Abstract
The variance associated with the distribution of sums of two unlike powers in arithmetic progressions is evaluated asymptotically.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Recently the authors [2] studied the distribution in arithmetic progressions of the numbers that are the sum of two positive cubes of integers, and established an asymptotic formula of Montgomery–Hooley type for the associated variance. As indicated on that occasion, a further development of our method supplies related results for sums of a square and an h-th power, for any \(h\geqslant 3\). Here we discuss the problem in broader generality and consider, for given numbers \(h\geqslant k\geqslant 2\), the sequence of numbers of the shape \(x^k+y^h\) as x and y range over the natural numbers. Our method is successful whenever the number
exceeds 1/2. Given such a pair k, h, let r(n) denote the number of solutions of \(x^k+y^h =n\) in natural numbers x and y, and let \(\rho (q,a)\) denote the number of incongruent solutions of the congruence . Finally, let
denote the area of the domain
For \(N\geqslant 1\), \(Q\geqslant 1\) we consider the variance
An asymptotic formula for V(N, Q) is expected to hold when Q is not too far from \(N^{\theta }\). One possible approach is a dispersion argument. Opening the square in (1.3), the expression
arises naturally and prominently impacts the behaviour of V(N, Q). It transpires that the case \(k=h=2\) is peculiar because, in marked contrast to all other cases, here r(n) is often so large that the order of magnitude of (1.4) is \(QN \log N\). This atypical case has been analysed by Dancs [3] in his thesis, in a slightly different setting; he replaces our r(n) with the number of solutions of \(n=x^2+y^2\) in integers x and y. Translated to our language, his main result asserts that there are real numbers \(c,c'\) such that whenever \(1\leqslant Q\leqslant N\) then
Here and later in this paper, we apply the following convention concerning the letter \(\varepsilon \): whenever \(\varepsilon \) occurs in a statement, it is asserted that the statement is true for any positive value assigned to \(\varepsilon \).
For sums of two cubes, the case \(k=h=3\), the situation is rather different. There are only about \(N^{2/3}\) numbers n not exceeding N that are the sum of two positive cubes, and for a typical such number one has \(r(n)=2\). Therefore, the sum \(\sum _{n\leqslant N} r(n)^2\) will be of size \(N^{2/3}\). This reflects in the shape of the asymptotic expansion of the variance for which we obtained ([2, Theorem 1.1])
uniformly in the range \(1\leqslant Q\leqslant N\).
The remaining pairs k, h with \(h\geqslant k\geqslant 2\) and \(\theta >1/2\) are
Thus, in all these cases we have \(h>k\), and as we shall see in Lemma 2.1, this implies that for the typical number that is the sum of a k-th power and an h-th power, one has \(r(n)=1\). Again, this changes the leading term in an asymptotic formula for V(N, Q).
Theorem 1.1
Suppose that k, h is one of the pairs satisfying (1.5), and let \(1\leqslant Q\leqslant N^{\theta }\). If \(k=3\), then
If \(k=2\) and \(h\geqslant 7\), then
If \(k=2\) and \(4\leqslant h\leqslant 6\), then the error term in (1.6) is to be replaced by
while for \(k=2\), \(h=3\) this error is
There is a large body of work concerned with the distribution of arithmetic sequences in residue classes, with a view toward an asymptotic formula for the associated variance. The historic papers of Montgomery [9] and Hooley [6] on the von Mangoldt function triggered interest in analogous results for other arithmetic functions of great familiarity in multiplicative number theory, such as the indicator function of the k-free numbers [17] and their l-tuplets [10], or the divisor function [11,12,13]. There are also axiomatic studies in work of Hooley [7, 8] and Vaughan [15, 16]. Any attempt to review all examples that have been detailed hitherto would take us far afield, but a common feature of previous work is that in all cases that we are aware of, the order of magnitude of the crucial expression (1.4) only mildly digresses from QN. In particular we do not know a single instance where the appropriate analogon of \(\sum _{n\leqslant N} r(n)^2\) is bounded above by \(N^{1-\delta }\), for some \(\delta >0\). This paper and its compagnion [2] provide a family of such examples, with \(\delta \) approaching 1/2. At \(\delta =1/2\), however, essential obstacles arise on which we comment in more detail.
