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constraints. Let it be stressed that what are sought are complete results listing all exceptions. For example, prior to Theorem 1.1 is the theorem of Cohen [Co1] (see also [JuVa]) that, given a ∈ F

_q

and n ≥ 2 (n ≥ 3, if a = 0), there exists a primitive element ξ of F

qⁿ

with T

n

(ξ) = a, except when n = 3, q = 4, and a = 0. Another stage in the scheme is described later in the introduction.

The proof of Theorem 1.1 derives from careful estimates in respect of expressions that combine Kloosterman sums over F

qⁿ

and over F

q

. Next, some properties of Kloosterman sums (and Gauss sums) will be developed.

For example, whereas the absolute value of a Kloosterman sum over F

q

is bounded by 2 √

q, on average, it is less than √

q (Corollary 3.2). By means of a sieving process, the proof is completed theoretically, without direct verification, except for a few small values of q (≤ 16) and n = 5 or 6, plus q = 2 when n = 8 (when a and b are not both zero). In fact, when a and b are both zero, then Theorem 1.1 follows from the work of W.-S. Chou and the author [ChCo] and is “best possible” in the sense that it fails for all pairs (q, n) with n < 5. Otherwise, the method will succeed, in principle, also when n = 4. We defer the study of this case to a further paper, because of the difficulty of identifying efficiently those values of q for which direct verification is required. The question addressed is sensible even when n = 3, but the method fails, and it may be difficult to resolve that case.

For n ≥ 2, the associated (irreducible) minimal polynomial (of degree n) over F

_q

of a primitive element ξ of F

_q

is itself referred to as primitive. Since the (F

_qⁿ

, F

_q

)-norm of such an element ξ is necessarily a primitive element of F

q

, and so, when q = 2 or 3, is uniquely determined, we have the following consequence of Theorem 1.1.

Corollary 1.2. Suppose that q is a prime power , n ≥ 5, and a

n−1

and a

₁

are given elements of F

_q

. Then, if either a

_n−1

= a

₁

= 0 or q ≤ 3, there exists a primitive polynomial of the form

(1.1) x

ⁿ

+ a

n−1

x

ⁿ⁻¹

+ . . . + a

1

x + a

0

.

More generally, Theorem 1.1 implies that there is a primitive polynomial of the form (1.1) for n ≥ 5 with both a

n−1

and the ratio a

1

/a

0

prescribed.

The Kloosterman sum technique should be instrumental in delivering the next stage in the programme beyond Theorem 1.1, namely, that for given values of a

n−1

, a

1

, and a

0

(with a

0

a primitive element of F

q

and n ≥ 6 or, perhaps, n ≥ 7), there is a primitive polynomial (1.1) with prescribed values of a

_n−1

, a

₁

and a

₀

. For other results, conjectures, and data on the existence of primitive elements of F

qⁿ

satisfying further constraints, see, for example, [HaMu], [Mu], [MoMu], [Ha], [CoHa1,2], [Co2]. I also acknowledge some preliminary notes by Dr Wun-Seng Chou (Taipei) on the “fixed traces”

question, out of which the considerations of this paper arose.

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From the results of [ChCo] we can exclude the case a = b = 0 of Theorem 1.1 in what follows. Indeed, by symmetry, we shall assume, without loss of generality, that a 6= 0.

2. Character sum formulation. The number of elements ξ of F

^∗_q

for which T

_n

(ξ) = a and T

_n

(1/ξ) = b can be expressed in terms of (standard) Kloosterman sums over F

_qⁿ

. The further constraint that ξ can be primitive heralds the introduction of multiplicative characters and more intricate expressions involving generalised (or twisted) Kloosterman sums. Obtaining the relevant formulae in their most transparent form is the object of this section. For the background on Kloosterman sums and Gauss sums for this section and Section 3, Chapter 5 of [LiNi] (including the Exercises) or some other source may be consulted.

Let M be a divisor of q

ⁿ

− 1. If ξ ∈ F

^∗_qn

is such that ξ = α

^d

, where d | M and α ∈ F

_qⁿ

, implies d = 1, we shall say that ξ is not any kind of Mth power in F

_qⁿ

. Given q, n, a, b as in Section 1, define N

_q,n

(M ; a, b) (= N

_q,n

(M ) = N (M )) to be the number of elements ξ of F

^∗_qn

, scaled (multiplied) by a factor q

²

(for convenience), such that T

_n

(ξ) = a, T

_n

(1/ξ) = b, and ξ is not any kind of M th power in F

_qⁿ

. To establish Theorem 1.1 in respect of q, n, a, b, it is necessary to show that N

q,n

(M ; a, b) for general divisors M of q

ⁿ

− 1.

Note, in particular, that the value of N (M ) depends only on the distinct prime divisors of M , i.e., on the square-free part of M .

Next, we lay down the basic material on characters. It will be amplified later. Let χ be the canonical additive character of F

_q

. Thus, for x ∈ F

_q

, χ(x) = exp(2πiT (x)/p), where here T denotes the absolute trace (from F

q

to F

p

). Moreover, every additive character b χ of F

q

is such that b χ(x) = χ(cx) (x ∈ F

_q

) for some c ∈ F

_q

; take c = 0 to obtain the trivial character χ

₀

. Further, let χ

⁰

= χ(T

n

) denote the lift of χ to F

qⁿ

. Passing to multiplicative characters of F

_qn

, we shall reserve the symbol ψ for such; more precisely, for any divisor d of q

ⁿ

− 1, ψ

_d

will denote a typical character of F

_qⁿ

of exact order d. Thus, ψ

1

is the trivial character.

Now, for any α, β ∈ F

qⁿ

and any multiplicative character ψ, we define the generalised Kloosterman sum K

_n

(α, β; ψ) (= K

_q,n

(α, β; ψ)) by

K

n

(α, β; ψ) = X

ξ∈F^∗_qn

χ

⁰

(αξ + βξ

⁻¹

)ψ(ξ).

