Proceedings of the Steklov Institute of Mathematics, Vol. 218, 1997, pp. 414–432. Translated from Trudy Matematicheskogo Instituta imeni V. A. Steklova, Vol. 218, 1997, pp. 415–432. Original Russian Text Copyright \begin{matrix}c \\ ◯ \end{matrix} 1997 by Tolev. English Translation Copyright \begin{matrix}c \\ ◯ \end{matrix} by ���� �����/Interperiodica Publishing (Russia).

On the Number of Representations of an Odd Integer as a Sum of Three Primes, One of Which Belongs

to an Arithmetic Progression

D. I. Tolev

Received February, 1997

1. INTRODUCTION AND STATEMENT OF THE RESULT

In 1937 I. M. Vinogradov [8] proved that for every sufficiently large integer N the equation

p_{1} + p_{2} + p_{3} = N  (1)

had solutions in prime numbers p_{1}, p_{2}, p_{3}. After the publication of Vinogradov’s remarkable work many mathematicians investigated the solvability of equation (1) in primes satisfying some additional conditions. Zulauf [9, 10] considered (1) with prime numbers lying in arithmetic progressions. Suppose that k, l are integers, (k, l) = 1, and denote

J_{k,l}(N) = ∑_{\begin{matrix}p_{1}+p_{2}+p_{3}=N \\ p_{1}≡l;(mod k) \end{matrix}} log p_{1} log p_{2} log p_{3}.  (2)

If k ≤ (log N)^{D} for some fixed D > 0, then the following asymptotic formula holds as N tends to infinity:

J_{k,l}(N) = \frac{N^{2}}{2φ(k)}𝔖_{k,l}(N) + O(\frac{N^{2}}{(log N)^{A}} ).

Here A > 0 is arbitrarily large,

𝔖_{k,l}(N) = ∏_{p∤kN}(1 + \frac{1}{(p − 1)^{3}} ) ∏_{\begin{matrix}p∤k \\ p|N \end{matrix}}(1 − \frac{1}{(p − 1)^{2}} ) ∏_{\begin{matrix}p|k \\ p∤N−l \end{matrix}}(1 − \frac{1}{(p − 1)^{2}} ) ∏_{\begin{matrix}p|k \\ p|N−l \end{matrix}} \frac{p}{p − 1},  (3)

and the constant in the O-term depends only on D and A.

In 1939 Van der Corput [5] used the method of I. M. Vinogradov and proved that there existed infinitely many arithmetic progressions consisting of three different primes. This result has not been proved yet for progressions of four or more terms. In 1981, however, Heath-Brown [3] proved that there existed infinitely many arithmetic progressions of four different terms, three of which are primes and the fourth is an integer with no more than two prime factors (counted according to multiplicity).

The main point in [3] is a result of Bombieri–Vinogradov type for the quantity

w(n) = ∑_{\begin{matrix}x<p_{2}, p_{3}≤2x \\ p_{1}+p_{3}=2p_{2} \\ p_{2}+n=2p_{3} \end{matrix}} u(p_{1})u(p_{2})u(p_{3}),

where u(n) = log n/ log 3x for n ≥ 5 and u(n) = 0 otherwise. Heath-Brown proved that for each ε > 0

∑_{\begin{matrix}d≤x^{1/2−ε} \\ (d,6)=1 \end{matrix}} μ^{2}(d)| ∑_{n:d|n} w(n) − X \frac{ω(d)}{d}| ≪ \frac{X}{(log x)^{4/3}} ,

where

X = 2 ∏_{p>2}(1 − \frac{1}{(p − 1)^{2}} ) × \frac{x^{2}}{(log 3x)^{3}}  and  ω(d) = ∏_{p|d} \frac{p}{p − 2}.