Our approach to estimates of the type provided by Theorem 1.1 uses a variant of the dispersion argument proposed by Goldston and Vaughan [4]. One is required to evaluate the sum
asymptotically, and this is within the competence of the circle method. In our earlier work on sums of two cubes, this approach was modified, and the circle method was brought into play only after r(n) was replaced by an arithmetic function that resembles the minor arcs contribution in the integral
the latter being valid for all n not exceeding N. This led to considerable technical simplifications. Similar ideas apply in the more general set-up of this paper as well. As is often the case with mixed exponents in representation problems, a proportion of the Farey intervals in our application of the Hardy–Littlewood method have to be treated as major for the smaller exponent, but as minor for the larger one. This new hurdle is overcome by a succession of pruning exercises that we execute in §4. We then obtain, in §5, an imperfect version of Theorem 1.1. This is of some interest on its own right, see Theorem 5.4 below. The deduction of Theorem 1.1 from Theorem 5.4 is then achieved in §6 by following the routines developed in [2, Section 4].
The Fourier integral that we estimate by the circle method appears in (5.4) below. The square root cancellation barrier for this integral is \(N^{\theta + 1/2}\), and it therefore appears to be very difficult to find asymptotic relations for V(N, Q) with an error estimate superior to \(N^{\theta + 1/2}\). An error of this size is dominated by the leading \(QN^{\theta }\) only if \(Q\geqslant N^{1/2}\). It should therefore be noted that in the cases \(k=2\), \(h\geqslant 7\) Theorem 1.1 indeed supplies a valid asymptotic formula whenever \(Q\geqslant N^{1/2+\varepsilon }\), and achieves square root cancellation in a certain range for Q. In the remaining cases, the result is somewhat weaker but then \(\theta \) is rather larger than 1/2. In fact, our methods are tuned to perform optimally for the smaller values of \(\theta \), leaving the other cases susceptible to some small improvement.
In §7 we consider the numbers representable as sums of a k-th power and an h-th power without multiplicities. Define \(r_0(n)=1\) whenever \(r(n)\geqslant 1\) and let \(r_0(n)=0\) otherwise. Then, for a typical natural number n one has \(r(n)=r_0(n)\). It is now natural to examine
Theorem 1.2
Suppose that k, h is one of the pairs satisfying (1.5). Let \(Q\leqslant N^{\theta }\). Then
This combines easily with the results of Theorem 1.1, and provides asymptotic formulae for \(V_0(N,Q)\). In particular, one finds that (1.6) holds with V(N, Q) replaced by \(V_0(N,Q)\). For an analogous result in the case \(k=h=3\) see [2, Theorem 1.4]. Perhaps surprisingly, for the numbers that are the sum of two squares, an asymptotic formula for \(V_0(N,Q)\) is not yet known.
2 Auxiliaries
We begin with elementary mean value estimates for r(n).
Lemma 2.1
Let k, h be a pair satisfying (1.5). Then
and
Proof
The linear mean of r(n) follows by the standard lattice point argument of Gauß. The sum of \(r(n)^2\) equals the number of solutions of
in positive integers x, y, u, v. The solutions with \(y=v\) (and a fortiori \(x=u\)) contribute \(\sum r(n)\). There are \(O(N^{2/h})\) choices for \(y\ne v\), and once these are chosen, a divisor argument shows that there are no more than \(O(N^\varepsilon )\) choices for x and u.\(\square \)
We frequently encounter a family of multiplicative functions that we now describe. Let \(l\geqslant 2\) be a natural number, and let \(\kappa _l\) be the multiplicative function that for primes p and integers \(\nu ,\lambda \) with \(\nu \geqslant 0\) and \(2\leqslant \lambda \leqslant l\) is defined by
We then have the immediate bounds
for all \(q\in {\mathbb {N}}\), and the estimate
Lemma 2.2
Let k, h be a pair satisfying (1.5). Then
and
Proof
By (2.2) we have \(q \kappa _2(q)^2\leqslant 1\). Hence, the cases of (2.4) where \(k=2\) are immediate from (2.3). If \(k=3\) and \(h=4\) or 5, then one checks from (2.1) that holds for all \(\nu \geqslant 1\) while (2.2) yields the bounds
that are superior when \(\nu \) is large. Similar to the argument in (2.3), the estimate (2.4) now follows after turning the sum into an Euler product.
Next we establish (2.5) in the case where \(k=3\), \(h=5\). Let
By (2.2),
and hence,
For \(1\leqslant \nu \leqslant 14\) one checks from (2.1) that , and so, for the same \(\nu \), we have \(K(p^\nu ) \leqslant (\nu +1) p^{(\nu -1)/2}\) and \(p^{-\nu } K(p^\nu )^2\leqslant (\nu +1)^2 p^{-1}\). It follows that the expression on the left-hand side of (2.5) does not exceed
This establishes (2.5) in the case \(k=3\), \(h=5\). By the obvious inequality \(\kappa _4(q) \leqslant \kappa _5(q)\) the case \(k=3\), \(h=4\) also follows.