In particular, we write K

_n

(α, β) for K

_n

(α, β; ψ

₁

), the (standard) Klooster- man sum.

As a final preliminary to the basic formula, we describe some further notation. In a sum P

u

(or double sum P

u,v

), the variable(s) will be assumed to run over all members of the ground field F

q

. If u runs over F

^∗_q

we will write P

u6=0

, etc. For any divisor M of q

ⁿ

− 1, abbreviate to T

d|M

a weighted

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sum of the form P

d|M

(µ(d)/φ(d)) P

(d)

ψ

d

, where µ is the M¨obius function, and the inner sum ranges over all φ(d) characters of order d. Further for any positive integer h, set

Θ(h) = φ(h)/h = Y

l|h, l prime

(1 − l

⁻¹

).

Moreover, a bar over a symbol signifies complex conjugation.

Proposition 2.1. Let q be a prime power and n a positive integer. Sup- pose that elements a, b of F

q

are given. Then, for any divisor M of q

ⁿ

− 1, we have

(2.1) N

q,n

(M ; a, b) = Θ(M ) X

u,v

χ(au + bv) \

d|M

K

n

(u, v; ψ

d

).

P r o o f. The characteristic function for the subset of F

^∗_qn

comprising elements that are not any kind of M th power can be expressed as an ex- tension of the Vinogradov formula, [Ju], Lemma 7.5.3. It takes the form Θ(M ) T

d|M

ψ

_d

(ξ). Moreover, that for the subset of F

_qⁿ

comprising elements with T

n

(ξ) = a is (1/q) P

c

χ(c(T

n

(ξ) − a)). Hence, taking account of the scaling factor q

²

, we have

N

_q,n

(M ; a, b) = X

ξ∈F^∗_qn

Θ(M ) X

u,v

χ(u(T

_n

(ξ) − a))χ(v(T

_n

(ξ

⁻¹

) − b)) \

d|M

ψ

_d

(ξ)

= Θ(M ) X

u,v

χ(au + bv) \

d|M

χ(T

_n

(uξ + vξ

⁻¹

))ψ

_d

(ξ), and the result follows from the definition of the Kloosterman sum.

From Proposition 2.1, N (M ) can immediately be estimated using the standard bound for Kloosterman sums that follows.

Lemma 2.2. Let ψ be a multiplicative character of F

_qⁿ

. Then K

n

(0, 0; ψ) =

n q

ⁿ

− 1 if ψ = ψ

₁

, 0 otherwise.

Further , if either ψ 6= ψ

1

or α, β ∈ F

qⁿ

are not both zero, then

|K

n

(α, β; ψ)| ≤ 2q

^n/2

.

Nevertheless, before applying Lemma 2.2, it is profitable to develop the formula (2.1). This is the aim in the next sequence of lemmas. They also involve Gauss sums, since some Kloosterman sums reduce to these. The Gauss sum G

_n

(ψ) is defined by

G

_n

(ψ) = X

ξ∈F^∗_qn

χ

⁰

(ξ)ψ(ξ).

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Lemma 2.3. (i) If α (6= 0), β ∈ F

qⁿ

, then

K

_n

(α, β; ψ) = ψ(α)K

_n

(1, αβ; ψ).

(ii) If β 6= 0, then K

n

(0, β; ψ) = ψ(β)G

n

(ψ).

(iii) If α 6= 0, then K

_n

(α, 0; ψ) = ψ(α)G

_n

(ψ).

Lemma 2.4. (i) G

n

(ψ

1

) = −1.

(ii) If ψ 6= ψ

₁

, then |G

_n

(ψ)| = q

^n/2

.

(iii) If q is odd and n = 1, then G

²₁

(ψ

₂

) = (−1)

^(q−1)/2

q.

(iv) If q is odd and n is even, then G

_n

(ψ

₂

) = −(−1)

^n(q−1)/4

q

^n/2

. (v) If q and n are odd and λ

₂

is the quadratic character on F

^∗_q

, then

G

₁

(λ

₂

)G

_n

(ψ

₂

) = (−1)

(n−1)(q−1)/4

q

^(n+1)/2

.

Note that Lemma 2.4 contains information relating to the specific characters ψ

1

and (when q is odd) ψ

2

, the quadratic character. Similarly, we can specialise Kloosterman sums to these cases, though the results are more complicated. First, when ψ = ψ

₁

and u, v are in the ground field F

_q

, then K

n

(u, v) can be expressed in terms of Kloosterman sums over F

q

. For t ∈ F

^∗_q

, set k

_t

:= K

₁

(1, t). Further, let D

_n

(X, c) be the Dickson polynomial of the first kind of degree n. Thus, D

_n

(X + c/X, c) = X

ⁿ

+ c

ⁿ

/X

ⁿ

.

Lemma 2.5. Suppose t ∈ F

q^∗

. Then

K

_n

(1, t) = (−1)

ⁿ⁻¹

D

_n

(k

_t

, q).

Define δ

q,n

:=

1 q − 1

X

t6=0

K

n

(1, t)

q

^n/2

, δ

_q,n^∗

:=

1 q − 1 X

t6=0

|K

n

(1, t)|

q

^n/2

. Then, from Lemma 2.2, it follows that

(2.2) |δ

_q,n

| ≤ δ

_q,n^∗

≤ 2.

The next result indicates that the numbers k

_t

are essentially uniformly distributed.

Lemma 2.6. For a given prime power q, P

t6=0

k

_t

= 1.

P r o o f.

X

t6=0

k

_t

= X

t,u6=0

χ

u + t

u

= χ(0) = 1,

since, evidently, for any c ∈ F

q

, the equation u + t/u = c (i.e., t = u(c − u)) has q − 2 solution pairs (t, u) ∈ (F

^∗_q

)

²

, unless c = 0, when there are q − 1 solutions.

