We establish a lower bound for the remainder term in the asymptotic expansion for the Dirichlet summatory function ofρ(n)

AN OMEGA THEOREM ON DIFFERENCES OF TWO SQUARES, II

M. K ¨UHLEITNER

Abstract. Letρ(n) denote the number of pairs (u, v)∈N×Z_withu2−v²=n.

Due to a formula of Sierpinski,ρ(n) is closely related to the classical divisor func- tiond(n). We establish a lower bound for the remainder term in the asymptotic expansion for the Dirichlet summatory function ofρ(n).

1. Introduction

As in part I of this paper [8], letρ(n) denote the number of pairs (u, v)∈N×Z with u²−v² =n. For the more general case where the square is replaced by a

“k”-th powerk ≥2 see Kr¨atzel [6], [7] and the recent paper of Nowak [9]. Due to an elementary formula of Sierpinski, our functionρ(n) is closely related to the classical divisor functiond(n) by

(1) ρ(n) =d(n)−2d n

2

+ 2d n 4 ,

whered(·) = 0 for non-integers, due to Sierpinski.

For a large real variablex, we consider the remainder termθ(x) in the asymp- totic formula

T(x) =X

n≤x

ρ(n) = x

2logx+ (2γ−1)x

2+θ(x), whereγ denotes throughout this paper the Euler-Mascheroni constant.

Upper bounds forθ(x) can be readily established as a trivial generalization of the corresponding results for the Dirichlet divisor problem. It is known that

D(x) =xlogx+ (2γ−1)x+ ∆(x) with

∆(x)x^23/73(logx)^461/146.

Received August 4, 1997.

1980Mathematics Subject Classification(1991Revision). Primary 11N37.

Key words and phrases. Divisor problem, Dirichlet summatory function, asymptotic expan- sion.

(See Huxley [5] for this upper bound and the textbook of Kr¨atzel [6] for an en- lightening survey of the theory of Dirichlet’s divisor problem and the definition of theO- and the Ω- symbols.)

Concerning lower estimates, the author proved in [8], on the basis of [1] and Hafner’s method [3], that

θ(x) = Ω+

(xlogx)^1/4(log logx)(3+2 log 2)/4exp (−Ap

log log logx) . The aim of the present article is an Ω⁻- result forθ(x), corresponding to that of Corr´adi and K´atai [1] for the divisor problem.

Theorem.

T(x) = x

2logx+ (2γ−1)x

2+θ(x), with

θ(x) = Ω−

x^1/4exp c(log logx)^1/4(log log logx)^−3/4 , wherec is a positive absolute constant.

2. Notations and Lemmas

For large real xwe define Px as the set of all primes less than or equal to x, and Qx the set of all square-free integers composed only of primes fromPx. We write |Px| for the cardinality ofPx andM = 2^|Px| for the cardinality ofQx. We then have

|Px| x

logx and Mexp c1 x logx

,

for some positive constantc1. The largest integer inQx is bounded bye^2x, since forq∈Qx, we have

logq≤X

p≤x

logp≤2x.

LetSx be the set of numbers defined by Sx=n

µ= X

q∈Qx

rq

√q whererq∈ {0,±1}and at least tworq6= 0o .

Finally let

η(x) = infn√

n+ 2µ withn∈N_o andµ∈Sx

o,

and

q(x) =−log (η(x)).

By a slight modification of the method used for the corresponding result in Gangadharan [2], one readily shows the following lemma.

Lemma 1. Forx→ ∞ we have

xq(x)exp c2 x

logx ,

for some positive constantc2.

Lemma 2. There exists a positive constantc3 such that X

q∈Qx

d(q)

q^3/4 exp c3x^1/4

logx .

Proof. By the definition ofQx, we have X

q∈Qx

d(q) q^3/4 = Y

p≤x

(1 + 2p^−3/4) = expX

p≤x

log (1 + 2p^−3/4)

≥expX

p≤x

p^−3/4+O(1)

exp c3x^1/4

logx

.

As in Gangadharan [2] define for real z, V(z) = 2(cos(z

2))²= 1 + e^iz+e^−iz

2 ,

and

Tx(u) = Y

q∈Q_x

V u√

q−5π 4

.

Lemma 3. We have

(1) 0≤Tx(u)≤2^M, for allu, (2) T_x⁰(u)M2^Me^x, for allu,

(3) Tx(u) =T0+T1,x+T2,x+T3,x where, T0= 1,

T1,x=e^5πi/4 2

X

q∈Qx

e^−iu√q T3,x= X

µ∈Sx

hµe^iuµ,

T2,x is the complex conjugate ofT1,x and|hµ| ≤1/4.

Proof. The proof of Lemma 3 is straightforward by the definition ofV(z) and Tx(u).