This leaves the cases \(k=2\), \(h\geqslant 3\). Here, by (2.2) and (2.1), we have
for all \(\nu \geqslant 1\), and also
The proof of (2.5) in these cases now proceeds as above.\(\square \)
3 Gauß and Weyl sums
For \(l\geqslant 2\) let
be the l-th power Gauß sum. By [14, Lemmata 4.3, 4.4 and 4.5], the bound
holds whenever \((a,q)=1\). The partial singular series relative to the parameter \(T\geqslant 1\) for the sum of a k-th power and an h-th power is the sum
We require the following mean value estimate.
Lemma 3.1
Let \(N\geqslant 1\), \(T\geqslant 1\). Then, for pairs k, h satisfying (1.5),
Proof
One opens the square and the definition (3.2). Then, by the dual of the large sieve inequality (see [2, Lemma 2.2], for example),
Via (3.1) and (2.4), we infer that
To deduce Lemma 3.1, split the sum over n into intervals \(M\leqslant n <2M\) and sum over \(M=2^\mu \). Since \(1/2<\theta <1\), the desired estimate is immediate.\(\square \)
The remainder of this section is primarily concerned with the exponential sum
that we examine by relating it to the more familiar Weyl sums
For the latter, we now define their major arc approximation. This entails the integral
for which partial integration provides the estimate
The next lemma is [14, Theorem 4.1].
Lemma 3.2
Let \(a\in {\mathbb {N}}\), \(q\in {\mathbb {N}}\), \(\alpha \in {\mathbb {R}}\) and write \(\beta =\alpha -a/q\). Then
From now on, let k, h be a pair satisfying (1.5). We require appropriate analogues of Lemma 3.2 for the sum \(g(\alpha )\). By (3.3),
We apply Lemma 3.2 to the sum over x. In the notation of that lemma, this yields
We define the function
and arrive at the following imperfect approximation for \(g(\alpha )\).
Lemma 3.3
Let \(a\in {\mathbb {N}}\), \(q\in {\mathbb {N}}\), \(\alpha \in {\mathbb {R}}\) and write \(\beta =\alpha -a/q\). Then
To proceed further, we apply an obvious substitution in (3.5) to infer
We apply Lemma 3.2 again to see that the above expression equals
Once more by obvious substitutions, the double integral here simplifies to
where \(\theta \) is defined by (1.1) and
is a special value of Euler’s Beta function. By (1.2) and a mundane computation, one finds that \(B=kh \theta C\) and then concludes from (3.1), (3.7) and (3.8) that
By Euler’s summation formula,
We define the sum
and observe, with later applications in mind, that the proof of [14, Lemma 2.8] provides the estimate
Further, we write
Then, by (3.9), (3.10), (3.12) and Lemma 3.3, we conclude as follows.
Lemma 3.4
Let \(\alpha \in {\mathbb {R}}\), \(q\in {\mathbb {N}}\) and \(a\in {\mathbb {Z}}\) with \(|\alpha -a/q|\leqslant 1\). Then
4 Pruning exercises
Let \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\), and let \({\mathfrak {N}}(q,a;X)\) denote the interval of all real \(\alpha \) with \(|q\alpha -a|\leqslant X/N\). Further, let \({\mathfrak {N}}(X)\) denote the union of \({\mathfrak {N}}(q,a;X)\) with \(1\leqslant a\leqslant q\leqslant X\) and \((a,q)=1\). Note that this union is disjoint. For convenience, we write \({\mathfrak {N}} = {\mathfrak {N}}\bigl (\frac{1}{4} N^{1/2}\bigr )\). When \(1\leqslant a\leqslant q\leqslant \frac{1}{4} N^{1/2}\), \((a,q)=1\) and \(\alpha \in {\mathfrak {N}}\bigl (q,a;\frac{1}{4} N^{1/2}\bigr )\), put
This defines a function \(\Phi :{\mathfrak {N}} \rightarrow (0,\infty )\). Our basic tool is a development of [1, Lemma 1].
Lemma 4.1
Let \(\Psi :{\mathbb {R}} \rightarrow [0, \infty )\) be a trigonometric polynomial
with real non-negative coefficients \(\psi _m\). Then, uniformly for \(\gamma \in {\mathbb {R}}\) and \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\), one has
Proof
Let \({\mathfrak {I}}\) denote the integral to be estimated. Since \(\Psi \) is a non-negative function, we have
The classical bound for Ramanujan’s sum [5, Theorem 272]
now shows that
For non-zero integers m one routinely finds that
and the lemma follows immediately.\(\square \)
Within this section we adumbrate \(f_l(\alpha ,N^{1/l})\) to \(f_l(\alpha )\). As a first application of Lemma 4.1, we take \(\Psi (\alpha )=|f_l(\alpha )|^2\) where \(\psi _m\) is the number of solutions of \(x^l-y^l = m\) with \(1\leqslant x,y\leqslant N^{1/l}\). Thus, the hypotheses of this lemma are satisfied with \(M=N\), so uniformly in \(\gamma \in {\mathbb {R}}\) we infer the estimate
Performing the same argument with \(\Psi (\alpha )=|f_l(\alpha )|^4\) yields
The principal object in this section is to estimate the integral
where \(g(\alpha )\) is the sum defined in (3.3) with k, h chosen in accordance with (1.5). There are several approaches, depending on the relative size of k and h, and on the size of X. For convenience, we put
and note at once that
If we pair this bound with the mean value
that in turn is implied by (3.3), Lemma 2.1 and orthogonality, we deduce a first result concerning J(X), namely
More sophisticated bounds for J(X) depend on (4.3) or (4.4).