When q is odd, Kloosterman sums with ψ = ψ

2

(the quadratic character)

are apparently easier to evaluate than standard sums. The following two

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results derive, in essence, from Lemma 3.5 and Corollary 3.6 of [ChCo]. For clarity, for any divisor e of q − 1, we denote a multiplicative character of order e of F

_q

by λ

_e

.

Lemma 2.7. Let q be an odd prime power. If n is odd and t ∈ F

^∗_q

is such that λ

2

(t) = −1, then K

n

(1, t; ψ

2

) = 0. If λ

2

(t) = 1 with s

²

= t (s ∈ F

q

) and p - n, then

K

_n

(1, t; ψ

₂

) = (χ(2ns) + χ(−2ns))G

_n

(ψ

₂

).

Otherwise, if n is odd, p | n, and λ

2

(t) = 1, or n is even and either p | n or λ

₂

(t) = −1, then

K

n

(1, t; ψ

2

) = 2G

n

(ψ

2

).

Lemma 2.8. Let q be an odd prime power and n be an even integer. Then X

t6=0

K

_n

(1, t; ψ

₂

) = −(−1)

^n(q−1)/4

(εq − 2)q

^n/2

, where

ε =

2 if p | n, 1 if p - n.

Given q, n, define

m = m(q, n) := q

ⁿ

− 1 q − 1 .

The significance of this number is that there are occasions when it is useful to distinguish characters ψ

d

for which d | m. This arises as follows. For any divisor d of q

ⁿ

−1, the restriction to F

^∗_q

of a character ψ

_d

(of F

^∗_qn

) will be denoted by b ψ

_d

. Then, in fact, b ψ

_d

= λ

_d∗

, where d

^∗

= (q − 1)/gcd((q

ⁿ

− 1)/d, q − 1).

In particular, b ψ

_d

= λ

₁

if and only if d | m. Moreover, if q is odd, then

(2.3) ψ b

_d

=

λ

₂

if n is odd, λ

₁

if n is even.

To present refinements of Proposition 2.1, we modify the notation T

d|M

used in its statement. First, for a divisor M

⁰

of M , write T

d|M, d

-

M⁰

to indicate a similar sum that excludes terms corresponding to divisors d of M

⁰

. This is modified further to T

d|M, d

-

M⁰, 6=2

to signify that the terms with d = 2 are excluded (if there remain any). This makes a difference only if q is odd, M is even, and M

⁰

is odd. A frequent choice of M

⁰

is M

^∗

:= gcd(M, m).

We come to the main theorems of this section; their statements depend

heavily on declared notation. Because the formulae have a different shape

according to whether a, b are zero or not, it is from this point on we insist

that a 6= 0. In the first case, we also suppose that b 6= 0.

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Theorem 2.9. Let q be a prime power and n (≥ 4) an integer. Suppose that a, b ∈ F

^∗_q

. Then, for any divisor M of q

ⁿ

− 1, we have

N

_q,n

(M ; a, b) = Θ(M ) n

q

ⁿ

+ 1 + ∆

₂

+ (−1)

ⁿ⁻¹

X

t6=0

k

_tab

D

_n

(k

_t

, q) (2.4)

+ \

d|M d

-

2

X

t6=0

K

1

(1, tab; b ψ

d

)K

n

(1, t; ψ

d

) − 2 \

d|M^∗ d

-

2

G

n

(ψ

d

)

+ \

d|M d

-

M^∗, 6=2

( b ψ

_d

(a) + b ψ

_d

(b))G

₁

( b ψ

_d

)G

_n

(ψ

_d

) o

,

where ∆

₂

= 0, unless q is odd and M is even. In the latter event, if n is odd,

∆

₂

= (−1)

(n−1)(q−1)/4

γq

^(n+1)/2

, where

γ =

 



4 if λ

₂

(a) = λ

₂

(b) = −1, but ab 6= n

²

, 2(1 − λ

2

(a)) − q if ab = n

²

,

0 otherwise,

and, if n is even, ∆

₂

= (−1)

^n(q−1)/4

γq

^n/2

, where γ =

(−1)

^(q+1)/2

+

¹₂

q −

¹₂

if ab = n

²

,

(−1)

^(q+1)/2

λ

2

(ab) −

¹₂

λ

2

(ab − n

²

)

q −

¹₂

otherwise.

P r o o f. We consider the terms on the right hand side of (2.1) (taking for granted the factor Θ(M )). First, by Lemma 2.2, the contribution of the terms with u = v = 0 is exactly q

ⁿ

− 1. Next, by Lemma 2.3(i), the contribution of the terms with uv 6= 0, on replacement of uv by t, becomes

X

t6=0

X

u6=0

χ

au + bt u

ψ b

_d

(u)K

_n

(1, t; ψ

_d

).

This yields the first two sums on the right hand side of (2.4) (with T

16=d|M

rather than T

d|M , d

-

2

) on replacing au by u and separating, with the aid of Lemma 2.5, the contribution arising from ψ

₁

.

Next, by Lemma 2.3(ii), the contribution of the terms of (2.1) with u = 0,

v 6= 0, is \

d|M

n X

v6=0

χ(bv) b ψ

_d

(v) o

G

_n

(ψ

_d

).

On interchanging ψ

d

with its conjugate and replacing bv by v, we obtain

\

d|M

ψ b

d

(b)G

1

( b ψ

d

)G

N

(ψ

d

) = 1 − \

16=d|M^∗

G

n

(ψ

d

) + \

d|M d

-

M^∗

ψ b

d

(b)G

1

(ψ

d

)G

n

(ψ

d

),

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by Lemma 2.4(i). There is a similar contribution from the terms with u 6= 0, v = 0.

As a consequence of the above, it suffices to assume that q is odd and M is even, and prove that the net contribution of the terms in (2.1) corresponding to ψ

2

is the designated expression for ∆

2

.