It is interesting to find a similar result for the quantity J_{k,l}(N). Let us denote

E = E(N, θ) = ∑_{k≤N^{θ}} max_{(l,k)=1}|J_{k,l}(N) − \frac{N^{2}}{2φ(k)}𝔖_{k,l}(N)|.  (4)

It is natural to expect that using the method of Heath-Brown one can prove that

E ≪ \frac{N^{2}}{(log N)^{A}}  (5)

for each θ < 1/2 and for an arbitrarily large A. However, at present we can verify (5) only for θ < 1/3. We shall prove the following

Theorem. Suppose that J_{k,l}(N), 𝔖_{k,l}(N), and E are defined by (2), (3) and (4), respectively. Then, for an arbitrarily large A and for each θ < 1/3 inequality (5) holds.

Using this theorem and some results from the theory of sieves, one can prove that for each sufficiently large odd N equation (1) has a solution in primes such that p_{1} + 2 has at most H prime factors, where H is a fixed integer. In order to prove this with smaller H, we need estimate (5) for θ as large as possible. At the end of the paper we present an approach that can be used to improve the upper bound of θ for which (5) holds true.

2. NOTATION AND THE OUTLINE OF THE PROOF

We denote real numbers by x, y, r, u, α, θ, prime numbers by p, p_{1}, p_{2}, p_{3}, and integers by m, n, d, t, a, q, k, l, s, h, f. We always have (k, l) = 1. As usual, μ(n), φ(n), τ(n) denote Möbius’ function, Euler’s function, and the number of positive divisors of n. We denote by (m, n) and [m, n] the greatest common divisor and the least common multiple of m and n respectively. For real x, y, however, (x, y) and [x, y] denote the open interval (closed interval) on the real line with endpoints x and y. The meaning is always clear from the context. Instead of m ≡ n; (mod k), we shall write, for simplicity, m ≡ n (k). We shall also use the notation e(x) = exp(2πix) and ∥x∥ = min_{n∈ℤ} |x − n|. The letter c denotes some positive real number, not the same in all appearances. This convention allows us to write

(log x)e^{−c\sqrt{log x}} ≪ e^{−c\sqrt{log x}},

for example.

Let N be a sufficiently large odd integer. Let A > 100 be a constant and B be a constant, sufficiently large in terms of A, which we shall define later by (32) and (64). Let us denote Q = (log N)^{B}, τ = NQ^{−1},

E_{1} = ⋃_{q≤Q} ⋃_{\begin{matrix}a=0 \\ (a,q)=1 \end{matrix}}^{q−1}(\frac{a}{q} − \frac{1}{qτ} , \frac{a}{q} + \frac{1}{qτ} ), E_{2} = (−\frac{1}{τ} , 1 − \frac{1}{τ} )\E_{1},

S_{k,l}(α) = ∑_{\begin{matrix}p≤N \\ p≡l (k) \end{matrix}}(log p)e(αp),  S(α) = S_{1,0}(α),  M(α) = ∑_{m=1}^{N} e(αm).

We have

J_{k,l}(N) = ∫_{−1/τ}^{1−1/τ}S_{k,l}(α)S^{2}(α)e(−Nα)dα = J_{k,l}^{(1)}(N) + J_{k,l}^{(2)}(N),

where

J_{k,l}^{(i)}(N) = ∫_{E_{i}} S_{k,l}(α)S^{2}(α)e(−Nα)dα, i = 1, 2.

Consequently

E ≤ E_{1} + E_{2},  (6)

where

E_{1} = E_{1}(N, θ) = ∑_{k≤N^{θ}} max_{(l,k)=1}|J_{k,l}^{(1)}(N) − \frac{N^{2}}{2φ(k)}𝔖_{k,l}(N)|,  (7)
E_{2} = E_{2}(N, θ) = ∑_{k≤N^{θ}} max_{(l,k)=1}|J_{k,l}^{(2)}(N)|.  (8)

In Section 3, we use the Hardy–Littlewood circle method and the theorem of Bombieri and Vinogradov (see [1] and [7]) to prove that

E_{1} ≪ \frac{N^{2}}{(log N)^{A}}  for each θ < \frac{1}{2}.  (9)

In Section 4, we estimate E_{2} using the method of Heath-Brown from [3]. Two conjectures are stated in which the real parameter u appears. It is proved that the validity of some of the conjectures for a certain u > 2 implies the estimate

E_{2} ≪ \frac{N^{2}}{(log N)^{A}}  for each θ < \frac{1}{u}.  (10)

Hence, by (6), (9), (10) we get (5) for each θ < 1/u. It is also shown that Lemma 3 of [3] immediately implies the validity of Conjecture 1 for u = 4, and therefore we have (5) provided that θ < 1/4.