3. Proof of the Theorem We start with the well known Voronoi identity for

∆1(x) ^def= Z x

0 ∆(t)dt= x

4+ x^3/4 2√

2π² X∞ n=1

d(n)

n^5/4sin 4π√ nx−π

4

+O(1).

Inserting this in

θ(x) = ∆(x)−2 ∆ x 2

+ 2 ∆ x 4 ,

and substitutingT = 4π√

x, we get E1(T) ^def= Z T

0 E(t)t dt

=T^3/2 X∞ n=1

d(n) n^5/4

sin (T√

n−π/4)−2^5/4sin (Tp

n/2−π/4) + 2^3/2sin (Tp

n/4−π/4) , with

E(t) = 2π√ 2π

θ(t²/16π²)−1/4 . Define

P(x) = exp a x

logx such that

q(x)≤P(x) and M²≤P(x), and let

σx= exp (−2P(x)).

Next define for fixedx,

γx= sup

u>0

−2π√

2π θ(u²/16π²) u^1/2+1/P(x) .

We may assume thatγx<∞, otherwise more than Theorem 1 would be true.

Thus

(2) γxu^1/2+1/P(x)+A+E(u)≥0,

for allu, where A= 2π√ 2π/4.

Let

Jx=σ_x^5/2Z ^∞

0 γxu^1/2+1/P(x)+A+E(u)

uexp (−σxu)Tx(u)du.

The next lemma provides an asymptotic expansion forJx.

Lemma 4. Forx→ ∞, Jx=e²Γ 5

2

γx−1 4Γ 5

2 X

q∈Qx

d(q)

q^3/4 +o(γx) +o(1).

Proof. Do deal with the first two terms of Jx, we observe that, for r = 1 or r= ³₂+_P(x)¹ ,

Z ^∞

0 u^rexp (−σxu)Tx(u)du= Γ(1 +r)σ_x^−(1+r)

+ X

i=1,2,3

Z ^∞

0 u^rexp (−σxu)Ti,x(u)du where 1≤r≤³

2 +_P¹_(x).

The part ofT1,x contributes exactly, e^5πi/4

2 Γ(1 +r) X

q∈Qx

1 (σx+i√

q)^1+r X

q∈Qx

q^−(1+r)/2

X

q∈Qx

1Mp

P(x) =o(σx^−5/2).

The contribution of T2,x = T1,x is obviously no more than this. Finally T3,x

contributes X

µ∈S_x

hµ

(σx+iµ)^1+r 3^Mη(x)^−(1+r)

exp (Mln 3 + (1 +r)(−logη(x))exp (3P(x)) =o(σ_x^−5/2).

Next we deal with the contribution ofE(u) toJx. Our first step is to integrate by parts to introduceE1(u) in the integral. Thus,

I^def= Z ^∞

0 E(u)uexp (−σxu)Tx(u)du=−Z ^∞

0 E1(u) d

du exp (−σxu)Tx(u) du,

sinceE1(u)u^3/2 for largeuand E1(0) = 0. Inserting the series representation forE1(u) and integrating term by term, noting that the series converges absolutely for everyuand uniformly on compact sets, we get

I=− X∞ n=1

d(n)

n^5/4Im (e^−πi/4In) +OZ ^∞

0

d

du exp (−σxu)Tx(u)du +OZ ^∞

0 u^1/2exp (−σxu)|Tx(u)|du ,

since

u^3/2 d

du exp (−σxu)Tx(u)

= d

du u^3/2exp (−σxu)Tx(u)

−3

2u^1/2exp (−σxu)Tx(u), and

In def

= Z ^∞

0 (e^iu√n−2^5/4e^iu

√

n/2+ 2^3/2e^iu

√

n/4) d du

u^3/2exp (−σxu)Tx(u) du.

Estimating the contributions of the error terms, we see that Z ^∞

0

d

du exp (−σxu)Tx(u)du≤Z ^∞

0

|Tx(u)⁰−σxTx(u)|exp (−σxu)du

≤4^Mσ_x⁻¹+ 2^M exp cp

P(x)

1 + exp (2P(x))

=o(σ⁻_x^5/2), and

Z ^∞

0 u^1/2exp (−σxu)|Tx(u)|du2^M Z ^∞

0 u^1/2exp (−σxu)du 2^Mσ⁻_x^3/2exp cp

P(x) + 3P(x)

=o(σ⁻_x^5/2).

We integrateIn by parts once more and expandTx(u) as in (3) of Lemma 3, to get

In=−i X

k=0,... ,3

Z ^∞

0

√

ne^iu√n−2^5/4 rn

2e^iu

√

n/2+ 2^3/2 rn

4e^iu

√

n/4

×u^3/2exp (−σxu)Ti,x(u)du

=I0(n) +I1(n) +I2(n) +I3(n),

for short. We shall show that the main term ofIn comes fromI1(n). In fact, the contribution ofI0(n) is

√

n|σx−i√

n|^−5/2n^−3/4, that ofI2(n) is

√ n X

q∈Qx

|σx−i(√ n+√

q)|^−5/2Mn^−3/4.