To this end, suppose first that n is odd; thus M

^∗

is odd. Observing that the weighting factor µ(2)/φ(2) = −1, from (2.4) we have

∆

₂

= −λ

₂

(a)(λ

₂

(ab) + 1)G

₁

(λ

₂

)G

_n

(ψ

₂

) − X

t6=0

K

₁

(1, tab; λ

₂

)K

_n

(1, t; ψ

₂

).

Now, for t ∈ F

^∗_q

, λ

2

(a) = 1 if and only if ψ

2

(a) = 1. Consequently, from Lemma 2.7, the product K

₁

(1, tab; λ

₂

)K

_n

(1, t; ψ

₂

) is zero unless λ

₂

(t) = λ

2

(tab) = 1. Evidently, therefore, ∆

2

= 0, unless λ

2

(ab) = 1. Moreover, if λ

₂

(ab) = 1 and c

²

= ab, c ∈ F

^∗_q

, then, by Lemma 2.7 again, we have

X

t6=0

K

1

(1, tab; λ

2

)K

n

(1, t; ψ

2

)

= G

1

(λ

2

)G

n

(ψ

2

) X

s6=0

χ(2ns)(χ(2sc) + χ(−2sc))

= G

₁

(λ

₂

)G

_n

(ψ

₂

) X

s6=0

{χ(2s(n + c)) + χ(2s(n − c))}.

The sum in this expression is q − 2 if c = ±n (i.e., ab = n

²

). Otherwise, it is −2. The result for n odd now follows from Lemma 2.4(v).

Finally, suppose n is even, so that M

^∗

is even. Then, by (2.3), b ψ

₂

= λ

₁

, and

∆

2

= 2G

n

(ψ

2

) − X

t6=0

K

1

(1, tab)K

n

(1, t; ψ

2

) (2.5)

= G

n

(ψ

2

)

2 − X

t6=0

(1 − λ

2

(t))K

1

(1, tab) − X

s6=0

χ(2ns)K

1

(1, s

²

ab)

= G

n

(ψ

2

)

1 + X

t6=0

λ

2

(t)K

1

(1, tab) − X

s6=0

χ(2ns)K

1

(1, s

²

ab)

, by Lemmas 2.7 and 2.6. Now, in (2.5), we have

X

t6=0

λ

2

(t)K

1

(1, tab) = X

t,u6=0

χ

u + tab u

λ

2

(t) = X

u6=0

χ(u) X

t6=0

χ

abt u

λ

2

(t)

= λ

₂

(ab) X

u6=0

χ(u)λ

₂

(u)G

₁

(λ

₂

)

= λ

₂

(ab)G

²₁

(λ

₂

) = (−1)

^(q−1)/2

λ

₂

(ab)q,

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by Lemma 2.4(iii). Also, in (2.5), we have S := X

s6=0

χ(2ns)K

₁

(1, s

²

ab) = 1 + X

s

X

u6=0

χ

u + s

²

ab u + 2ns

= 1 + X

u6=0

χ

u

1 − n

²

ab

X

s

χ

ab u

s − nu

ab

₂

= 1 + X

u6=0

χ

u

1 − n

²

ab

1 2

X

v6=0

χ

ab u v

(1 + λ

2

(v))

+ 1

.

Now, if ab = n

²

, it follows that S = 1+q−1+ 1

2 X

v6=0

(1+λ

₂

(v)) X

u6=0

χ(abvu) = q− 1 2

X

v6=0

(1+λ

₂

(v)) = 1 2 (q+1).

Hence, in this case,

∆

₂

= G

_n

(ψ

₂

)

1 2 + (−1)

^(q−1)/2

q − 1 2 q

,

as required (by Lemma 2.4(iv)). On the other hand, if ab 6= n

²

, then S = 1

2 X

u6=0

χ

u

1 − n

²

ab

−1 + X

v6=0

χ

ab u v

λ

₂

(v)

= 1 2

1 + G

1

(λ

2

) X

u6=0

χ

u

1 − n

²

ab

λ

2

(abu)

= 1

2 (1 + λ

₂

(ab − n

²

)G

²₁

(λ

₂

)) = 1

2 (1 + (−1)

^(q−1)/2

λ

₂

(ab − n

²

)q).

Again, by Lemma 2.4(iv), this yields the stated expression for ∆

2

. Here now is the corresponding result with b = 0.

Theorem 2.10. Let q be a prime power and n (≥ 4) an integer. Suppose that a ∈ F

^∗_q

. Then, for any divisor M of q

ⁿ

− 1, we have

N

_q,n

(M ; a, 0) = Θ(M ) n

q

ⁿ

− q + 1 + ∆

₂

+ (−1)

ⁿ

X

t6=0

D

_n

(k

_t

, q) (2.6)

+ \

d|M^∗ d

-

2

h

(q − 2)G

_n

(ψ

_d

) − X

t6=0

K

_n

(1, t; ψ

_d

) i

+ \

d|M d

-

M^∗, 6=2

ψ

d

(a)G

1

( b ψ

d

) h

G

n

(ψ

d

) + X

t6=0

K

n

(1, t; ψ

d

) io

,

(10)

where ∆

2

= 0, unless q is odd, M is even and p | n. In the latter event, we have

∆

2

=

−(−1)

^n(q−1)/4

q

^n/2+1

if n is even,

−λ

2

(a)(−1)

(n−1)(q−1)/4

q

^(n+3)/2

if n is odd.

