a mean-value theorem on sums of two k-th powers of numbers in residue classes
TRANSCRIPT
Abh. Math. Sem. Univ. Hamburg 60 (1990), 249-256
A Mean-Value Theorem on Sums of Two k-th Powers of Numbers in Residue Classes
By G. KUBA
1 Introduction
For positive integers k > 2 and n, let rk(n) denote the number of ways to write n as a sum of two k-th powers of natural numbers. To study the "average order" of this arithmetic function, one considers Dirichlet's summatory function
Rk(x) = ~ rk(n), n~<x
where x is a large real variable. Of course, the evaluation of Rk(X) is a problem of lattice point theory in the classical sense of Landau. (See the recent textbook of KRJiTZEL [2] for an enlightening survey of the field.) It has been proved by KR~,TZEL [3] that
2(1 oo I'~(~) X~ --X ~ +ClX~-~ ' y ' n - l - ~ �9 , ~ sm(2znx~ - ~ ) + o(x=) Rk(X) = 2kF(2)
n=l
with an effective constant cl depending on k. Consequently, for k > 3,
r2(~) x~ - x~ + O(x~-b ) Rk(x) = 2kF@
and
r2(~) x~ - x~ + n_+(x~-~). Rk(x) = 2kF(2)
(For k = 2, this is the famous circle problem of Gauss; see again [2] for its history; of course, the term involving the infinite series is meaningless in this case.)
Applying the modern "Discrete Hardy-Littlewood method" in the shape due to HUXLEY [1], M/tiLLER and NOWAK [4] improved the error term to O(x7/ltk(logx)45/22). (This is the sharpest known estimate for the circle problem, too.)
This paper is part of a research project supported by the Austrian "Fonds zur Fgrderung der wissensehaftlichen Forschung" (Nr. P7514-PHY).
250 G. Kuba
2 Subject and Results of this Paper
It is an usual idea in the theory of arithmetic functions to impose some kind o f congruence conditions to some o f the parameters involved (For instance, think o f the prime number theorem in ari thmetic progressions !). In this article, we are thus going to consider representations n -- u k + O k (U,/) E ~ ) with u and v lying in prescribed residue classes. For positive integers 11, I2 and m, and k > 2 and n as earlier, define
rk(ll,12,m;n) = #{(u,v) E N 2 : uk-al-l) k = n, u =-- II (mod m), v = 12 (mod m)},
and again for a large real variable x,
Rk(Ib t2, m; x) = ~] rk(lb 12, m; n). n<_x
We will establish a result uniform in 11,12 and the module m.
Theorem. (i) For 0 < 1l, I2 <_ m and x ~ co,
F2(~) x Rk(lbl2,m;x) = ~ (~-;)~- + (1
11 12 x t
m
where
k ~ - X ! _ • X 7 X 45 r(b § , F (l,,I2,m;x) +
2~1+~ . ,
F k ( I b l 2 , m ; x ) = 2 n - l - [ ( s i n ( (x~ -- ll) -- ~ ) + sin( ( x ~ - - / 2 ) - - n = l
)),
the O-constant depending only on k.
(ii) For k > 3,
r2(~) ( x__)~ + (1 - - - Rk(ll , lz, m ; x ) = ~ mk
and
r2(~) ( x ) ~ + ( 1 Rk(lbl2,m;x) = ~ mk
m tt~ II~
ll /_2~t x ~! x ! _ • m m " m k
uniformly in lbl2 N m <_ x 1/k. In particular 1 ,
t 1 - - < lim supIFk(Ibl2,m;x) 1 < 2~(t + ~) 10 - x~oo
for k > 3 and arbitrary lb 12 and m.
1The value 1~ here is not the best possible one and in fact of little importance. We give it only to illustrate the uniformity in lb 12 and m.