In Section 5 we prove Conjecture 2 for u = 3 which implies the proof of the Theorem. An approach used to prove some of the conjectures for u > 2 and u which is close to 2 is also presented.

3. THE ESTIMATE OF E_{1}

It is clear that

J_{k,l}^{(1)}(N) = ∑_{q≤Q} ∑_{\begin{matrix}a=0 \\ (a,q)=1 \end{matrix}}^{q−1} I(a, q),  (11)

where

I(a, q) = ∫_{−1/(qτ)}^{1/(qτ)} S_{k,l}(\frac{a}{q} + α)S^{2}(\frac{a}{q} + α)e(−N(\frac{a}{q} + α))dα.  (12)

If

q ≤ Q,  (a, q) = 1,  |α| ≤ \frac{1}{qτ}  (13)

then we have

S(\frac{a}{q} + α) = \frac{μ(q)}{φ(q)}M(α) + O(Ne^{−c\sqrt{log N}} )  (14)

(see [6, Lemma 3.1] for the proof).

Let us consider S_{k,l}(a/q + α) for a, q, α satisfying (13). We cannot find an asymptotic formula for this sum for a particularly large k (unless we use some hypotheses which have not been proved so far). We shall find, however, an asymptotic formula with an error term which is small on the average.

We have

S_{k,l}(\frac{a}{q} + α) = ∑_{\begin{matrix}1≤m≤q \\ (m,q)=1 \\ m≡l ((k,q)) \end{matrix}} e(\frac{am}{q})T (α) + O(q log N),  (15)

where

T (α) = ∑_{\begin{matrix}p≤N \\ p≡l (k) \\ p≡m (q) \end{matrix}}(log p)e(αp).

Using the elementary theory of congruences, one can easily prove that if the integers k, l, m, q satisfy (k, l) = (m, q) = 1 and m ≡ l ((k, q)), then there exists an integer f = f(k, l, m, q) such that (f, [k, q]) = 1 and such that for any integer n the congruence n ≡ f ([k, q]) is equivalent to the system n ≡ l (k), n ≡ m (q). Hence, we have

T (α) = ∑_{\begin{matrix}p≤N \\ p≡f ([k,q]) \end{matrix}}(log p)e(αp).

Let us denote

Δ(x, h) = max_{y≤x} max_{(l,h)=1}| ∑_{\begin{matrix}p≤y \\ p≡l (h) \end{matrix}} log p − \frac{y}{φ(h)}|.  (16)

We use Abel’s formula and obtain

T (α) = − ∫_{0}^{N}( ∑_{\begin{matrix}p≤y \\ p≡f ([k,q]) \end{matrix}} log p)\frac{d}{dy} e(αy) dy + ( ∑_{\begin{matrix}p≤N \\ p≡f ([k,q]) \end{matrix}} log p)e(αN)

= − ∫_{0}^{N}(\frac{y}{φ([k, q])} + O(Δ(N, [k, q])))\frac{d}{dy} e(αy) dy + (\frac{N}{φ([k, q])} + O(Δ(N, [k, q])))e(αN)

= \frac{1}{φ([k, q])}(− ∫_{0}^{N} y(\frac{d}{dy} e(αy)) dy + Ne(αN)) + O((1 + |α|N)Δ(N, [k, q])).

We integrate by parts, and then use (13) and the well-known formula

∫_{0}^{N} e(αy) dy = M(α) + O(1).

As a result we get

T (α) = \frac{M(α)}{φ([k, q])} + O((1 + \frac{N}{qτ} )Δ(N, [k, q])).