Lemma 4.2
Let k, h be one of the pairs satisfying (1.5), and let \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Then
Proof
Let
Hence, by (4.5),
where we wrote
For any complex numbers \(z, z'\) one has \(2|zz'| \leqslant |z|^2+ |z'|^2\), and so,
We put
By symmetry in \(\gamma \) and \(\gamma '\) it now follows that
The trivial bound \(K(\gamma )\ll N(1+N\Vert \gamma \Vert )^{-1}\) implies that
and we arrive at the preparatory bound
By (4.9) and Schwarz’s inequality,
while Hua’s Lemma [14, Lemma 2.5] yields
both bounds being valid uniformly in \(\gamma \). This shows that
and the lemma is available from (4.10).\(\square \)
The bounds obtained so far are useful for large X. The next two lemmata are of preparatory nature for an argument that gives good bounds for J(X) when X is smaller.
Lemma 4.3
Let k, h be a pair satisfying (1.5) and let \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Then
Proof
We follow through the initial phase of the proof of Lemma 4.2 leading to (4.10). In this way, we arrive at the provisional bound
By Schwarz’s inequality, the integral on the right-hand side is reduced to the integrals in (4.3) with \(l=k\) and \(l=h\), and the lemma follows immediately.\(\square \)
Define the function \(g^*:{\mathfrak {N}}\rightarrow {\mathbb {C}}\) by taking \(g^*(\alpha )= g^*(\alpha ;q,a)\) whenever \(\alpha \in {\mathfrak {N}}(q,a;\frac{1}{4}N^{1/2})\) with \(1\leqslant a\leqslant q\leqslant \frac{1}{4} N^{1/2}\) and \((a,q)=1\).
Lemma 4.4
Let k, h be a pair satisfying (1.5), and let \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Then
Proof
Let \({\mathscr {B}}(q)= [-1,1]\) when \(\frac{1}{2} X< q\leqslant X\), and when \(q\leqslant \frac{1}{2} X\) put
Then, writing I for the integral to be estimated,
so that
the expression on the right being real and non-negative. We insert this in (4.11), bring the sum over a inside and estimate it by (4.1). This manoeuvre yields
First consider the portion of the sum on the right where \(\frac{1}{2} X<q\leqslant X\). Here \({\mathscr {B}}(q)=[-1,1]\), so we can bring the sum over q inside the integral and use the trivial uniform bound \(|v_k(\beta , (N-y^h)^{1/k})|\leqslant N^{1/k}\). We then see that this portion of (4.12) is bounded above by
Here we single out terms with \(y_1=y_2\) and apply (4.2) for the remaining choices of \(y_1,y_2\). Then, by (2.2) and (2.3) we see that the above expression is bounded by
which is sufficient.
Our treatment of the portion where \(q\leqslant \frac{1}{2} X\) is similar but relies on the bound \(v_k(\beta , (N-y^h)^{1/k})\ll |\beta |^{-1/k}\) that is again uniform in y, and which follows from (3.6). This portion of (4.12) therefore does not exceed
We can now proceed as in (4.13) to obtain the same final estimate. \(\square \)
Lemma 4.5
Let k, h be a pair satisfying (1.5) and \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Then
Proof
Directly from (4.5) we have
while Schwarz’s inequality shows that
Combining the last two inequalities implies that
The second integral on the right-hand side is estimated in Lemma 4.4, and contributes an acceptable amount. To bound the first integral on the right-hand side, note that Lemma 3.3 yields \(g(\alpha )-g^*(\alpha ) \ll N^{1/h}X^{1/2+\varepsilon }\) for \(\alpha \in {\mathfrak {L}}(X)\), and then we apply Lemma 4.3 to find that
This establishes the lemma.\(\square \)
We have completed the estimation of J(X) but simplify the results for readier use.