A Mean-Value Theorem 251
3 Proof of Clause (i) of the Theorem
We put X = 7,x 2 = kin, # = kin. Then it is clear that
Rk(ll,12, m;x) = Rk(X) def #{(r,s) C No 2 : (r + 2) k + (s + ~)k ~ X} ~---
y . 1 + y . 1 + o<r_<(xp/k-;, r>(~)'/k--ZO<s_<(X)'/k--~
0<s_<_( 1 X} l/t:_# (r+2)k+(s+#) k <_X
+ 1 + o (1 ) =
O<r<.( ~2 )llk-j,s>( ~2 )l/k-l* (r+2)k +(s+~)k~X
- - ( lp( (X) �89 - - y ) + l p ( ( 2 ) � 8 9 2) ) (2){ + I + I I + 0(1),
here W(z) = z - [z] - �89 and I , I1 are given by
= y ' [ ( x - (r + ,~)k) ~ _ ~ 1 , I
( ~ ) �89 --2 <r<(X--/* k) l/k--)L
= ~ [ ( x - (s + ~)k) ~ _ , q . I I
(x)�89
We apply the Euler summation formula to I and I I and observe that
f~x (x-~)'/~-; ((X)�89 2)((2) �89 _ y ) + )~_~ ((X-- (t + ;Ok) ~ - l . t ) d t + -
f( X--;tk)~lk--t, + y(~)~_.~ ~ ( ( x - (t + ~)b~ -,Z)dt=
F2@ - 2kr( -~ X~ - (2 + y)X~ + 0 (1 ) ,
by a simple geometric interpretation as an area. Thus, after a short computa- tion, we arrive at
F2(-~) Rk(X) = 2kF('~")' X~ + (1 - 2 - #)X{ -
- d(2, ~) - J(#, 2) - S (2, #) - S(~, 2) + 0 (1 ) , (1)
with J(a,b) clef f (x-bkyk-" = (X -- (t + a)k)k -1 (t + a) k-l,#(t) dt ,
252 G. Kuba
and S(a, b) d~f Z ~p((X -- (n + a)k) } -- b)
( X)l/k--a<n<_(X--bk)l/k--a
for {a,b} = {2,#}. By the second mean-value theorem and an obvious substitution,
f0 1
J (a ,b ) X } ( 1 - k }-l k-1 } = u ) U ~ ( X u - a ) d u + O ( 1 ) . (2)
Our next step is to establish an upper bound for the sums S(a, b).
Proposition 1. For {a, b} = {2, p}, and X ~ ~ ,
7 S(a ,b ) << Xm( logX)~ .
P r o o f The essential ingredient in the deduction of this estimate is HUXLEY'S [1] "Discrete Hardy-Littlewood method" in the shape presented in MOLLER and NOWAK [4],
First of all, choose Ml in such a way that M1 ~ X 7~Ilk. By the trivial estimate,
S(a ,b) = Z tp(f(n)) + O ( X 7 / , , k ) , (3) (x)~/k<n<X1/k--Ml
with f ( t ) = (x -- (t + a)k) 1/k -- b.
We note that
f ' ( t ) = - ( t q- a ) k - l ( x -- (t + a)k) ~-1 ,
f " ( t ) = - - (k -- 1)(t + a ) k - 2 x ( x - - (t q- a )k ) ~ - 2 ,
f " ( t ) = - ( k - 1)(t + a ) k - 3 X ( X -- (t + a ) k ) } - 3 ( ( k -- 2)X + (k + 1)(t + a)k ) .
As an easy consequence, for (X)a/k < t < X '/k -- M , ,
! !
f ' ( t ) ~. X ~ - - J ( X } - t) }-2 , (4)
and 1 I 1
f " ( t ) ~ X ~ - ~ ( X ~ -- t) }-3 ,
since X - (t + a) k ~ ( X } - (t + a ) ) X }(k-~) ~ ( X } - - t ) X ~ (k -1 ) .
It remains to estimate
(5)
s" d~ ( X w~ <n<_X~/k--M l
A Mean-Value Theorem 253
To deal with this last sum, let us put g(u) = f ( [X l/k] - u ) , then (with c = 1 - (�89
S * = E v(g(m)) + 0(1). M1 <rn~cX I/k
We split up this interval of summation by a geometric sequence Mj = 2J-IMI, for j = 1 . . . . , J , with J such that 2JM1 = cX 1/k, and M~ chosen in such a way that this is possible. Put lj =]Mj_bMj] ( j = 1 . . . . . J). To each of these subintervals Ij we now apply
Lemma 1. (see [4] Theorem B) Let M, M' and U be positive real parameters > 1, M' < 2M << U 1/2, and let the real function g be three times continuously differentiable on [M, 2M]. Furthermore, suppose that (for r = 2, 3 ),
c~-lUM -r-1 <_ Ig(r)(u)l < crUM -r-1 ,
on [M, 2M], with suitable constants c2, c3. Finally, let again V (z) = z - [z] - �89 Then it follows that
Z tp(g(m)) << u 7/22 (1 + M -1U19/44(tog U) 9/44) 1/2(10g U) 45/22.
