Sums of squares in $\mathbb {Z}[\sqrt{k}]$

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

F

ERNANDO

C

HAMIZO

Sums of squares in Z[

√

k]

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{1 (1997),}

p. 25-39

<http://www.numdam.org/item?id=JTNB_1997__9_1_25_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Sums of squares in

Z[~k]

par FERNANDO CHAMIZO

RÉSUMÉ. Nous étudions une généralisation du fameux _problème du

cer-cle aux anneaux d’entiers quadratiques réels : nous nous intéressons à

C(N,M) = 03A3n~N 03A3m~M r(n

+

m~k),

le nombre de _{représentations} de

n+m~k

comme somme de deux carrés dans

Z[~k]

(où k &#x3E; 1 et sans facteur

carré). En utilisant la théorie spectrale dans _PSL2(Z)BH, nous obtenons

une formule asymptotique avec terme erreur _pour C(N, M), démontrant _que

certaines _techniques d’estimations de fonctions L _automorphes fournissent précisément des majorations de ce terme erreur.

ABSTRACT. We study a _{generalization} of the classical circle _problem to

real quadratic rings. Namely we _{study C(N, M) =}

_{03A3n~N 03A3m~M}

_{r(n +}

m~k)

where _r(n +

m~k)

is the number of _{representations} of n +

m~k

as

a sum of two _squares in

_Z[~k]

_(with k &#x3E; 1 and _{squarefree). Using spectral} theory in

_PSL2(Z)BH,

we _get an _asymptotic formula with error term for

C(N, M), showing that some _techniques on the estimation of automorphic

L-functions can be _applied to _{get upper} bounds of the error term.

1. Introduction.

The classical circle

_problem

asks about the

_asymptotic

behaviour of

r(n)

where

_r(n)

is the number of

_{representations}

as a sum of two squares. The

problem

comes back to Gauss

_[Ga]

and it was

conjectured

by

Hardy

[Ha]

that for _every a &#x3E;

_1/4

Several methods of

_exponential

sums have been

applied along

this

_century

to _{extend the range of a in which}

_(1.1)

holds true. The best known result is due to M.N.

_Huxley

_[Hu]

who has

_proved

_(1.1)

for every a &#x3E;

_23/73.

A natural

_{generalization}

of the circle

_problem

to the

_ring

with k

squarefree, k

&#x3E;

_1,

leads to consider

where

_r(n

+

m.;k)

is the number of

_{representations}

of n +

mf

as a sum

of two _squares in

_Curiously,

with this formulation circle

_{problem in}

reduces to an

_{ellipsoid problem}

that can be

_completely

solved. Other

formulations due to W. Schall

_[Sc]

and U. Rausch

_[Ra]

lead to difficult and

interesting problems

about the distribution of the

_conjugates

of an

_algebraic

number. In this paper we

_study

i.e. we substitute the square 1

n, m

N in the definition

of C (N)

by

the

rectangle

To

_get

an

optimal

_asymptotic

formula for thin

_{enough rectangles (M}

small)

appears as a difficult

problem

involving

harmonic

_analysis

in the

hy-perbolic plane.

In

_fact,

in the

_limiting

case M =

_2,

the

problem

_can be

in-terpreted

as a

hyperbolic

circle

problem

which has

_strong

resemblances with

the classical circle

_problem

_(see

Ch. 12 of

_{[Iw2], [Ph-Ru]}

and

_[Chl]),

and in

some _{ranges the estimation of the} error term in the

_asymptotic

formula for

C(N, M)

is related to the estimation of some

_automorphic

L-functions.

In

_[Chl]

it was stated as an

_example

that the

_asymptotic

of

can be decided

using

the

spectral theory

of

automorphic

forms

(the

needed

results are summarized in section 2

below).

In this paper we

_study

the

uniformity

in m to

_get

an

_asymptotic

formula with error term for

_{C (N, M).}

Using

well known results in lattice

_{point theory}

_(when

M is

_large)

and

aver-age results for Hecke

eigenvalues (when

M is

small),

we obtain in section 3

bounds for the error term.