Lemma 4.6
Let \(k=2\), \(h\geqslant 3\) and \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Put \(\sigma (3)=1/12\), \(\sigma (4)=1/16\) and \(\sigma (h)=0\) for \(h\geqslant 5\). Then
Proof
First suppose that \(h\geqslant 5\). For \(X\geqslant N^{2/h}\) Lemma 4.2 gives \(J(X)\ll N^{1/h}\). For \(X\leqslant N^{2/h}\) the desired estimate is contained in Lemma 4.5.
Next suppose that \(h=3\) or 4. If \(X\geqslant N^{1/2-\sigma (h)}\), we use (4.8) and obtain \(J(X) \ll N^{1/h+\sigma (h)+\varepsilon }\). For \(X\leqslant N^{1/2-\sigma (h)}\) the desired bound is again a consequence of Lemma 4.5.\(\square \)
Lemma 4.7
Let \(k=3\) and \(h=4\) or 5. Further let \(1\leqslant X\leqslant \frac{1}{4} N^{1/2}\). Then
Proof
This follows from Lemma 4.2 for \(X\geqslant N^{1/3+ 1/(4h)}\), and from Lemma 4.5 for the remaining X.\(\square \)
5 Imperfect variance
We launch our attack on Theorem 1.1 by first considering the expression
that may be viewed as an imperfect version of the variance V(N, Q). One opens the square and finds that
where
and
Our ultimate goal in this section is an asymptotic formula for U(N, Q, T). It is easy to extract a main term from \(U_0(N,T)\).
Lemma 5.1
Let \(T\geqslant 1\). Then
Proof
We square out the expression in (5.2), and first consider the cross term
Here, by (3.1), (3.2) and (2.3), we have the trivial bound
Hence, by Lemma 2.1 and partial summation, we see that the cross term is bounded by
which is acceptable. This leaves the sum involving \(|\mathfrak {s}(n;T)|^2\), and here Lemma 3.1 provides an acceptable estimate. \(\square \)
The next theorem provides an estimate for S. Recall the data \(\sigma (h)\) defined in Lemma 4.6.
Theorem 5.2
Let k, h be one of the pairs satisfying (1.5) and suppose that \(1\leqslant Q\leqslant N^\theta \). If \(k=2\) and \(1\leqslant T\leqslant N^{1/h}\), then
If \(k=3\), \(h=4\) and \(1\leqslant T\leqslant N^{1/8}\), then
and if \(k=3\), \(h=5\) and \(1\leqslant T\leqslant N^{1/20}\), then
The initial steps in the proof of this theorem are identical to the work in [2, Section 3]. Form the exponential sums
and
Then
We follow Goldston and Vaughan [4, Section 3] and examine the integral in (5.4) by the circle method. This depends on a mean square estimate for \(G(\alpha )\). By (5.3) and (3.3),
so that
By Lemma 3.1 and orthogonality, one finds that
Then, by (4.7), (5.5) and (5.6), it follows that
Consider a typical interval \({{\mathfrak {M}}}(r,b)\) associated with the element b/r of the Farey dissection of order \(2N^{1/2}\), namely, when \(1\leqslant b\leqslant r\leqslant 2N^{1/2}\) and \((b,r)=1\),
where \(r_\pm \) is defined by \(br_\pm \equiv \mp 1\pmod r\) and \(2N^{1/2}-r<r_\pm \leqslant 2N^{1/2}\) and \(b_\pm \) is defined by \(b_\pm =(br_\pm \pm 1)/r\). We observe that
lies in \([1/(4rN^{1/2}),1/(2rN^{1/2}))\). For \(r\leqslant \frac{1}{4} N^{1/2}\) this implies that
The analysis of the function F relative to the Farey intervals performed in [2, Section 3] we invoke with \(R=2N^{1/2}\) (in the notation of [2]). Then, by (5.8), we infer the bound
for all \(\alpha \) covered by the Farey intervals. Consequently, by (5.4) and (5.7),
We are reduced to estimating the integral on the right hand side. We choose a parameter Y with \(1\leqslant Y\leqslant \frac{1}{4}N^{1/2}\), write \({\mathfrak {K}}= {\mathfrak {N}}(Y)\) for the core major arcs, and put . By (4.6) we see that \(\Phi (\alpha )\ll Y^{-1}\) holds uniformly on \({\mathfrak {k}}\), so that (5.6) provides us with the bound
Further, since \({\mathfrak {k}}\) is covered by no more than \(\log N\) sets \({\mathfrak {L}}(X)\) with \(Y\leqslant X\leqslant \frac{1}{4} N^{1/2}\), it follows from (4.5) that
If \(k=2\), then by (5.5) and Lemma 4.6, the last two bounds combine to give
When \(k=3\), the same argument based on Lemma 4.7 yields
On the core major arcs, we use approximations to \(G(\alpha )\) provided by Lemma 3.4. Here we follow the pattern of our previous work [2, Section 3] quite closely, beginning with (3.12) of that memoir. In the wider context of our current analysis, this formula still reads
but now stems from (5.3), (3.12) and (3.2).
