In the last few years a number of authors have investigated nonlinear problems in additive prime number theory for short intervals

LXXXV.2 (1998)

On the sum of a prime and the kth power of a prime

by

Claus Bauer (Freiburg)

1. Introduction and statement of results. In the last few years a number of authors have investigated nonlinear problems in additive prime number theory for short intervals. Perelli and Pintz [7] and Mikawa [5] have shown independently that in an interval [x, x + y] with x^7/24+ε ≤ y ≤ x, all but _c y(log x)^−c integers can be represented as the sum of a prime number and a square of a natural number, where c is any positive constant.

A similar result was achieved by Perelli and Zaccagnini [8] for the sum of a prime number and the kth power of a natural number for a fixed integer k ≥ 2. Zhan and Liu [13] have proved the following result: Define

E_k(x) = |{n : n ≤ x, 2 | n, n 6≡ 1 (mod p) ∀p > 2 with p − 1 | k, n 6= p₁+ p^k₂ for all prime numbers p1, p2}|.

Then

E₂(x + y) − E₂(x)  y(log x)^−A

for x^7/16+ε≤ y ≤ x. We are going to generalize this result for all k ≥ 2 by proving the following theorem:

Theorem 1. For any k ≥ 2, any A > 0 and any ε > 0, E_k(x + y) − E_k(x)  y(log x)^−A

for x^{127(1−2k1)+ε} ≤ y ≤ x, where the -constant depends at most on k, A and ε.

Applying a standard argument we will derive this estimate from the following theorem. Let Λ(n), µ(n) and φ(n) denote the von Mangoldt, the

1991 Mathematics Subject Classification: 11F32, 11F25.

This article forms a part of the author’s doctoral dissertation supervised by Professor Dr. D. Wolke from the Department of Mathematics at the University of Freiburg, Ger- many. During the preparation of this article the author was holding a common scholarship by the Chinese State Education Commission and the German Academic Exchange Service (DAAD).

[99]

M¨obius and the Euler function respectively and write e(α) = e^2πiα. For any fixed integer k and any integer d ∈ {1, k} define

Xq _∗ m=1

= Xq (m,q)=1m=1

, C_d(q, a) = Xq _∗ m=1

e

m^da q

 ,

A(q, n) = Xq _∗

a=1

C1(q, a)Ck(q, a)e

−an q



, σ(n, R) = X

q≤R

A(q, n) φ²(q) ,

R(n) = X

n=m₁+m^k₂ x−y<m₁≤x y/2^k<m^k₂≤(2^k+1)y/2^k

Λ(m1)Λ(m2), P (n) = X

n=m₁+m^k₂ x−y<m₁≤x y/2^k<m^k₂≤(2^k+1)y/2^k

1.

We are going to show

Theorem 2. For any fixed k ≥ 2, any A > 0 and any ε > 0, X

x<n≤x+H

|R(n) − σ(n, P )P (n)|² Hy^2/k(log x)^−A

for P = (log x)^B1, where B₁ = B₁(A) is a sufficiently large constant, x^7/12+ε ≤ y ≤ x and y^1−1/2k+ε ≤ H ≤ y. The -constant depends at most on k, A and ε.

Our results are weaker than Perelli and Zaccagnini’s analogous results in [8], who in our notation can choose H in Theorem 2 as small as max(y^1−1/k+ε, x^1/2+ε) and therefore obtain an estimate for the correspond- ing exceptional set for y as small as max(x^{127(1−1k)+ε}, x^1/2+ε). This is due to the fact that we need a mean value estimate for nonlinear trigonometric sums over primes and not just over natural numbers as given by Perelli and Zaccagnini. We can only establish this estimate for a range of H longer than the one in [8].

2. Notation and structure of the proof. Furthermore, we will use the following notation:

D₁(α) = X

x−y<m≤x

Λ(m)e(mα), D_k(α) = X

y/2^k<m^k₂≤(2^k+1)y/2^k

Λ(m)e(m^kα),

I1(α) = X

x−y<m≤x

e(mα), Ik(α) = X

y/2^k<m^k₂≤(2^k+1)y/2^k

e(m^kα), m ∼ M ⇔ M ≤ m < 2M.

c and ε denote positive constants which depend at most on k and can take different values on different occasions. By kxk we denote the distance from

x to the nearest integer. We set

L = log x, Q = HL^−B2, P = L^B1,

where B₁ and B₂ will be determined in the sequel. Without further refer- ences we shall make use of the relations log x  log y  log H. The major arcs M and the minor arcs m are defined by

M = [

q≤P

[q (a,q)=1a=1

a q − 1

Q,a q + 1

Q



, m =



− 1

Q, 1 − 1 Q



\ M.