_Finally,

in section 4

_(that

we consider the main

part

of this

_paper)

we show that there is a relation between the estimation

of

_automorphic

L-functions and

_bounding

the error term in our

problem.

Combining

a method due to W. Luo

_{(see [Lu])}

with other

_results,

we obtain

several bounds

_improving

in some _{ranges the}

_previously

obtained.

The author would like to thank the referee and the editorial board for

the

_celerity

in the

_acceptance

of this _paper. The author also wants to thank

specially

the

_{encouragement}

_{given by}

E. Valenti.

2. Notation and review on some known results.

The main _{purpose of this section is} to

_expand

_{C(N, M)}

in terms of

essentially,

described in

_[Chl]

but to ease references we

prefer

to

_give

here

a

_specific

statement of the results that we need with a _very brief comment about the

_underlying

ideas.

We shall follow the notation of

_[Iw2]

and we also refer the reader to that

_monography

for any basic result on

_spectral

_theory

used in this paper.

Nevertheless we find convenient to refresh

_briefly

at least the most basic

concepts.

We shall denote

_by

Maass _cusp forms for

_PSL2(Z);

i.e.

automor-phic

functions

_vanishing

at

_infinity

which are normalized

_{eigenfunctions}

of

Laplace-Beltrami

operator.

The

_eigenvalues

will be denoted

_by

_1/4

+

_t~,

taking (by

notational convenience in Lemma

_{2.3) tj}

with a different

_sign

for and

We shall assume that the are Hecke

_eigenforms,

i.e.

_{they satisfy}

= where

T",

is the Hecke

_operator

On

_PSL2(Z)BM

there is

_only

one Eisenstein

_series,

and

E~(~,1/2

+

_it)

is an

_{eigenfunction}

of

_Tm

whose

_eigenvalue

is

_{qt(m) =}

.

The

_hyperbolic

circle

_problem

will

_play

an

_important

role in the

subse-quent

arguments.

As the classical lattice

_point

_problem

consists of

_counting

the

_images

of a

_point

under

_integral

translations,

the

hyperbolic

circle

prob-lem deals with

-

-where _p is the Poincaré distance

_{given by}

p(z, w)

= arc

cosh (1 + 2u(z, w))

with

u(z, w} -

4ImzImw’

°

41mzImw*

By simplicity

it is convenient to write arc cosh

(X/2)

instead of

_R,

_defining

Let us define

--We shall relate

_{c(N, m)}

to the

_hyperbolic

circle

_problem

thanks to the

following

elementary

result whose

_proof

reduces to

_expand

_(a

+ +

LEMMA 2.1. Ifm is

_odd,

= 0 and ifrrt is _even, is

the number of

_integral

solutions of a,

b,

c, d of

A calculation proves

where _a,

_b,

_c, d are the entries of the

_{matrix q}

E

_{SL2 (Z) (see}

Lemma 3.8 of

[Chl]

for a

generalization

of this

fact) .

On the other

hand,

the Hecke

opera-tor

_{T rn}

can be

_used,

_roughly

_speaking,

to pass from matrices of determinant

one to matrices of determinant m,

namely

With this idea in

_mind,

one can _prove

_(see

[Iw2]

or

[ChI])

LEMMA 2.2. For

_{N, m &#x3E;}

1

H(X; z,w)

is

_automorphic

as a function of z or _w, so it can be

_analysed

with the

_{spectral theory}

of The lack of

_regularity

of H

requires

some

smoothing

before

applying

the

_pretrace

formula. These

_steps

were worked out in Lemma 2.3 of

_[Chl]

_(see

also

_Proposition

2.3 of

_[Ch2]).

With the

_arguments

used

_there,

it is deduced

LEMMA 2.3. Given

_0A1,~~6M

and X &#x3E;

_3,

it holds

with

where an associated

_Legendre

function

(see

[Gr-Ry])

and S =

arc cosh

_(X/2)

+

_Ao

for some

Do,

-0

Ao

A.