We wish to use this for \(\alpha \in {\mathfrak {K}}\) and therefore write \({\mathfrak {K}}\) as the disjoint union of intervals \({\mathfrak {K}}(r,b)= \{\alpha : |r\alpha - b|\leqslant Y/N\}\) with \(1\leqslant b\leqslant r\leqslant Y\) and \((b,r)=1\). Suppose that \(\alpha \) is in one of these arcs \({\mathfrak {K}}(r,b)\). Then
where in view of (5.12) we have
Here we estimate a typical summand on the far right when \(c/t \ne b/r\). If \(\alpha \in {\mathfrak {K}}(r,b)\), then \(|\alpha -b/r|\leqslant Y(rN)^{-1}\). We proceed subject to the condition that \(T\leqslant Y\), this will turn out to be the case later. It then follows that
Hence, by (3.1), (3.11) and (3.12), we have
At this stage, we may follow through the argument in [2] that leads from (3.15) to (3.20) of that paper. Instead of [2, (3.17)] we then encounter the sum
and by (2.3) and Cauchy’s inequality, this sum is bounded \(O(T^{1+\varepsilon })\), as is required to proceed to [2, (3.18)], and we then arrive at the estimates
and
Lemma 5.3
One has
Proof
We begin with the set of all \(\alpha \) where \(D(\alpha ;r,b)\ll r^{\theta }+T^{1+\varepsilon }\) holds. This set makes a contribution to the integral in question that does not exceed
On the remaining set we have \(r>T\) and \(D(\alpha ;r,b) \ll |W(\alpha ;r,b)|\). This follows from (5.14) and (5.15). By (3.1), (3.11), (3.12) and (2.4) we see that
This confirms the estimate proposed in Lemma 5.3.\(\square \)
Our last auxiliary estimate is now almost immediate. For \(\alpha \in {\mathfrak {K}}(r,b)\) we see from (5.13) that
We multiply by \(\Phi (\alpha )\). Then, by Lemma 3.4,
Now we integrate over \({\mathfrak {K}}\) and obtain
We apply (2.3) and combine the result with Lemma 5.3 to confirm the bound
We are ready to derive an estimate for S. Indeed, \({\mathfrak {N}}\) is the union of \({\mathfrak {k}}\) and \({\mathfrak {K}}\), so we have to combine the results in (5.9) and (5.16) with either (5.10) or (5.11). We first consider the cases with \(k=2\) and choose \(Y=N^{1/h}\). Then, we are allowed to take \(1\leqslant T\leqslant N^{1/h}\), and find that
For \(k=3\), the choices \(Y=Y_5=N^{1/20}\) and \(Y=Y_4=N^{1/8}\) give
in the range \(1\leqslant T\leqslant Y_h\). In both cases, Theorem 5.2 is now available.
Asymptotic formulae for U(N, Q, T) are also available.
Theorem 5.4
Let k, h be one of the pairs satisfying (1.5) and suppose that \(1\leqslant Q\leqslant N^\theta \). If \(k=2\) and \(1\leqslant T\leqslant N^{1/h}\), then
If \(k=3\) and \(1\leqslant T \leqslant N^{1/(8h)}\), then
We remark here that for \(Q\leqslant N^\theta \) and \(k=2\), \(h\geqslant 6\) one has \(QTN^{2/h} \leqslant N^{(1/2)+(4/h)} \leqslant N^{1+1/h}\), so the term \(QTN^{2/h}\) can be ignored except when \(h\leqslant 5\). Iterating an earlier comment relating to the error term in Theorem 1.1, note here that for \(k=2\) errors of size \(N^{1+1/h}\) correspond to square root cancellation in the integral representation of S, and are probably hard to improve.
For a proof of Theorem 5.4 use Lemma 2.1 within Lemma 5.1 to see that
where we choose T in accordance with Theorem 5.2. Then \(T\leqslant N^{1/h}\) in all cases, so the term \(T^2\) in the error term is redundant. By (5.1) we get
First suppose that \(k=3\) and \(1\leqslant T \leqslant N^{1/(8h)}\). Then, by Theorem 5.2, one has \(S\ll N^{2\theta +\varepsilon }T^{-1}\), and Theorem 5.4 follows from (5.17).