Thus we arrive at

(2.1) X

x<n≤x+H

|R(n) − σ(n, P )P (n)|²

= X

x<n≤x+H

1−1/Q\

−1/Q

D₁(α)D_k(α)e(−nα) dα − σ(n, P )P (n)  ²

 X

x<n≤x+H

  \

M

D₁(α)D_k(α)e(−nα) dα − σ(n, P )P (n)  ²

+ X

x<n≤x+H

  \

m

D₁(α)D_k(α)e(−nα) dα  ²

=: X

M

+X

m

.

3. The minor arcs. In order to estimate the contribution of the integral over the minor arcs, we shall establish Lemma 3.3 below. For this purpose we will first give some results and definitions from [4]. For any positive integers x, y and r with 1 ≤ r ≤ x, x^ε≤ y ≤ x and any real number α = a/q + θ/q², (a, q) = 1, |θ| ≤ 1, we have:

(3.1) X

x<n≤x+y

τ^c(n)τ^c(n + r)  y(log x)^c,

(3.2) X

n≤y

τ^c(n) min

 x, 1

knαk



 (xyq^−1/2+ xy^1/2+ x^1/2y + (xyq)^1/2)(log xyq)^c (see (3.3) and (3.4) of [4]).

For any arithmetic function g(n) we define

∇(g(n); v1) = g(n)g(n + v1),

∇(g(n); v₁, . . . , v_j) = ∇(∇(g(n); v₁, . . . , v_j−1); v_j).

Thus

(3.3) ∇((g₁g₂)(n); v₁, . . . , v_j) = ∇(g₁(n); v₁, . . . , v_j)∇(g₂(n); v₁, . . . , v_j) and for g(n)  G(n),

(3.4) ∇(g(n); v₁, . . . , v_j)  ∇(G(n); v₁, . . . , v_j).

For a polynomial f (n) with real coefficients we set

∆(f (n); v₁) = f (n + v₁) − f (n),

∆(f (n); v₁, . . . , v_j) = ∆(∆(f (n); v₁, . . . , v_j−1); v_j).

For f (n) = βn^k and two polynomials f₁(n) and f₂(n) we thus obtain

(3.5)

∆(f (n); v₁, . . . , v_k−1)

= βk!v₁. . . v_k−1n + βk!

2

X

a1+...+a_k−1=k ai≥1

v^a₁¹. . . v_k−1^ak−1,

∆(f (n); v₁, . . . , v_k) = βk!v₁. . . v_k,

∆((f1+ f2)(n); v1, . . . , vk−1)

= ∆(f₁(n); v₁, . . . , v_k−1) + ∆(f₂(n); v₁, . . . , v_k−1).

For positive numbers x and y, an arithmetic function g(n) which only takes positive values and a polynomial f (n) with real coefficients we fur- thermore define

S = X

x<n≤x+y

g(n)e(f (n)).

Thus for each integer j ≥ 1 we have

|S|^2j  y^2j−j−1 (3.6)

×X

v₁

. . .X

v_j

X

n

∇(g(n); v₁, . . . , v_j)e(∆(f (n); v₁, . . . , v_j)), where the v_irun over all integers and for any fixed v₁, . . . , v_j the summation over n is restricted by the inequalities

(3.7) x < n + σ(j) ≤ x + y, where σ(j) runs over the set

(3.8) Σ(j) =n X

z∈Z

z : Z is any subset of {v₁, . . . , v_j} o

. Finally,

(3.9) X

v1y

. . . X

vjy

X

n∼N

∇(τ^c(n)τ^c(n + r); v₁, . . . , v_j)  y^jN (log y)^c for N^ε  y  N and r  N .

The above statements can all be found in [4], (3.5)–(3.10), Lemmas 3.1 and 3.2 or they follow straight from the definitions.

In the next three lemmas we use L to denote log y (and not log x as before).