Remarks:

_i)

Actually,

the condition X &#x3E; 3 can be

replaced by

X &#x3E; c where

c is _any constant

_greater

than

_2,

but the D-constant in the error term is

unbounded when c --&#x3E; 2. The same situation occurs with the factor 3 in

the statement of Lemma 2.4 and in the results based on this lemma.

ii)

Note that it holds

_(use

8.723.1 of

_[Gr-Ry]

and Lemma 2.4

_(c)

of

_[Chl])

for t E R far _away form zero, and

g(0)

G

_(see

8.713.2 of

[Gr-Ry]).

The Eisenstein series +

_it)

and

_{E. (i, 1/2}

+

_it)

are

essen-tially Epstein zeta-functions, namely

where

_{rk (n) =}

_{#{(a, b)}

E

Z’ : a 2+

kb2 =

n}.

_Using

standard

_arguments

involving

an

_approximate

functional

_equation

of the

_Epstein

zeta-functions

(compare

with

_[Go]

or with Theorem 4.2 of

[Iv] ,

_they

can be

_approximated

in s =

1/2

+ it

by

a Dirichlet

_polynomial

of

_length

less than

_Itll+E

and

coefficients an «

nE,

then

Hence the contribution of the continuous

_spectrum

in is and it follows at once from Lemma 2.2 and Lemma 2.3

where

and

with g

as in Lemma 2.3

taking

X =

N /mVk.

Remarks:

_i)

Note that

_{s(M) -~}

1 when M --&#x3E; oo, indeed

s(M)

= 1 ₊

o (M-1+E).

.

ii)

It is

_important

to note that

_although

_Ao

in Lemma 2.3

_depends

on X

and

_A,

once N and A are fixed the same

Ao

can be chosen for _every

_g(tj)

appearing

in the summation over m in The reason of this fact

is that

_Ao

is obtained

_by

the mean value theorem

_using

the

_monotonicity

of

_{f (t)}

=

H(t

₊

N /mVk;

i)

and the function

F(t) =

is

also monotonic.

iii)

In the

_{applications,}

d -1

will be bounded

_by

a _power of N and

consequently

calculating

the _supremum of

EA

(N, T)

it is

_enough

to consider

T less than a certain power of N because otherwise

EA

(N,

T)

is absorbed

by

the other error terms. We shall

_implicitly

use this fact in the

proofs.

3. The

_asymptotics

of

_{C(N, M).}

As we mentioned in the

introduction,

when N = M it is

possible

to

get

a

_sharp

estimate of the error term as a _{consequence of} a similar result for

rational

_ellipsoids.

This is the content of the

_following

result

PROPOSITION 3.1. If N

M Vk

then

holds &#x3E; 1 and does not hold for any a 1.

Remark: Note that this

_implies

that

Proof.

_By

Lemma

_2.1,

_C(N,

_M)

is one half of the

_integral

solutions of

but the

_{inequalities a2 + kb2 &#x3E;}

c2 + k d2 &#x3E;

_{2 1 cd I Vk-}

and our

hypoth-esis N

_implies

that the second

_inequality

in

_(3.1)

is

_superfluous.

Moreover the values of

_{a, b, c, d}

with ad - bc = 0 contribute

O(N1+E),

_so,

up to a term of that

_order,

_{C (N, M)}

is one half of the number of lattice

points

in the

_ellipsoid

and when

_{a ~}

1 the

_proposition

follows from the classical results of Landau

[La]

(see

also

_§21

in

_[Fr)).

To deal with the case a =

1,

it is

enough

to

prove

By

the work of Kloosterman

_[Kl],

for

_infinitely

_many values of

_N,

with

suitably

chosen

_prime

_factors,

the

_{equation a2}

+ ~b2

+

C2

+

kd2 -

N has

more than cN

llplN

(1

+ » N

_log

_log

N solutions. If ad - bc = 0 then

a =

dl a’,

_c =

dlc’,

b =

d2a’,

d =

d2c’

with

gcd(a’,

c’)

= 1 and

hence all of the solutions

_excepting

at most

_{O(N~) satisfy}

ad - 0. 0 Now we shall state an

_asymptotic

formula for

C (N, M)

when M is small

in

_comparison

with N.