P r o o f. Again the contribution of the terms in (2.1) with u = v = 0 is q

ⁿ

− 1. That from the terms with u = 0, v 6= 0 is

\

d|M

G

_n

(ψ

_d

) X

v6=0

ψ b

_d

(v) = −(q − 1) + (q + 1) \

16=d|M^∗

G

_n

(ψ

_d

),

by Lemmas 2.3 and 2.4, plus the fact that b ψ

_d

is non-trivial if and only if d - M

^∗

. The contribution from the terms with u 6= 0, v = 0 is

\

d|M

G

_n

(ψ

_d

) X

u6=0

χ(au) b ψ

_d

(u) = 1− \

16=d|M^∗

G

_n

(ψ

_d

)+ \

d|M d|M^∗

ψ

_d

(a)G

₁

( b ψ

_d

)G

_n

(ψ

_d

).

This, again, has used the fact that b ψ

d

is trivial whenever d | M

^∗

. The contribution of the terms with uv 6= 0 is

\

d|M

X

t,u6=0

χ(au) b ψ

_d

(u)K

n

(1, t; ψ

d

)

= − \

d|M^∗

K

n

(1, t; ψ

d

) + \

d|M d

-

M^∗

ψ

d

(a)G

1

( b ψ

d

)K

n

(1, t; ψ

d

).

This yields the remaining terms on the right-hand side of (2.6), apart from the evaluation of the terms arising from ψ

₂

.

Indeed, we now describe the contribution of the terms involving ψ

₂

(when q is odd and M is even). If n is even, then M

^∗

is even and

∆

2

= −(q − 2)G

n

(ψ

2

) + X

t6=0

K

n

(1, t; ψ

2

)

=

−(−1)

^n(q−1)/4

q

^n/2+1

if p | n,

0 otherwise,

by Lemma 2.8. On the other hand, if n is odd and so M

^∗

is odd, then

∆

₂

= −λ

₂

(a)G

₁

(λ

₂

)

G

_n

(ψ

₂

) + X

t6=0

K

_n

(1, t; ψ

_d

)

= −λ

₂

(a)G

₁

(λ

₂

)

G

_n

(ψ

₂

) + X

s6=0

χ(2ns)G

_n

(ψ

_d

)

, by Lemma 2.7. Thus, λ

2

= 0, unless p | n, in which event,

∆

₂

= −qλ

₂

(a)G

₁

(λ

₂

)G

_n

(ψ

₂

) = −λ

₂

(a)(−1)

(n−1)(q−1)/4

q

^(n+3)/2

,

(11)

by Lemma 2.4(v). Hence, everything is proved.

3. Inequalities. The sums in the identities of Theorems 2.9 and 2.10 can obviously be estimated by means of the fundamental bounds for Kloost- erman sums and Gauss sums stated in Lemmas 2.2 and 2.4(ii). Thus, for example, with regard to the main error terms in Theorem 2.9, we have

X

t6=0

K

1

(1, tab; b ψ

d

)K

n

(1, t; ψ

d

)

≤ 4(q − 1)q

^(n+1)/2

, d > 1.

Crucially, we are able to halve this estimate; it then becomes similar to the corresponding estimate for the main error terms in Theorem 2.10, namely

G

1

( b ψ

_d

) X

t6=0

K

_n

(1, t; ψ

_d

)

≤ 2(t − 1)q

^(n+1)/2

, d > 1.

Lemma 3.1. For any multiplicative character λ of F

^∗_q

, we have X

t6=0

|K

₁

(1, t; λ)|

²

=

q(q − 1) − 1 if λ = λ

₁

, q(q − 2) if λ 6= λ

₁

. P r o o f.

X

t6=0

|K

1

(1, t; λ)|

²

= X

t,u,v6=0

χ

u + t

u − v − t v

λ

u v

= (q − 1)

²

+ X

t6=u

X

u,v6=0 u6=v

χ

u − v − t(u − v) uv

λ

u v

= (q − 1)

²

+ X

t6=u

X

y6=0 z6=0,1

χ

y − t(z − 1)

²

yz

λ(z)

= (q − 1)

²

+ X

y6=0 z6=0,1

λ(z) X

t6=0

χ

y − t(z − 1)

²

yz

= (q − 1)

²

− X

z6=0,1

λ(z) X

y6=0

χ(y) = (q − 1)

²

+ X

z6=0,1

λ(z)

=

(q − 1)

²

+ (q − 2) if λ = λ

₁

, (q − 1)

²

− 1 if λ 6= λ

1

, and the result follows.

Corollary 3.2. For λ as in Lemma 3.1, X

t6=0

|K

₁

(1, t; λ)| < (q − 1) √

q.

(12)

Hence, if d (> 1) is a divisor of q

ⁿ

− 1 and ab ∈ F

^∗_q

, then

X

t6=0

K

₁

(1, tab; b ψ

_d

)K

_n

(1, t; ψ

_d

)

< 2(q − 1)q

^(n+1)/2

. P r o o f. By Cauchy’s inequality and Lemma 3.1,

X

t6=0

|K

₁

(1, t; λ)| ≤ p

q − 1 X

t6=0

|K

₁

(1, t; λ)|

²

_1/2

< p

q − 1 p

q(q − 1)

= (q − 1) √ q.

Granted Lemma 2.2, the other inequality is immediate.

Taking λ = λ

1

in Corollary 3.2, we see that δ

^∗_q

:= δ

_q,1^∗

(see Section 2) satisfies δ

^∗_q

< 1 (cf. (2.2)). Indeed, for a range of prime values of q tested (with 7 < q < 200), δ

^∗_q

lay within the interval (0.82, 0.89).

Moreover, we can effectively improve further on Corollary 3.2 under the following circumstances: q, n odd, d even, d | 2m (so that b ψ

d

= λ

2

).

Lemma 3.3. Let q be an odd prime power and χ be the canonical additive character on F

_q

. Then

X

s

|χ(s) + 1| = 2q p cosec

π 2p

. P r o o f. Suppose q = p, an odd prime. Then

X

s

|χ(s) + 1| = X

s

2|cos(πs/p)| = 2

1 + 2

(p−1)/2

X

s=0

cos(πs/p)

= 2 cosec(π/(2p)).

This implies the result for a prime power q, since the elements of F

_q

are uniformly distributed with respect to absolute trace.