The contribution ofI3(n) is bounded by I3(n)√

n X

µ∈Sx

|σx−i(√

n−µ)|^−5/2







√n3^M(η(x))^−5/2, ifn≤2 max{|µ|:µ∈Sx}

n^−3/43^M, else.

This max{|µ|:µ∈Sx}is bounded byMe^cx for some positive constantc. Hence the total contribution toI is bounded by

X

n≤2Me^cx

d(n) n^5/4

√n3^Mexp −5 logη(x) 2

+O 3^Mσ_x^−5/4 X

n>2Me^cx

d(n) n²

!

3^Mσ_x^−5/4 X

n≤2Me^cx

n^−3/4++O(3^Mσ_x^−5/4) 3^Mσ_x^−5/4(Me^cx)^1/4+

=o(σx^−5/2).

Therefore, I=−1

2 X∞ n=1

d(n) n^5/4Im

i X

q∈Qx

Z ^∞

0

√ne^{iu(√n−√q)}−2^5/4 rn

2e^iu(

√

n/2−√ q)

+ 2^3/2 rn

4e^iu(

√

n/4−√ q)

u^3/2exp (−σxu)du

+o(σ_x^−5/2)

=−1 2

X

q∈Qx

d(q)

q^5/4 −2^5/4 d(2q)

(2q)^5/4 + 2^3/2 d(4q) (4q)^5/4

Z ^∞

0

√q u^3/2exp (−σxu)du

+OX^∞

n=1

d(n) n^5/4

X

q∈Qx n6=q

Z ^∞

0

√ne^{iu(√n−√q)}u^3/2exp (−σxu) du

.

For this last error term we get a bound exactly as above for I3(n) with M replacing the factor 3^M, since

√n−√ q(√

n+√

q)⁻¹e^−xexp (−P(x)), forn≤2 max{q:q∈Qx} 2e^2x andn6=q.

We get, I=−1

2Γ 5 2

σ_x^−5/2 X

q∈Qx

(d(q)−d(2q) +1 2d(4q)

q^−3/4+o(σ⁻_x^5/2)

=−1 4Γ 5

2

σ_x^−5/2 X

q∈Qx

d(q)q^−3/4+o(σ_x^−5/2),

since

d(q)−d(2q) +1

2d(4q) = 1 2d(q).

This completes the proof of Lemma 4.

Sinceσx>0 andJx>0 by (2), we have exp

cx^1/4 logx

X

q∈Qx

d(q)q^−3/4γx, by Lemma 2 and the last assertion by Lemma 4.

Thus by the definition ofγxthere is a sequenceuxwhich tends to infinity withx, such that

−θ(u²_x)u^1/2_x explogux

P(x) +cx^1/4 logx

,

sinceθ(u) is bounded for boundedu, which follows for smallufrom θ(u) =−u

2logu−(2γ−1)u 2, and is obvious for the other values ofu.

Consider first the values ofux for which

(3) logux

P(x) ≤cx^1/4 logx. Taking logarithms on both sides, we have

log logux x logx.

Since y^1/4(logy)^−3/4 is an increasing function of y for sufficiently large y, we have from (3)

(log logux)^1/4

(log log logux)^3/4 x^1/4 logx, from which the desired estimate follows.

Consider now those values ofxfor which

(4) cx^1/4

logx≤ logux

P(x) . We may assume that

(log logux)^1/4

(log log logux)^3/4 logux

P(x) ,

otherwise the estimate holds obviously. Taking logarithms on both sides gives log logux x

logx,

from which the estimate follows as above. This proves the theorem.

References

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2.Gangadharan K. S., Two classical lattice point problems, Proc. Cambridge Phil. Soc. 57 (1961), 699–721.

3.Hafner J. L.,New omega theorems for two classical lattice point problems, Invent. Math.63 (1981), 181–186.

4. ,New omega theorems in a weighted divisor problem, J. Number Theory28(1988), 240–257.

5.Huxley M. N.,Exponential sums and lattice points II, Proc. London Math. Soc.66(3)(1993), 279–301.

6.Kr¨atzel E.,Lattice Points, Berlin, 1988.

7. ,Primitive lattice points in special plane domains and a related three dimensional lattice point problem I, Forschungsergebnisse FSU Jenam, N/87/11, 1987.

8.K¨uhleitner M.,An omega theorem on differences of two squares, Acta Math. Univ. Comeni- anaeLXI(1)(1992), 117–123.

9.Nowak W. G.,On differences of twok-th powers of integers, (to appear).

M. K¨uhleitner, Institut f¨ur Mathematik, Universit¨at f¨ur Bodenkultur, Gregor Mendel Straße 33, A-1180 Wien, Austria;e-mail: [email protected]