_Estimating

the error term we shall

employ sharp

average bounds for

[

and

but,

as we shall see in the next

section,

in some _{ranges it is}

possible

to

_proceed

in a better _way

studying

the cancellation induced

by

the oscillation of when m varies.

Although

the bound

(uniformly

for m is

only

known

conjecturally

under

Ramanujan-Petersson

_conjecture

_IÀj(m)1

=

O (me) ,

it is

possible

_to

prove the

following

result

_(the

same situation occurs, in a different

_context,

in Theorem 4.3 of

THEOREM 3.2. If N &#x3E;

_3Mv~’k-/2,

then for _{any e} &#x3E; 0

Proof.

_By

Lemma 2.4 it is

_enough

to _prove for some choice of A and

every T &#x3E; 1

As we remarked in Lemma 2.3

On the other hand Theorem 8.3 of

_[Iw2]

assures

Hence,

by

Cauchy’s inequality,

for T 2T

Using Proposition

7.2 of

_[Iw2]

we deduce

which combined with

_(3.3)

_implies

Finally, choosing A

=

N-1~3,

1

(3.2)

is

proved.

0

4.

_{C(N, M)}

and

_automorphic

L-functions.

The

_growth

of the Riemann zeta-function in the critical

_strip

is

_closely

related to the

_study

of the sums

EmM

m -it.

For

instance,

an _upper

bound of order

_{ItlE M1/2+E}

for 1 M « ~

implies

Lindel6f

_hypothesis.

In the same _{way, the} sums

are related to

analytic

properties

of the

automorphic

(Hecke)

L-function

The estimation of these L-functions

_{(beyond convexity)}

is more difficult

than the one of the Riemann zeta-function because there is not an

_analogue

of van der

Corput’s

method on

_exponential

sums after

_twisting

with

_{Aj (m).}

H. Iwaniec and other authors have

_developed

an

amplification technique

that allows to surpass

convexity

arguments

getting

individual bounds from average results

(see

[Iwl]).

After Lemma

2.2,

Lemma 2.3 and the

_asymptotic

_expansion

of

_P-1

summation over m leads in some _ranges to

_partial

sums of

H(1/2

+ let us name them as

In this way it is found a relation between the error term in the

_asymptotic

formula for

_C(N,

_M)

and the estimation of these

_automorphic

L-functions.

Lindel6f

_hypothesis

in s and in

_spectral

aspect is in this context

which

_suggests

The

_{previous conjecture,}

as Lindel6f

_conjecture,

is out of reach

_by

current

methods,

but there are some _{average results}

_supporting

it that will be useful

in our

_arguments,

namely,

from the work of W. Luo

[Lu]

it is deduced

(apply

Theorem 1 after

_expanding

_Hq(M)

_using

the

_{multiplicativity}

of

_{A; (m) )}

ll.jl::::::.L

Note that this

_{gives Conjecture}

4.1 in _{average whenever}

MqT2

is the

_leading

term of the

_right

hand side.

In the

_{spectral analysis}

_{of C (N, M)}

the

_partial

sums _appear

mul-tiplied by

the factor _{then the average result}

_(4.1)

can be

_only

used if we take control of the _cusp forms evaluated in these

special

values.

The

_inequality

G is

_conjectured

_{(see (0.8)}

in

_~Iw-Sa~,

in fact it

can be understood as a

_stronger

form of Lindel6f

hypothesis)

but for our

CONJECTURE 4.2. For T &#x3E; 1 _{and every} i &#x3E; 0

Remark:

Recently

N. Pitt has

_proved

_{(personal communication)}

a

Hardy-Voronoi

_type

summation formula

_implying

the

_previous

_conjecture

for

in-finitely

many

special

values of z but not

including z

= nor z = i.