We substitute the last expression for T (α) into (15) and find out that under condition (13) we have

S_{k,l}(\frac{a}{q} + α) = \frac{c_{k,l}(a, q)}{φ([k, q])}M(α) + O(Q(log N)Δ(N, [k, q])),  (17)

where

c_{k,l}(a, q) = ∑_{\begin{matrix}1≤m≤q \\ (m,q)=1 \\ m≡l ((k,q)) \end{matrix}} e(\frac{am}{q}).  (18)

Formulas (14), (17) and the trivial estimate

|S_{k,l}(\frac{a}{q} + α)| ≪ \frac{N log N}{k}

imply that under condition (13) we have

S_{k,l}(\frac{a}{q} + α)S^{2}(\frac{a}{q} + α)e(−N(\frac{a}{q} + α)) = \frac{μ^{2}(q)}{φ^{2}(q)} \frac{c_{k,l}(a, q)}{φ([k, q])}e(−N \frac{a}{q})M^{3}(α)e(−Nα)

+ O(\frac{N^{3}}{k} e^{−c\sqrt{log N}}) + O(\frac{QN^{2} log N}{φ^{2}(q)} Δ(N, [k, q])).

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

For the integral I(a, q) defined by (12) we find

I(a, q) = \frac{μ^{2}(q)}{φ^{2}(q)} \frac{c_{k,l}(a, q)}{φ([k, q])}e(−N \frac{a}{q}) ∫_{−1/(qτ)}^{1/(qτ)} M^{3}(α)e(−Nα) dα

+ O(\frac{N^{2}}{k} e^{−c\sqrt{log N}}) + O(\frac{Q^{2}N log N}{qφ^{2}(q)} Δ(N, [k, q])).

Hence,

∑_{\begin{matrix}a=0 \\ (a,q)=1 \end{matrix}}^{q−1} I(a, q) = \frac{μ^{2}(q)}{φ^{2}(q)} \frac{b_{k,l}(q)}{φ([k, q])} ∫_{−1/(qτ)}^{1/(qτ)} M^{3}(α)e(−Nα) dα

+ O(\frac{N^{2}}{k} e^{−c\sqrt{log N}}) + O(\frac{Q^{2}N log N}{qφ(q)} Δ(N, [k, q])),  (19)

where

b_{k,l}(q) = ∑_{\begin{matrix}a=0 \\ (a,q)=1 \end{matrix}}^{q−1} c_{k,l}(a, q)e(−N \frac{a}{q})

and where c_{k,l}(a, q) is defined by (18). It is easy to see that b_{k,l}(q) is a multiplicative function of q and that for the prime p we have

b_{k,l}(p) = \{ \begin{matrix}1 & for p ∤ kN \\ 1 − p & for p ∤ k, p | N \\ −1 & for p | k, p ∤ N − l \\ p − 1 & for p | k, p | N − l. \end{matrix}  (20)

Consequently,

μ^{2}(q)|b_{k,l}(q)| ≤ φ(q).  (21)

We also have

∫_{−1/(qτ)}^{1/(qτ)} M^{3}(α)e(−Nα) dα = \frac{1}{2}N^{2} + O(q^{2}τ^{2})  (22)

(see the proof of Theorem 3.3 in [6]). Using (19), (21), (22), we get

∑_{\begin{matrix}a=0 \\ (a,q)=1 \end{matrix}}^{q−1} I(a, q) = \frac{N^{2}}{2φ(k)}λ_{k,l}(q)+O(\frac{q^{2}τ^{2}}{φ(q)φ([k, q])})+O(\frac{N^{2}}{k} e^{−c\sqrt{log N}})+O(\frac{Q^{2}N log N}{qφ(q)} Δ(N, [k, q])),

(23)

where

λ_{k,l}(q) = \frac{μ^{2}(q)b_{k,l}(q)φ(k)}{φ^{2}(q)φ([k, q])}.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