Lemma 3.4. Let q be an odd prime power. Define κ

_q

by

κ

q

=

 

 

 

 4

π if q = p or p

²

, 4

π

1 + 2

p

²

otherwise.

Then X

s6=0

|χ(s) + 1| < κ

_q

(q − 1).

P r o o f. Using Lemma 3.3 and setting x = π/(2p), we obtain X

s6=0

|χ(s) + 1| = 2

q

p cosec x − 1

< 2

q

p(x − x

³

/6)

− 1 < 4

π (q − 1),

(13)

provided

q p

²

< 12

π

1 − 2

π

1 − x

²

6 .

Because x = π/(2p) ≤ π/6, the right side of this inequality exceeds 1.3, and so the inequality holds whenever q = p or p

²

. The adjustment shown suffices for general values of q.

Lemma 3.5. For any α ∈ F

^∗_qn

and multiplicative character ψ, K

n

(1, α; ψ) = ψ(α)K

n

(1, α; ψ).

P r o o f. Replace ξ by α/ξ in the definition of K

n

(1, α; ψ).

The climax of the preceding few lemmas comes next. In it, for a character ψ, set

S(ψ) := X

t6=0

K

₁

(1, t; b ψ)K

_n

(1, t; ψ).

Lemma 3.6. Let q be an odd prime power and n an odd integer. Suppose that a, b ∈ F

^∗_q

and d is an even divisor of 2m. If λ

2

(ab) = −1, then S(ψ

d

) + S(ψ

_d

) = 0, whereas if λ

₂

(ab) = 1, then

|S(ψ

_d

) + S(ψ

_d

)| < 2κ

_q

(q − 1)q

^(n+1)/2

,

where κ

q

is as defined in Lemma 3.4; in particular , κ

q

= 4/π whenever q = p or p

²

.

P r o o f. By Lemma 3.5, and the fact that b ψ

_d

= λ

₂

, S(ψ

_d

) + S(ψ

_d

) = X

t6=0

(1 + λ

₂

(t))K

₁

(1, tab; λ

₂

)K

_n

(1, t; ψ

_d

).

If λ

₂

(ab) = −1, then either λ

₂

(t) = −1 or λ

₂

(tab) = −1 whenever t ∈ F

^∗_q

. Hence, the above expression is zero by Lemma 2.7. If λ

2

(ab) = 1, with c

²

= ab (c ∈ F

^∗_q

), then, again by Lemma 2.7,

|S(ψ

d

) + S(ψ

_d

)| = 2|G

1

(λ

2

)|

X

s6=0

χ(2cs)K

n

(1, s

²

; ψ

d

)

= √ q

X

s6=0

(χ(2cs) + χ(−2cs))K

n

(1, s

²

; ψ

d

)

≤ 2q

^(n+1)/2

X

s6=0

|χ(2cs) + χ(−2cs)|, by Lemma 2.2. The result follows from Lemma 3.4, since

X

s6=0

|χ(2cs) + χ(−2cs)| = X

s6=0

|χ(s/2) + χ(−s/2)| = X

s6=0

|χ(s) + 1|.

(14)

Lemma 3.5 can be exploited for other characters χ, but not with such generality. The consequences tend to depend on the order of ab more specif- ically.

The remaining inequality is of a different nature and derives from a sieving process. The aim is to obtain a lower bound for N

_q,n

(M ) that may not be good asymptotically (as q → ∞) but is especially effective for small values of q.

As noted in Section 2, given q, n, a, b, the value of N (M ) (M is a divisor of q

ⁿ

−1) depends only on the distinct prime factors of M . Accordingly, divisors M

1

, . . . , M

r

(r ≥ 1) of M will be called complementary divisors of M with common divisor M

₀

if the set of distinct prime divisors of lcm{M

₁

, . . . , M

_r

} is the same as that of M , and, for any pair (i, j) with 1 ≤ i 6= j ≤ r, the set of distinct prime divisors of gcd(M

i

, M

j

) is that of M

0

. When r = 1, we have M

₁

= M

₀

= M . Though easy to prove (see [ChCo], Proposition 6.1), the following inequality is extremely useful.

Proposition 3.7. Let q be a prime power and n (≥ 1) an integer. Sup- pose that a, b ∈ F

_q

. Let M

₁

, . . . , M

_r

be complementary divisors of M (itself a divisor of q

ⁿ

− 1) with common divisor M

0

. Then, with N (M ) = N

_q,n

(M ; a, b), etc., we have

N (M ) ≥ n X

^r

i=1

N (M

_i

) o

− (r − 1)N (M

₀

).

Proposition 3.7 motivates a generalisation of the ratio Θ(M ) defined in Section 2. With the same notation, define

Θ(M

₁

, . . . , M

_r

) :=

n X

^r

i=1

Θ(M

_i

) o

− (r − 1)Θ(M

₀

).

This number represents the coefficient of q

ⁿ

on the right hand side of the inequality when Theorems 2.9 or 2.10 are applied to each of the constituents.

To be useful, therefore, it is vital that Θ(M

₁

, . . . , M

_r

) be positive. Indeed, preferably the ratio b Θ(M

₁

, . . . , M

_r

) := Θ(M

₁

, . . . , M

_r

)/Θ(M

₀

) should not be too small. For this reason, when q is odd we suppose (except in one place) that M

₀

is even, because of the effect of the prime 2 otherwise. Consequently, we say that the complementary divisors are regular if qM

₀

is even.

4. Criteria for success. These criteria are effective only if n ≥ 4, which we assume from now on. Again, assume also that the prime power q and a (6= 0), b ∈ F

q

are given. Also, from now on, fix M as q

ⁿ

− 1; M

1

, . . . , M

r

will be regular complementary divisors of M with common divisor M

₀

. We

supplement the previously defined integer m = (q

ⁿ

− 1)/(q − 1) with the

definitions m

_i

:= gcd(M

_i

, m), i = 0, 1, . . . , r. For any positive integer h, let

(15)

W (h) = 2

^ω(h)

be the number of square-free divisors of h (where ω(h) is the number of distinct prime factors of h).