Next suppose that \(k=2\) and \(1\leqslant T\leqslant N^{1/h}\). Now \(2\theta -1= 2/h\), and the term \(O(N^{2/h+\varepsilon })\) in (5.17) can be ignored. Further, one has \(QN^{1/2}\leqslant N^{1+1/h}\), and the cases with \(k=2\) of Theorem 5.4 again follow from (5.17) and Theorem 5.2.
6 From the imperfect to the perfect
In this section we establish Theorem 1.1. The argument is very similar to the one presented in §4 of our previous communication [2] but there are some subtleties, and we therefore proceed in moderate detail. Let
and consider
Lemma 6.1
Let \(1\leqslant T\leqslant Q\leqslant N^\theta \). Then
Proof
The method of proof of [14, Lemma 2.12] yields
We split the partial singular series as \({\mathfrak {s}}(n;T) = {\mathfrak {s}}^*_q(n;T) + {\mathfrak {s}}^\dagger _q(n;T)\) where
and \({\mathfrak {s}}^\dagger _q(n;T)\) is defined in the same way, but with the complementary condition \(t\not \mid q\).
Let \(\Xi ^\dagger (N,T;q,a)\) be the sum defined in (6.1), but with \({\mathfrak {s}}(n;T)\) replaced by \({\mathfrak {s}}^\dagger _q(n;T)\). We shall use this notation also in situations where \(\Xi \) and \({\mathfrak {s}}\) are decorated by symbols other than \(\dagger \). We show that \(\Xi ^\dagger (N,T;q,a)\) is small on average. Let \(1\leqslant a\leqslant q\). Then
The total contribution from terms with \(m=0\) is
For \(m\geqslant 1\) we have \(a+mq \gg mq\), and so, by partial summation,
Hence the contribution to \(\Xi ^\dagger (N,T;q,a)\) from terms with \(m\geqslant 1\) is bounded by
Here we used (2.3) in the final estimate. By similar estimates within (6.5), we also see \( \Xi ^\dagger (N,T;q,a) \ll (a^{\theta -1} + q^{\theta -1}) T^{1+\varepsilon } \) and then have
Now let
This sum depends on n only modulo q. Thus, with the convention concerning decorations of \(\Xi \) in mind,
Much as before, we find that whenever \(q\leqslant N^\theta \) then
and consequently,
By (6.7) and Cauchy’s inequality, and then by orthogonality,
Recall that \(Q\leqslant N^{\theta }\). Hence
In the sums over q and t we write \(q=ts\). By (2.4) the double sum is
This proves that
We collect the results obtained so far in a single estimate. By (6.3), (6.4) and (6.7), we have
It follows that
and from (6.6) and (6.8) we then conclude
By (6.9) and (6.2), we are reduced to replacing
To realize this, we may follow the argument given in [2] very closely. By orthogonality, we see that
Here, by (3.11), the sum on the far right over n is \( O(\Vert b/q\Vert ^{-\theta })\). Further, by (6.3) and (3.1),
Observing orthogonality, it now follows that the sum
is bounded above by
as is readily confirmed from (2.5).
The elementary evaluation
together with (6.10) and (2.5) implies the bound
Equipped with the estimate obtained for the sum in (6.11), we deduce that
This bound coupled with (6.9) implies Lemma 6.1.\(\square \)
The endgame begins by writing (1.3) as
where
Then, squaring out and estimating the cross term by Cauchy’s inequality, we find
Theorem 1.1 is now easily deduced. When \(k=3\) we choose \(T=N^{1/8h}\). Then, for \(T\leqslant Q\leqslant N^\theta \), Lemma 6.1 gives \(\Delta \ll N^{2\theta }T^{-1}\), and Theorem 5.4 yields \(U=CQN^{\theta } +O(N^{2\theta }T^{-1})\). Now (6.12) gives
in the range \(N^{1/8h}\leqslant Q\leqslant N^\theta \). For \(1\leqslant Q\leqslant N^{1/8h}\) the asymptotic formula (6.13) reduces to the upper bound \(V(N,Q)\ll N^{2\theta -1/(8h)+\varepsilon }\) that we have just confirmed for \(Q= N^{1/8h}\), and hence remains true for smaller values of Q. This completes the proof of Theorem 1.1 in the case \(k=3\).
The other cases are similar, and we merely indicate the choices of parameters. For \(k=2\), \(h\geqslant 7\), we take \(T=N^{1/h}\). Then, for \(T\leqslant Q\leqslant N^{\theta }\) we have \(T^3 \leqslant (N/Q)^{2\theta } \) so that Lemma 6.1 again gives \(\Delta \ll N^{2\theta }T^{-1}\ll N^{1+1/h}\). As above, we first deduce that
holds for \(T\leqslant Q\leqslant N^\theta \), and then see that the range \(1\leqslant Q\leqslant T\) is covered for trivial reasons.