420 TOLEV

It is easy to see that

\frac{φ(k)}{φ([k, q])} = \frac{φ((k, q))}{φ(q)},  (24)

which, as a function of q, is multiplicative. Hence, λ_{k,l}(q) is multiplicative with respect to q too. We use Euler’s identity (see [2, Theorem 286]) and also (3), (20), (21), (24) and get

∑_{q≤Q} λ_{k,l}(q) = ∑_{q=1}^{∞} λ_{k,l}(q) + O(∑_{q>Q} |λ_{k,l}(q)|) = ∏_{p}(1 + λ_{k,l}(p)) + O(∑_{q>Q} \frac{φ((k, q))}{φ^{2}(q)})
= 𝔖_{k,l}(N) + O(∑_{q>Q} \frac{φ((k, q))}{φ^{2}(q)}).  (25)

From (11), (23)–(25) we find

J_{k,l}^{(1)}(N) = \frac{N^{2}}{2φ(k)}𝔖_{k,l}(N) + O(\frac{N^{2}}{φ(k)} ∑_{q>Q} \frac{φ((k, q))}{φ^{2}(q)}) + O(∑_{q≤Q} \frac{q^{2}τ^{2}φ((k, q))}{φ^{2}(q)φ(k)})

+ O(\frac{N^{2}}{k} e^{−c\sqrt{log N}}) + O(∑_{q≤Q} \frac{Q^{2}N log N}{qφ(q)} Δ(N, [k, q])).

We substitute the last expression for J_{k,l}^{(1)}(N) into (7) and use the well-known estimate φ(n) ≫ n(log log n)^{−1} to obtain

E_{1} ≪ (N^{2}Σ_{1} + τ^{2}Σ_{2} + Q^{2}NΣ_{3}) log^{2} N + N^{2}e^{−c\sqrt{log N}},  (26)

where

Σ_{1} = ∑_{k≤N^{θ}} ∑_{q>Q} \frac{(k, q) log q}{kq^{2}} ,  Σ_{2} = ∑_{k≤N^{θ}} ∑_{q≤Q} \frac{(k, q)}{k},  Σ_{3} = ∑_{k≤N^{θ}} ∑_{q≤Q} \frac{1}{q^{2}} Δ(N, [k, q]).

Let us estimate Σ_{1}. We have

Σ_{1} = ∑_{d≤N^{θ}} d ∑_{k≤N^{θ}} ∑_{\begin{matrix}q>Q \\ (k,q)=d \end{matrix}} \frac{log q}{kq^{2}} ≪ ∑_{d≤N^{θ}} d ∑_{k≤N^{θ}/d} ∑_{q>Q/d} \frac{log(qd)}{kdq^{2}d^{2}}

≪ log N ∑_{d≤Q} \frac{1}{d^{2}} ∑_{k≤N^{θ}/d} \frac{1}{k} ∑_{q>Q/d} \frac{log q}{q^{2}} + log N ∑_{Q<d≤N^{θ}} \frac{1}{d^{2}} ∑_{k≤N^{θ}/d} \frac{1}{k} ∑_{q=1}^{∞} \frac{log q}{q^{2}}

≪ log^{2} N ∑_{d≤Q} \frac{1}{d^{2}} × \frac{log Q}{Q/d} + log^{2} N ∑_{Q<d} \frac{1}{d^{2}} ≪ \frac{log^{3} N}{Q} .  (27)

For the sum Σ_{2} we find

Σ_{2} = ∑_{d≤Q} d ∑_{k≤N^{θ}} \frac{1}{k} ∑_{\begin{matrix}q≤Q \\ (k,q)=d \end{matrix}} 1 ≪ ∑_{d≤Q} d ∑_{k≤N^{θ}/d} \frac{1}{kd} \frac{Q}{d} ≪ Q log^{2} N.  (28)

We shall now consider Σ_{3}. Obviously,

Σ_{3} = ∑_{h≤QN^{θ}} ω(h)Δ(N, h),  (29)

where

ω(h) = ∑_{k≤N^{θ}} ∑_{\begin{matrix}q≤Q \\ [k,q]=h \end{matrix}} \frac{1}{q^{2}} .