On the other

_hand,

the best known L°°-bound for is

due to H. Iwaniec and P. Sarnak

_(see

_[Iw-Sa]),

which

_implies

_(by

Proposi-tion 7.2 of

_[Iw2])

In order to relate and it will be convenient to use the

following

result

LEMMA 4.3.

_{Let g}

as in Lemma 2.4 and T

_2T,

then

where G is

_decreasing

in m and

Proof.

_By

8.723.1 of

_[Gr-Ry]

which substituted in

_(4.3)

and in the definition of

_gives

with

_verifying

the

_{required properties}

and

Finally,

Theorem 8.3 of

_[Iw2]

_proves the desired bound for E. D

Although

the different level of

_difficulty

and

_depth

of

_Conjecture

4.1 and

Conjecture

4.2

_(the

former is an out of reach Lindel6f

_hypothesis

and there

are

_partial

results toward a

proof

of the

latter)

both of them

imply

almost

the same result in our

problem.

THEOREM 4.4. For N &#x3E;

3M Vk/2

and any e &#x3E; 0 we have under

Conjec-ture 4.1

and under

_Conjecture

4.2

ProoL

_By

Lemma 2.4 it is

_enough

to _{prove for} some 0 A 1

and

Assume

_{firstly Conjecture}

_4.1,

then Lemma 4.3

_(after

_partial

_summation)

and

_{by Proposition}

7.2 of

_[Iw2]

Choosing A

=

N-1/3 M-1/3,

(4.4)

is

proved.

Now,

let us assume

Conjecture

4.2,

then

by Cauchy’s

inequality

for some M’

M/2.

and the second term in the

_parenthesis

is absorbed

_by

the other

obtain-ing

(4.5).

On the other

_hand,

if T we have as in the

proof

of

Theorem 3.2

which

_gives

a better bound. 0

THEOREM 4.5. For N &#x3E; and _any c &#x3E; 0 we have

_{unconditionally}

where L = min

(L1, L2)

with

Remark: When M is _very small in

_comparison

with N this is not the best

_possible

result obtainable from

_(4.1);

in

_fact,

_taking

_q

_{large enough}

it is

_possible

to deduce

_(NM)B

for 1 M

N f( 8)

with 0 as close as we wish to

_{2/3 (and f (0)}

--&#x3E; 0 as 0 ~

2/3),

but _we do _not consider

worthy

to

_complicate

the statement of the

_previous

theorem

_covering

all

the cases because in these short ranges Theorem 3.2

_gives

similar bounds.

For

_instance,

when M = Theorem 4.5 could be

slighty

improved

_to

NO .759 ...

_using

_(4.1)

_{with q}

=

7,

but Theorem 3.2

gives

in _{this range}

L K

N°~’s6....

Proof

_By

Lemma 4.3 and

_Proposition

7.2 of

_[Iw2]

with

Using

the first bound of

_(4.2),

_{Cauchy’s inequality}

and

_partial

summation

give

for some M’

M/2,

and so it is bounded

_by

_(4.1)

with _q = 1

_getting

Choosing A

=

N -6 ~23 M -fi /23

_we have

Therefore

_by

Lemma

_2.4,

it can be taken

On the other

_hand,

_applying

H61der

_inequality

in

_(4.6)

with

_exponents

for some M"

_M/2.

Using

(4.1)

with q

= 2

and

_{choosing A}

=

N -12 /41 ~ -12 /41

_we

_get

which

_{implies by}

Lemma 2.4

Some calculations _prove that when the first term in

_(4.7)

is the main _term,

the bound

_(4.8)

is better. The same situation occurs with the last term in

(4.8),

hence both of them can be omitted in the final result. 0

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[Fr]

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Fernando CHAMIZO

Departamento de Matemiticas Facultad de Ciencias

Universidad Aut6noma de Madrid 28049-Madrid, Spain