Lemma 3.1. Let am and bm for m ≥ 0 be real numbers satisfying am τ^c(m) and b_m τ^c(m). Then for every fixed number k ≥ 2 and any A > 0 there exists a B3= B3(A) > 0 such that for B ≥ B3 the estimate

(3.10)

2y\

y

 X

t<m^kn^k≤t+H, m∼M

a_mb_ne(m^kn^kα) 

²dt  H²y^2/k−1L^−A holds for α = a/q + θ/q², (a, q) = 1, |θ| ≤ 1, L^B ≤ q ≤ HL^−B, y^1−1/k ≤ H ≤ y, L^B ≤ M ≤ 2Hy^1/k−1L^−B. The -constant depends at most on k and A. The lemma also holds if the summation range of n is shortened.

P r o o f. Let K = 2^k−1 and J1 denote the left-hand side in (3.10). By Cauchy’s inequality and (3.1) we thus see

J₁ ML^c X

m∼M 2y\

y

 X

t<m^kn^k≤t+H

b_ne(m^kn^kα)  ²dt

= ML^c X

m∼M

X

n₁

X

n2, n16=n2

y<m^kn^k₁≤2y+H

b_n1b_n2e(m^k(n^k₁− n^k₂)α)

T\2

T1

1 dt

+ O



ML^cMH X

ny^1/k/M

τ^c(n)

 ,

where T₁ = max(m^kn^k₁− H, m^kn^k₂− H) and T₂ = min(m^kn^k₁, m^kn^k₂). Set n1− n2 = r, n2 = n and g(m, n, r) = H − m^kr(n^k−1+ . . . + (n + r)^k−1).

As TT2

T11 dt = 0, if not m^k|n^k₁ − n^k₂| ≤ H, we can assume that |r|  HM^−kM^k−1y^−(k−1)/k = HM⁻¹y^−(k−1)/k, and also r > 0. By R_l(n) we denote a polynomial in at least the variable n whose degree relative to n is not greater than l. For a sufficiently large B, by using (3.1), H¨older’s inequality, (3.5) and (3.6) we obtain

|J1|^K/2

ML^c X

0<rH/(M y^(k−1)/k)

X

m∼M

X

y^1/k/M ny^1/k/M m^kr(n^k−1+...+(n+r)^k−1)≤H

g(m, n, r)

× b_nb_n+re(m^krkn^k−1α + m^kR_k−2(n)α) 

^K/2+ H^Ky^K(2−k)/2kL^−KA/2

 M^K/2(Hy^(1−k)/kM⁻¹M )^K/2−1

× L^c X

0<rH/(M y^(k−1)/k)

X

m∼M

 X

y^1/k/M ny^1/k/M m^kr(n^k−1+...+(n+r)^k−1)≤H

g(m, n, r)

× bnbn+re(m^krkn^k−1α + m^kRk−2(n)α) 

^K/2+ H^Ky^K(2−k)/2kL^−KA/2

 H^K/2−1M^K/2y(1−k)(K/2−1)/k

y^1/k M

_K/2−k+1 L^c

× X

0<rH/(M y^(k−1)/k)

X

m

X

v1

. . .X

vk−2

X

n

∇(g(m, n, r)bnbn+r; v1, . . . , vk−2)

× e(m^krk!v₁. . . v_k−2nα + m^kR₀(n)α) + O(H^Ky^K(2−k)/2kL^−KA/2), where |v₁|  y^1/kM⁻¹, . . . , |v_k−2|  y^1/kM⁻¹, y^1/kM⁻¹ n + σ(k − 2)  y^1/kM⁻¹, m^kr((n + σ(k − 2))^k−1+ . . . + (r + n + σ(k − 2))^k−1) ≤ H and m ∼ M . Applying H¨older’s inequality again as well as (3.3), (3.4) and (3.9) we find that

(3.11) |J1|^K2/2

 H^K2/2−Ky^K2(2−k)/2kM^K(k−1)

 H

M y^(k−1)/k · y^(k−1)/k M^k−1

_K−1 L^c

× X

0<rH/(M y^(k−1)/k)

X

v1

. . .X

vk−2

X

n

 X

m

∇(g(m, n, r); v1, . . . , vk−2)

× e(m^krk!v₁. . . v_k−2nα + m^kR₀(n)α)  ^K + H^K2y^K2(2−k)/2kL^−K2A/2,

where the summations are as stated before. Applying (3.5) and (3.6) to the inner sum over m we obtain

 X

m

 ^K

 M^K−kX

u₁

. . . X

u_k−1

X

m

∇(∇(g(m, n, r); v₁, . . . , v_k−2); u₁, . . . , u_k−1)