Next, consider the cases \(k=2\), \(4\leqslant h\leqslant 6\). We wish to conclude from Lemma 6.1 that \(\Delta \ll N^{2\theta }T^{-1}\) with T as large as is possible. This requires us to take \(T\leqslant N^{1/h}\) and \(T\leqslant (N/Q)^{2\theta /3}\). However, at least when \(h\leqslant 5\) the term \(QTN^{2/h}\) needs attention, and we require that \(QTN^{2/h}\leqslant N^{2\theta }T^{-1}\) as well. This reduces to \(T\leqslant (N/Q)^{1/2}\). However, for \(h\geqslant 4\) we have \(\frac{2}{3} \theta \leqslant \frac{1}{2}\), so this last condition for T is implied by the previous ones, and we can choose
Then, again via Theorem 5.4 and (6.12), we conclude that
holds for \(T\leqslant Q\leqslant N^\theta \), and this yields the bound in Theorem 1.1. The smaller values of Q are again a trivial range.
This leaves the case \(k=2\), \(h=3\). We follow the same strategy but this time have \(\frac{2}{3} \theta > 1/2\) and therefore choose
Theorem 5.4 gives
and Lemma 6.1 asserts that
7 The proof of Theorem 1.2
We define \(\eta (n)\) through
Then \(\eta (n)\geqslant 0\) and takes integer values, so that
By Lemma 2.1, this implies that
Now
where
To bound \(E_0\), one opens the square and finds that
Hence
Here the terms with \(n=m\) contribute
while the terms with \(n\ne m\) contribute no more than
This shows that
Further, by (7.1), (7.2) and Cauchy’s inequality,
Theorem 1.2 now follows.
References
Brüdern, J.: A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103(1), 27–33 (1988)
Brüdern, J., Vaughan, R.C.: A Montgomery–Hooley theorem for sums of two cubes. European Journal of Mathematics 7(4), 1616–1644 (2021)
Dancs, M.: On a Variance Arising in the Gauss Circle Problem. Ph.D. Thesis, The Pennsylvania State University (2002)
Goldston, D.A., Vaughan, R.C.: On the Montgomery–Hooley asymptotic formula. In: Greaves, G.R.H., et al. (eds.) Sieve Methods, Exponential Sums, and their Applications in Number Theory (Cardiff, 1995). pp. 117–142. London Mathematical Society Lecture Note Series, vol. 237. Cambridge University Press, Cambridge (1996)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, New York (1979)
Hooley, C.: On the Barban–Davenport–Halberstam theorem. I. J. Reine Angew. Math. 274/275, 206–223 (1975)
Hooley, C.: On the Barban–Davenport–Halberstam theorem. IX. Acta Arith. 83(1), 17–30 (1998)
Hooley, C.: On the Barban–Davenport–Halberstam theorem. X. Hardy-Ramanujan J. 21, 9 (1998)
Montgomery, H.L.: Primes in arithmetic progressions. Michigan Math. J. 17, 33–39 (1970)
Parry, T.: An average theorem for tuples of \(k\)-free numbers in arithmetic progressions. Mathematika 67(1), 1–35 (2021)
Pongsriiam, P., Vaughan, R.C.: The divisor function on residue classes I. Acta Arith. 168(4), 369–382 (2015)
Pongsriiam, P., Vaughan, R.C.: The divisor function on residue classes II. Acta Arith. 182(2), 133–181 (2018)
Pongsriiam, P., Vaughan, R.C.: The divisor function on residue classes III. Hardy–Ramanujan J. 56(2), 697–719 (2021)
Vaughan, R.C.: The Hardy–Littlewood Method. 2nd edn. Cambridge Tracts in Mathematics, vol. 125. Cambridge University Press, Cambridge (1997)
Vaughan, R.C.: On a variance associated with the distribution of general sequences in arithmetic progressions. I. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356(1738), 781–791 (1998)
Vaughan, R.C.: On a variance associated with the distribution of general sequences in arithmetic progressions. II. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356(1738), 793–809 (1998)
Vaughan, R.C.: A variance for \(k\)-free numbers in arithmetic progressions. Proc. London Math. Soc. (3) 91(3), 573–597 (2005)
Acknowledgements
The authors express their gratitude to an anonymous referee for a detailed scrutiny of our manuscript, and for a number of very useful comments.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
JB supported by grant BR3048/2 supplied by Deutsche Forschungsgemeinschaft. RCV supported in part by Simons Foundation Grant OSP1857531.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Brüdern, J., Vaughan, R.C. Sums of two unlike powers in arithmetic progressions. European Journal of Mathematics 8 (Suppl 1), 182–213 (2022). https://doi.org/10.1007/s40879-022-00560-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-022-00560-6