We have

ω(h) = ∑_{d≤Q} ∑_{k≤N^{θ}} ∑_{\begin{matrix}q≤Q \\ (k,q)=d \\ kq=hd \end{matrix}} \frac{1}{q^{2}} ≪ ∑_{d≤Q} ∑_{\begin{matrix}q≤Q \\ q≡0 (d) \end{matrix}} \frac{1}{q^{2}} ≪ ∑_{d≤Q} ∑_{q≤Q/d} \frac{1}{q^{2}d^{2}} ≪ 1.  (30)

We apply Bombieri–Vinogradov’s theorem (see [1, 7]). Since θ < 1/2 and Q = (log N)^{B}, we get from (16), (29), (30)

Σ_{3} ≪ \frac{N}{Q^{3}} .  (31)

By virtue of inequalities (26)–(28), (31) and the definition of Q and τ we find that

E_{1} ≪ \frac{N^{2}}{(log N)^{A}} for each θ < \frac{1}{2}

provided that

B ≥ A + 5.  (32)

4. THE ESTIMATE OF E_{2}

We use the identity

S_{k,l}(α) = ∑_{p≤N}(log p)e(αp) × \frac{1}{k} ∑_{h=1}^{k} e(\frac{h(p − l)}{k}) = \frac{1}{k} ∑_{h=1}^{k} e(−\frac{hl}{k})S(α + \frac{h}{k})

and get

J_{k,l}^{(2)}(N) = \frac{1}{k} ∑_{h=1}^{k} e(−\frac{hl}{k}) ∫_{E_{2}} S(α + \frac{h}{k})S^{2}(α)e(−Nα) dα.

Consequently,

E_{2} ≤ E_{3} = ∑_{k≤N^{θ}} \frac{1}{k} ∑_{h=1}^{k} ∫_{E_{2}} |S(α + \frac{h}{k})S^{2}(α)| dα.  (33)

Instead of E_{3} we can operate with a similar sum in which 1 ≤ h ≤ k, (h, k) = 1. We have to note that this is not necessary for the proof of our theorem, but it may be useful if we have to improve the upper bound of θ for which the theorem holds true.

We have

E_{3} = ∑_{d≤N^{θ}} ∑_{k≤N^{θ}} ∑_{\begin{matrix}h=1 \\ (h,k)=d \end{matrix}}^{k} ∫_{E_{2}} |S(α + \frac{h}{k})S^{2}(α)| dα
= ∑_{d≤Nθ} ∑_{k≤Nθ/d} \frac{1}{kd} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∫_{E_{2}} |S(α + \frac{h}{k})S^{2}(α)| dα ≪ (log N)E_{4},  (34)

where

E_{4} = ∑_{k≤N^{θ}} \frac{1}{k} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∫_{E_{2}} |S(α + \frac{h}{k})S^{2}(α)| dα.

Clearly,

E_{4} ≪ (log N) × max_{1/2≤G≤N^{θ}}(\frac{1}{G}E_{5}),  (35)

where

E_{5} = E_{5}(G, N) = ∑_{G<k≤2G} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∫_{E_{2}} |S(α + \frac{h}{k})S^{2}(α)| dα.

We set

W = \frac{N}{G(log N)^{A+3}} ,  (36)
E_{3} = \{α ∈ E_{2} | min(|S(α + \frac{h}{k})|, |S(α)|) ≤ W \},  E_{4} = E_{2} \ E_{3}.  (37)

Obviously,

E_{5} ≤ E_{6} + E_{7},  (38)

where

E_{6} = ∑_{G<k≤2G} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∫_{E_{3}} |S(α + \frac{h}{k})S^{2}(α)| dα,  (39)
E_{7} = ∑_{G<k≤2G} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} ∫_{E_{4}} |S(α + \frac{h}{k})S^{2}(α)| dα.  (40)

If α ∈ E_{3}, then we have

|S(α + \frac{h}{k})S^{2}(α)| ≤ W (|S^{2}(α + \frac{h}{k})| + |S^{2}(α)|),

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

and, hence,

∫_{E_{3}} |S(α + \frac{h}{k})S^{2}(α)| dα ≤ W ∫_{0}^{1}(|S^{2}(α + \frac{h}{k})| + |S^{2}(α)|) dα ≪ W N log N.