× e



mnr(k!)²v₁. . . v_k−2u₁. . . u_k−1α

+ nr(k!)²

2 v₁. . . v_k−2

 X

a₁+...+a_k−1=k a_i≥1

u^a₁¹+ . . . + u^a_k−1^k−1



α + T (m)α

 ,

where |u₁|  M, . . . , |u_k−1|  M , m+σ^∗(k−1) ∼ M , (m+σ^∗(k−1))^kr((n+

σ(k − 2))^k−1+ . . . + (r + n + σ(k − 2))^k−1) ≤ H and T (m) depends on m, but not on n. Substituting the last estimate in (3.11), using partial summation,

∇(∇(g(m, n, r); v1, . . . , vk−2); u1, . . . , uk−1) ≤ H^K2/2 and P

A<n<Be(nα)  min(B − A, 1/kαk) we find that (3.12) |J₁|^K2/2

 H^K2/2−Ky^K2(2−k)/2kM^K(k−1)

 H M^k

_K−1

M^K−kL^c

× X

0<rH/(M y^(k−1)/k)

X

v1

. . .X

vk−2

X

u1

. . .

. . . X

u_k−1

X

m

X

n

∇(∇(g(m, n, r); v₁, . . . , v_k−2); u₁, . . . , u_k−1)

× e



nr(k!)²

2 v₁. . . v_k−2u₁. . . u_k−1(2m + u₁+ . . . + u_k−1)α

  + H^K2y^K2(2−k)/2kL^−K2A/2

 H^K2−1y^K2(2−k)/2kL^c X

0<rH/(M y^(k−1)/k)

X

v₁

. . .X

v_k−2

X

u₁

. . .

. . . X

uk−1

X

m

min

y^1/k M , 1

r^(k!)₂²v₁. . . v_k−2u₁. . . u_k−1(2m + u₁+ . . . + u_k−1)α



+ H^K2y^K2(2−k)/2k,

where the summations are as stated before. The expression inside k k can only be zero if at least one ui or one vi is 0. (The expression in brackets is equal to m + (m + u₁+ . . . + u_k−1) and so 6= 0 because m + σ(k − 1) ∼ M.) Thus the contribution of these terms to (3.12) is

 y^1/k

M · H

M y^(k−1)/k

y^(k−3)/k

M^k−3 M^k−1+y^(k−2)/k M^k−2 M^k−2

 M (3.13)

 HL^−K2A/2−c.

The number of terms which satisfy 0 6= n = r(k!)²

2 v1. . . vk−2u1. . . uk−1(2m + u1+ . . . + uk−1)

is ≤ τ^2k−2(n), because r, uiand vj respectively divide n and for fixed r, ui

and v_j there is at most one possible choice for m. We can derive from

n  H

M y^(k−1)/k

y^1/k M

_k−2

M^k−1M = HM y^−1/k and (3.2) that these terms do not contribute to (3.12) more than

 X

0<nHM y^−1/k

τ^c(n) min

y^1/k M , 1

knαk



 HL^c(q^−1/2+ M^1/2y^−1/2k+ H^−1/2M^−1/2y^1/2k+ H^−1/2q^1/2)

 HL^−K2A/2−c,

if B is chosen arbitrarily large. Now the lemma follows from the last estimate, (3.12) and (3.13).

Lemma 3.2. Let a_m denote real numbers which satisfy a_m τ^c(m). For any integer k ≥ 2 and any A > 0 there exists a B₄= B₄(A) > 0 such that for B ≥ B4 the estimate

(3.14)

2y\

y

 X

t<m^kn^k≤t+H, m∼M

ame(m^kn^kα) 

²dt  H²y^2/k−1L^−A

holds for α = a/q + θ/q², (a, q) = 1, |θ| ≤ 1, L^B ≤ q ≤ HL^−B, y^1−1/kL^B ≤ H ≤ y and M^2k−2 ≤ y^1/2kL^−B. The -constant depends at most on k and A.

Remark. Under the conditions of Lemma 3.2,

2y\

y

 X

t<m^kn^k≤t+H, m∼M

(log n)ame(m^kn^kα) 

²dt  H²y^2/k−1L^−A. The lemma and the remark also apply if the summation range of n is shortened.