We derive criteria for N

q,n

(M ; a, b) to be positive. First, we suppose b 6= 0.

Proposition 4.1. Let q be a prime power and n (≥ 4) be an integer.

Suppose that a, b ∈ F

_q

. Let M

₁

, . . . , M

_r

be regular complementary divisors of M with common divisor M

₀

. Assume that Θ := Θ(M

₁

, . . . , M

_r

) is positive. If the following condition (labelled A

q,n

(M

1

, . . . , M

r

)) is satisfied, then N

_q,n

(M ; a, b) is positive:

(4.1) q

^(n−3)/2

> 2

W (M

₀

) − W (m

₀

) + δ

^∗_q

1 − 1

q

(W (m

₀

) − η

₁

) + 1

q

^3/2

(W (m

₀

) − 1)

+ η

1

3 2 √

q − η

2

2 − κ

q

1 − 1

q

(W (m

0

) − 1) + 1

+ Θ

⁻¹

X

r

i=1

Θ(M

_i

)

2 W (M

_i

) − W (M

₀

) − W (m

_i

) + W (m

₀

)

+

δ

^∗_q

1 − 1

q

+ 1

q

^3/2

(W (m

_i

) − W (m

₀

))

− η

2

2 − κ

q

1 − 1

q

(W (m

i

) − W (m

0

))

, where κ

q

(∼ 4/π) and δ

_q^∗

(< 1) are as in Section 3, and

η

₁

=

1 if q is odd and n is even,

0 otherwise; η

₂

=

n 1 if q and n are odd, 0 otherwise.

P r o o f. By Proposition 3.7 and Theorem 2.9, N (M ) = ΘN (M

₀

) +

X

r i=1

Θ(M

_i

) n \

d|Mi

d

-

M₀

X

t6=0

K

₁

K

_n

− 2 \

d|mi

d

-

m₀

G

_n

(4.2)

+ \

d|M_i d

-

M0

( b ψ

_d

(a) + b ψ

_d

(b))G

₁

G

_n

o

,

where we have employed an abbreviated notation based on (2.4). Further, N (M

0

) itself is given by (2.4) with M = M

0

, M

^∗

= m

0

. In particular, since the complementary divisors are regular, all contributions relating to ψ

₂

are counted within N (M

₀

). Now, for each of the φ(d) characters ψ

_d

, we have an absolute bound of 2q

^(n+3)/2

for P

t6=0

K

1

K

n

+ ( b ψ

d

(a) + b ψ

d

(b))G

1

G

n

(16)

in (4.2), by Corollary 3.2 and Lemma 2.4(iii). Indeed, for those d that are divisors of m

i

, but not m

0

(so that b ψ

d

is trivial), we have the improve- ment that the factor 2 may be replaced by 2(1 − q

⁻¹

)δ

_q^∗

, and we have a further contribution (bounded absolutely by 2q

^n/2

) from the minor term

−2 T

d|mi, d

-

m0

G

_n

.

More significantly, when q and n are odd (so that η

₂

= 1) and d is an even divisor of 2m

i

, we can, on average (when ψ

d

is paired with ψ

_d

), replace the factor 2 by κ

_q

(1 − q

⁻¹

), by Lemma 3.6. Because each d in T

d|Mi, d

-

M0

, say, appears with a weighting factor µ(d)/φ(d) (so that only square-free d matter, and each carries an absolute weight 1/φ(d)), and because the main term in ΘN (M

₀

) is Θq

ⁿ

, we deduce that (4.1) is correct insofar as the terms under the sum P

_r

i=1

are concerned.

The estimate for the remaining terms of N (M

0

) is similar. The only modifications are as follows. There is no contribution from d = 1 to

−2 T

d|M0, d

-

2

G

_n

. Also, when q is odd, then M

₀

is even and we need to adjust the terms with ψ

2

using the expression for ∆

2

in Theorem 2.9. Thus, if n (and so m

₀

) are even, then η

₁

= 1 and |∆

₂

| <

³₂

q

^n/2+1

, whereas, if n (and so m

0

) are odd, the contribution in the worst case (when ab = n

²

) is halved to q

^(n+1)/2

. This completes the proof.

When b = 0, we analogously employ Theorem 2.10.

Proposition 4.2. Let q be a prime power and n (≥ 4) an integer. Sup- pose that a ∈ F

^∗_q

. Let M

1

, . . . , M

r

be regular complementary divisors of M with common divisor M

0

. Assume that Θ is positive. If the following condition (labelled B

_q,n

(M

₁

, . . . , M

_r

)) is satisfied, then N

_q,n

(M ; a, 0) is positive:

(4.3) q

^(n−3)/2

− q − 1 q

^(n+3)/2

>

2 − 1

q

(W (M

₀

) − W (m

₀

) − η

₂

) + (−1)

(n−1)(q−1)/4

λ

₂

(a)ε

₂

+ 1

√ q

(W (m

0

) − 1 − η

1

) +

1 − 1

q

δ

q,n

+ (−1)

^n(q−1)/4

ε

1

+ Θ

⁻¹

X

r i=1

Θ(M

_i

)

2 − 1 q

(W (M

_i

) − W (M

₀

) − W (m

_i

) + W (m

₀

))

+ 1

√ q

3 − 4

q

(W (m

_i

) − W (m

₀

))

where η

1

and η

2

are as in Proposition 4.1, δ

q,n

(with absolute value at most

2) is as defined in Section 2, and ε

₁

and ε

₂

are zero, unless q is odd and

(17)

p | n, in which case, ε

1

=

1 if n is even,

0 otherwise; ε

2

=

n 1 if n is odd, 0 otherwise.