M<_m<_M'
To apply this estimate from the "Discrete Hardy-Littlewood method" to each of our sums
S; = Z lp(g(m)), mclj
we observe that, for u E I j , u ~ M j, and therefore,
and
1 1 !_ g"(u) ~. X~-~M] 2,
!_i__ !--3 g'(u) =z X k ~2 M] ,
in view of (4) and (5). Thus we may apply Lemma 1 with M = Mj and !_! !~-1
U = Uj = X k ~2 M ] - . (Because of Mj << X 1/k, it is easy to verify that 1/2 Mj << Uj .) We obtain
S; ,(< U;/22 (1 + M-~ 1 U) 9/44(log Uj)9/44) I/2 (log Uj)45/22 <<
!___i !+I M;1/2(X~_ ~ M) +1)47/88 (log X) 189/88 << (X k k2 M] )7/22(10gX)45/22 +
We now sum over j = 1 . . . . , J , observing that a finite geometric series is << its largest term. After a short computation, we arrive at
J
E s 7 , , <<Xm(logX) . j----1
This obviously completes the proof of Proposition 1.
254 G. Kuba
It remains to establish asymptotic expansions for the integrals J(a, b).
Proposition 2. For {a, b} = {2, p}, and X -* oe,
k~- X ~ - g Z n - l - { ( s i n ( 2 n n ( X ~ a) ~k ) + 0 ( 1 ) . J(a ,b)= F(~) ~ t , , �9 i t
2~7~t+~ n=I
P r o o f We make use of the well-known Fourier expansion
to infer from (2) that
1 sin(2nnz) = - - (z Z) 7~ n n=l
J(a,b)= 1x lf (1 uk) - u k - a " ' . . . . sm(2:~n(X~u-- a)) du + 0(1). (6) 7~ n
n = l
To evaluate these integrals, we employ a quite classical technical lemma (cf. e.g. WICTON [5] for a simple proof; KR.~TZEL [2] gives a better result based on a much deeper argument).
Lemma 2. (WILTON [5] Lemma 12) Let oJ be a real constant, 0 < e9 < 1, and 0 a real-valued continuous function on [0, 1], such that df exists and is bounded on ]0, 1 [. Then, for a large real parameter W,
~ 1(I -- u)~O-l~)(u)e iwu du = 4)(1)F(co)W-O fl ( w - ~ ) + O(W -I) .
To use this result for our purpose, we multiply by e -2~na and take the imaginary parts. We choose to = I , W = 2nnX ~/k, and
Thus we get
~ ( U ) = u k - I ( I + u + u 2 + . . . + u k - I ) ~ -1.
fo l(1 u k ) ~ - l u k - I �9 - - s m ( 2 n n ( X ~ u -- a)) d u =
i t 1 1 i , l 7~ sm(2nnX~ -- 2nna -- ~-s + O(n- lX -~) = k~- F(-s
In view of (6), this immediately yields the assertion of Proposition 2.
Inserting the two Propositions into (1), we complete the proof of part (i) of our Theorem.
A Mean-Value Theorem 255
4 Proof of Clause (ii) of the Theorem
The upper bound for Fk(ll, t2, m; x) is obvious. To establish the lower bound, let
oo
h(y) = h(2, p;y) de_f Z ( A n cos(2nny) + Bn sin(2nny)) n = l
with An = n -1-~ (-- sin(2rcn2 + ~ ) -- sin(2~zn~ + ~-~)),
Bn = n -1-r (cos(2rrn2 + ~--~) + cos(2nnp + ~-~)).
Then it is clear that (with the definitions at the beginning of section 3)
Fk(ll,12,m;x) = h(2,#;X~).
Since h(y) is periodic with period 1, it suffices to show that
1 sup Ih(2,#;y)l _>_ - - . (7)
0_<y<_l 10
Assuming the contrary, we infer from Parseval's identity that
0, 02 > Z(A2, + B 2) = 2 n -2-~ (1 + cos(2rcn(2 - /~ ) ) ) . n = l n = l
Considering only the terms n = 1 and n = 2 of the last sum, this would imply that, for n = 1 and 2,
1 + cos(2un(2 - #)) < 0,01 n 3 < 0, 1,
i.e. 2 - # and 2 ( 2 - # ) differ at most by ~arccos 0,9 = 0,0717830... from half an odd integer, an obvious contradiction (since 0 < 2,/~ < 1 ).
This implies (7) and thus completes the proof of our theorem.
References
[1] M.N, HUXLEY, Exponential Sums and Lattice Points. Proc. London Math. Soc. (3) 60 (1990), 471-502.
[2] E. KRT, TZEL, Lattice Points. VEB Dt. Verlag d. Wiss. Berlin 1988.
[3] E. KRXTZEL, Bemerkungen zu einem Gitterpunktsproblem. Math. Ann. 179 (1969), 90-96.
[4] W. M~LLER and W.G. NOWAK, Lattice Points in Planar Domains: Applications of Huxley's "Discrete Hardy-Littlewood Method". To appear in a Springer Lecture Notes volume (ed. E. Hlawka), in 1990.
[5] J.R. W1LTON, An Extended Form of Dirichlet's Divisor Problem. Proc. London Math. Soe., II. Ser. 36 (1933), 391-426.