P r o o f (of Lemma 3.2). Let J₂ denote the left-hand side in (3.14). Fol- lowing the same lines as in the proof of Lemma 3.1 we arrive at

J₂=

2y\

y

X

m1∼M

X

m2∼M

X

n₁

X

n2

m₁n₁6=m₂n₂ t<(m_in_i)^k≤t+H

a_m1a_m2

× e(((m₂n₂)^k− (m₁n₁)^k)α) dt + O



H X

ny^1/k

τ^c(n)



= X

m₁∼M

X

m₂∼M

am₁am₂

×X

n₁

X

n2

0<|(m₂n₂)^k−(m₁n₁)^k|≤H y<(mini)^k≤2y+H

(H − |(m₂n₂)^k− (m₁n₁)^k|)

× e((m₂n₂− m₁n₁)((m₁n₁)^k−1+ . . . + (m₂n₂)^k−1)α) + O(Hy^1/kL^c).

Let r = m₂n₂− m₁n₁, δ = (m₁, m₂), m₁ = δm^∗₁, m₂ = δm^∗₂, n = n₁ and without loss of generality assume r > 0. Then (m^∗₁, m^∗₂) = 1 and δ | r.

Writing r = δr^∗ and noting that

r^∗δ = (m^∗₂n₂)^k− (m^∗₁n₁)^k

(m^∗₁n1)^k−1+ . . . + (m^∗₂n2)^k−1δ  Hy^(1−k)/k, we arrive at

J₂= X

δM

X

0<r^∗Hy^1/k−1δ⁻¹

X

m^∗₁∼M δ⁻¹

X

m^∗₂∼M δ⁻¹ (m^∗₁,m^∗₂)=1

a_m^∗_1δa_m^∗_2δ (3.15)

×X

n

(H − kδ^kr^∗(m^∗₁n)^k−1+ Pk−2(m^∗₁n))

× e(kδ^kr^∗(m^∗₁n)^k−1α + P_k−2(m^∗₁n)α) + O(Hy^1/kL^c),

where Pk−2(m^∗₁n) is a polynomial in m^∗₁n, δ and r^∗ with only positive co- efficients, and its degree relative to m^∗₁n is not greater than k − 2. The summation over n is given by

m^∗₁δn ≡ −r^∗δ (mod m^∗₂δ), 0 < kδ^kr^∗(m^∗₁n)^k−1+ P_k−2(m^∗₁n) ≤ H, y < (m^∗₁δn + r^∗δ)^k≤ 2y + H, y < (m^∗₁δn)^k ≤ 2y + H.

Using (3.1) we see that the terms with δ > L^D do not contribute more thanX

L^D<δM

 H X

L^D<δ

τ^c(δ) X

0<r^∗Hy^1/k−1δ⁻¹

X

m^∗₁∼M δ⁻¹

X

m^∗₂∼M δ⁻¹

τ^c(m^∗₁)τ^c(m^∗₂)

× X

(m^∗₁δn)^ky

X

n₂ r^∗=m^∗₂n2−m^∗₁n

1

 H X

L^D<δ

τ^c(δ) X

0<r^∗Hy^1/k−1δ⁻¹

X

ny^1/kδ⁻¹

τ^c(n)τ^c(n + r^∗)

 H²y^2/k−1 X

L^D<δ

τ^c(δ)

δ²  H²y^2/k−1L^−A,

if D is sufficiently large. So we can concentrate on the case δ < L^D. Without loss of generality we assume δ = 1 since in the other cases the proof does not change fundamentally. As a consequence we suppose the m₁ and m₂ to be relatively prime and write n = T + vm2with v ≥ 0, 0 ≤ T  M , T ≡ −m1r (mod m₂) and m₁m₁≡ 1 (mod m₂). Then one can see that it is enough to estimate the following expression which we denote by J₂ again:

(3.16) J2= X

0<rHy^1/k−1

X

m1∼M

X

m2∼M

am₁am₂

×X

v

(H − kr(m1m2v)^k−1+ Pk−2(m1m2v))

× e(kr(m1m2v)^k−1α + Pk−2(m1m2v)α) + O(H²y^2/k−1L^−A), where v runs over

0 < kr(m₁m₂v + m₁T )^k−1+ P_k−2(m₁m₂v + m₁T ) ≤ H, y  (m1m2v + m1T + r)^k y, y  (m1m2v + m1T )^k  y.

So the maximal range of summation over v is given by (3.17) 0 < r(m1m2v)^k−1 ≤ H, y  (m1m2v)^k y.