P r o o f. This is similar to Proposition 4.1. But note the extra (mi- nor) term on the left hand side of (4.3) springing from terms of the shape

−Θ(M

_i

)(q −1) on the right side of (2.6). Further, for d a divisor of M

_i

(say), but neither of m

_i

nor M

₀

, the contribution from b ψ

_d

(a)G

₁

(G

_n

+ P

t6=0

K

_n

) is (1 + 2(q − 1))q

^(n+1)/2

= (2 − q

⁻¹

)q

^(n+3)/2

, and, for d a divisor of m

_i

, but not m

0

, that of (q −2)G

n

− P

t6=0

K

n

is ((q −2)+2(q −1))q

^n/2

= (3−4/q)q

^n/2+1

. In this way, the result is proved.

We shall sometimes abbreviate A

q,n

(M

1

, . . . , M

r

), say, to A

q,n

. We also use R

_q,n

(M

₁

, . . . , M

_r

) (possibly abbreviated) and L

_q,n

to denote the right and left sides of (4.1) or (4.3) as appropriate to the context. Note that, even for B

_q,n

, L

_q,n

is essentially q

^(n−3)/2

as the other term is generally negligible.

Although these conditions seem complicated, for larger values of q and n, it suffices to use the coarser estimate obtained by selecting only the terms involving M

₀

, M

₁

, . . . , M

_r

. Note finally that, because the terms in respect of divisors of m are generally diminished in B

_q,n

by a factor of order √

q, the condition A

q,n

is essentially more stringent than B

q,n

and therefore, as a rule, B

_q,n

holds whenever A

_q,n

does.

5. Theorem 1.1 for “almost all” pairs (q, n). In this section we shall take r = 1 and show that A

_q,n

(M ) and B

_q,n

(M ) hold for all but finitely many pairs (q, n). Nevertheless, in interpreting this conclusion, caution must be exercised because the number of potential exceptions is huge. Hence, properly understood, this is merely the first stage in the application of the theory. Note that, in A

_q,n

(M ), say, all the “Θ-terms” are absent.

We begin with a weak, but convenient, lemma to bound the function W . Note that 2 · 3 · 5 · 7 · 11 · 13 = 30030.

Lemma 5.1. Set γ := 64/30030

^1/4

< 4.9. Suppose that h is an integer indivisible by a prime p. Then

W (h) ≤ γ

p

h

^1/4

, where γ

p

=

1

2

p

^1/4

γ if p < 16, γ if p > 16.

In particular , γ

₂

< 2.9, γ

₃

< 3.2, γ

₅

< 3.7.

P r o o f. Granted that W is multiplicative, the proof is easy.

Now, we resume the assumption that q, n, a, b are given as in Section 4

with M = q

ⁿ

− 1, etc. We shall denote ω(M ) by ω

q,n

. From now on we shall

also suppose n ≥ 5, as in Theorem 1.1.

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Lemma 5.2. Let q be a prime power and n ≥ 5. Suppose that A

q,n

(M ) does not hold. Then n ≤ 9 and q < 2

2(ω+1)/(n−3)

, where ω = ω

_q,n

. Moreover , if n has the specified value, then

ω

q,5

≤ 17, ω

q,6

≤ 17, ω

q,7

≤ 6, ω

q,8

≤ 11, ω

q,9

≤ 7.

Further , identical conclusions hold in respect of the condition B

_q,n

(M ).

P r o o f. For both A

q,n

and B

q,n

, for most of the proof it suffices to use the rough bounds

R

_q,n

(M ) < 2

W (M ) − 1 q

, L

_q,n

> q

^(n−3)/2

− 1 q

⁴

. It follows that, if the relevant condition fails, then

(5.1) q

^(n−3)/2

< 2W (M ).

From Lemma 5.1, (5.1) implies that

(5.2) q

^(n−6)/4

< 2γ

_p

< 9.8.

Suppose n ≥ 10. It follows from (5.2) that q ≤ 9. Indeed, substituting bounds for γ

₃

, γ

₂

, we deduce that q ≤ 7. Moreover, n = 10 if q = 7 or 5 (using γ

₅

); n ≤ 11 if q = 4; n ≤ 12 if q = 3; n ≤ 16 if q = 2. But ω

_2,11

= 2 and ω

2,n

≤ 4 for 13 ≤ n ≤ 16; consequently, by (5.1), if q = 2, then n = 10 or 12. For q = 3, ω

_3,10

= ω

_3,11

= 3, ω

_3,12

= 5; hence (5.1) is false in each case. Further, for q = 4, 5 or 7, ω

q,n

≤ 5 in the relevant cases (with n = 10 or 11). Hence, again (5.1) cannot hold. This leaves only the aforementioned possibilities, that q = 2 and n = 10 or 12.

Now, for A

_2,n

(M ), we have R

2,n

= δ

^∗₂

W (M ) + 1

√ 2 (W (M ) − 1) = √

2W (M ) − 1

√ 2 since, trivially, δ

₂^∗

= 1/ √

2. Since ω

_2,10

= 3, ω

_2,12

= 5, we deduce that R

_2,10

= 2

^7/2

− 1/ √

2 < L

_2,10

= 2

^7/2

, R

_2,12

= 2

^9/2

− 1/ √

2 < L

_2,12

= 2

^9/2

. Hence, A

2,n

(M ) holds in both cases.

Similarly, for B

_2,n

(M ), we have R

_2,n

= 1

√ 2

W (M ) − 1 + δ

_2,n

2 < W (M )

√ 2 and it is evident that B

2,n

(M ) holds for n = 10, 12.

Accordingly, we may assume that ω ≤ 9 and (5.1) holds. In particular, q < 2

2(ω+1)/(n−3)

. We establish the displayed bounds for ω in the most delicate case, namely, when n = 5, so that (5.1) has the form

q < 2W (M ).