In the sequel we still assume the m_i and u_j to be pairwise coprime. By induction we will show that for 1 ≤ j ≤ k − 1, J = 2^j−1 and a sufficiently large B the following holds:

|J₂|^J  H^J−1y^1−J(y^1/k)^2J−j−1L^c X

0<r1Hy^1/k−1

. . . (3.18)

. . . X

0<r_jHy^j/k−1/(r₁...r_j−1)

X

m1∼M

. . . X

m2J∼M

a_m1. . . a_m2J

×X

n

g(r1, . . . , rj, m1, . . . , m2J, m1. . . m2Jn)

× e(k . . . (k − j + 1)r₁. . . r_j(m₁. . . m_2Jn)^k−jα + P_k−j−1(m₁. . . m_2Jn)α) + O(H^2Jy^−Jy^2J/kL^−JA), where the maximal range of summation over n is given by

(3.19) 0 ≤ r₁. . . r_j(m₁. . . m_2Jn)^k−j  H, y  (m₁. . . m_2Jn)^k  y, and g(r1, . . . , rj, m1, . . . , m2J, m1. . . m2Jn)  H^J is a polynomial in the given variables.

For j = 1, (3.18) follows from (3.16) and (3.17). Suppose that (3.18) holds for a j with 1 ≤ j ≤ k − 2. By using Cauchy’s inequality we get

(3.20) |J2|^2J

 H^2J−2y^2−2J(y^1/k)^4J−2j−2L^cHy^j/k−1

× X

0<r₁Hy^1/k−1

. . . X

0<r_jHy^j/k−1/(r₁...r_j−1)

X

m1∼M

. . .

. . . X

m_2J∼M

X

u₁∼M

. . . X

u_2J∼M

am₁. . . am_2Jau₁. . . au_2J

×X

n

X

u

g(r₁, . . . , r_j, m₁, . . . , m_2J, m₁. . . m_2Jn)

× g(r₁, . . . , r_j, u₁, . . . , u_2J, u₁. . . u_2Ju)e(k . . . (k − j + 1)r₁. . . r_j

× ((u₁. . . u_2Ju)^k−jα−(m₁. . . m_2Jn)^k−jα)+P_k−j−1(u₁. . . u_2Ju)α

− P_k−j−1(m₁. . . m_2Jn)α) + O(H^4Jy^−2J(y^1/k)^4JL^−2JA),

where the summations over n and u are both given by (3.19). Setting r_j+1= u₁. . . u_2Ju − m₁. . . m_2Jn, we obtain

(3.21) (u₁. . . u_2Ju)^k−j− (m₁. . . m_2Jn)^k−j

= rj+1(k − j)(m1. . . m2Jn)^k−j−1+ Pk−j−2(m1. . . m2Jn), where Pk−j−2 is a polynomial at least in m1. . . m2Jn and with degree ≤ k − j − 2 with respect to this variable. By employing the definition of r_j+1 and (3.19) we also have

r_j+1= (u₁. . . u_2Ju)^k−j− (m₁. . . m_2Jn)^k−j (m₁. . . m_2Jn)^k−j−1+ . . . + (u₁. . . u_2Ju)^k−j−1 (3.22)

 H

r1. . . rj

y^(j+1)/k−1.

We shall assume without loss of generality that rj+1 ≥ 0. Keeping in mind that the m₁. . . m_2J and u₁. . . u_2J were supposed to be coprime we write (3.23) n = S + gu₁. . . u_2J,

where m₁. . . m_2JS ≡ −r_j+1 (mod u₁. . . u_2J), 0 ≤ S  M^2J and g ≥ 0.

From (3.19) and (m₁. . . m_2JS)^k  M^4×2k−3k  yL^−2Bk we can derive (3.24) y  (m₁. . . m_2Ju₁. . . u_2Jg)^k  y.

From (3.19) and (3.23) we further conclude that

(3.25) 0 ≤ r₁. . . r_j+1(m₁. . . m_2Ju₁. . . u_2Jg)^k−j−1  H.

Taking into account (3.23) we can write

g(r1, . . . , rj, m1, . . . , m2J, m1. . . m2Jn)

× g(r₁, . . . , r_j, u₁, . . . , u_2J, m₁. . . m_2Jn + r_j+1) := g(r1, . . . , rj+1, m1, . . . , m2J, u1, . . . , u2J, m1. . . m2Ju1. . . u2Jg), and so from (3.20) to (3.25) we obtain

|J2|^2J H^2J−1y^1−2J(y^1/k)^4J−j−2L^c (3.26)

× X

0<r1Hy^1/k−1

. . . X

0≤rj+1Hy^(j+1)/k−1/(r1...rj−1rj)

× X

m₁∼M

. . . X

m_2J∼M

X

u₁∼M

. . . X

u_2J∼M

a_m1. . . a_m2Ja_u1. . . a_u2J

×X

g

g(r₁, . . . , r_j+1, m₁, . . . , m_2J,

u₁, . . . , u_2J, m₁. . . m_2Ju₁. . . u_2Jg)

× e(k . . . (k − j)r1. . . rjrj+1(m1. . . m2Ju1. . . u2Jg)^k−j−1 + P_k−j−2(m₁. . . m_2Ju₁. . . u_2Jg)α)

+ O(H^4Jy^−2J(y^1/k)^4JL^−2JA),

where the summation ranges are given by (3.24) and (3.25). Using (3.1) and (3.19) it follows that in (3.20) and therefore also in (3.26) the contribution of the terms with rj+1= 0 is

 H^4J−1y^1−2J(y^1/k)^4J−j−2L^c

× X

0<r1Hy^1/k−1

. . . X

0<rjHy^j/k−1/(r1...rj−1)

 H

r₁. . . r_j

_1/(k−j)

 H^4Jy^−2J(y^1/k)^4J−1L^c

× X

0<r1Hy^1/k−1

. . . X

0<rj−1Hy^{(j−1)/k−1}/(r1...rj−2)

1 r1. . . rj−1

 H^4Jy^−2J(y^1/k)^4JL^cy^−1/k.

Now (3.18) follows from (3.26) in the case of j+1 if we rename the u₁, . . . , u_2J as m2J+1, . . . , m4J. Choosing j = k − 1 in (3.18) and setting K = 2^k−2, we derive the following result from (3.19) and by using partial summation:

|J2|^K  H^2K−1y^−K(y^1/k)^2KL^c (3.27)

× X

0<r1Hy^1/k

. . . X

0<rk−1Hy^−1/k/(r1...rk−2)

X

m₁∼M

. . . . . . X

m2K∼M

τ^c(m₁. . . m_2K)

× min

 H

r₁. . . r_k−1m₁. . . m_2K, 1

kk!r₁. . . r_k−1m₁. . . m_2Kαk



+ H^2Ky^−K(y^1/k)^2KL^−KA

 H^2K−1y^−K(y^1/k)^2KL^c

× max

N HM^2Ky^−1/kL^c

X

n∼N

τ^c(n) min

H N, 1

knαk



+ H^2Ky^−K(y^1/k)^2KL^−KA,

because r₁. . . r_k−1m₁. . . m_2K HM^2Ky^−1/kL^c. For N ≥ L^D1 we find (3.28) X

n∼N

τ^c(n) min

H N, 1

knαk



 (Hq^−1/2+ (HN )^1/2+ HN^−1/2+ (Hq)^1/2)L^c

 (Hq^−1/2+ HM^Ky^−1/2k+ HL^−D1/2+ (Hq)^1/2)L^c

 HL^−KA−c

by applying (3.2) for sufficiently large B and D₁. For N ≤ L^D1 and D₁ fixed according to the preceding discussion, we obtain the following for a sufficiently large B:

X

n∼N

τ^c(n) min

H N, 1

knαk



 L X

nL^D1

1 (3.29) knαk

 Lq X

n≤L^D1

1  HL^−KA−c.

The lemma now follows from (3.27)–(3.29).

From Lemmas 3.1 and 3.2 we derive

Lemma 3.3. Let α = a/q + θ/q², (a, q) = 1 and |θ| ≤ 1. For every fixed k ≥ 2 and every A > 0 there exists a B₅= B₅(A) > 0 such that for B ≥ B₅,

2y\

y

 X

t<m^k≤t+H

Λ(m)e(m^kα) 

²dt  H²y^2/k−1L^−A

for L^B ≤ q ≤ HL^−B and y^1−1/2kL^B ≤ H ≤ y, where the -constant depends at most on k and A.

P r o o f. Set

M (s) = X

n≤X

µ(n)n^−s, X = 2y^1/2k, Re(s) > 1.