Using the last estimate, (36), and (39), we get

E_{6} ≪ \frac{N^{2}G}{(log N)^{A+2}} .  (41)

Let us consider E_{7}. We can find a good estimate for the part of E_{7} corresponding to the subset of E_{4} in which |S(α + h/k)| ≥ NG^{−1}(log N)^{R} for some constant R > 0 (see [3, p. 410]). However, it is impossible to proceed in a similar way with the part of E_{7} corresponding to the subset of E_{4} in which |S(α)| is large. In fact, as we shall see later, the main contribution to (40) comes from this subset. Fortunately, |S(α)| cannot be too large in E_{4} because

E_{4} ⊂ E_{2},  sup_{α∈E_{2}} |S(α)| ≪ \frac{N}{(log N)^{B/2−4}}  (42)

(see [6, Theorem 3.1]).

We can assume that G ≥ (log N)^{B/2−A−8}, otherwise, according to (36), (37), (42), the set E_{4} is empty. We divide E_{4} into O(log^{2} N) sets of the type

E_{5}(L_{1}, L_{2}) = \{α ∈ E_{4} | L_{1} < |S(α + \frac{h}{k})| ≤ 2L_{1}, L_{2} < |S(α)| ≤ 2L_{2}\},

where, according to (37) and (42), L_{1} and L_{2} satisfy

W ≤ L_{1} ≤ N log N,  W ≤ L_{2} ≤ \frac{N}{(log N)^{B/2−5}} .  (43)

Therefore

E_{7} ≪ log^{2} N × max_{L_{1},L_{2}}(L_{1}L_{2}^{2}E_{8}),  (44)

where the maximum is over L_{1}, L_{2} satisfying (43),

E_{8} = E_{8}(G, L_{1}, L_{2}) = ∑_{G<k≤2G} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} mes(E_{6})  (45)

and mes(E_{6}) denotes the measure of the set

E_{6} = E_{6}(L_{1}, L_{2}) = \{α ∈ [0, 1] | L_{1} ≤ |S(α + \frac{h}{k})|, L_{2} ≤ |S(α)|\}.

We proceed as in [3, p. 411, 412] and find that

mes(E_{6}) = ∑_{s=0}^{N−1} mes(E_{7}) = ∑_{s∈𝔑_{2}} mes(E_{7}),  (46)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

where E_{7} = E_{6} ∩ [s/N, (s + 1)/N] and where 𝔑_{2} is the set of all integers s ∈ {0, 1, …, N − 1} such that |S(α_{s})| ≥ L_{2} for some α_{s} ∈ [s/N, (s + 1)/N]. Let 𝔑′, 𝔑′′ be the subsets of even, respectively odd, integers from 𝔑_{2}. Suppose that 𝔑′ = {s_{1}, s_{2}, …, s_{h}} and let α_{i}, 1 ≤ i ≤ h, be numbers such that α_{i} ∈ [s_{i}/N, (s_{i} + 1)/N] and |S(α_{i})| ≥ L_{2}. It is clear that if i ≠ j, then |α_{i} − α_{j}| ≥ 1/N. Applying Theorem 2.3 of [4], we get

L_{2}^{2} × h ≤ ∑_{i=1}^{h} |S(α_{i})|^{2} ≪ N^{2} log N.

We estimate the numbers of elements of 𝔑′′ in the same way and find that

#𝔑_{2} ≪ \frac{N^{2} log N}{L_{2}^{2}} ,  (47)

where #𝔑_{2} denotes the number of elements of 𝔑_{2}.

We have

mes(E_{7}) ≤ mes\{α ∈ [\frac{s}{N} , \frac{s + 1}{N}] | L_{1} ≤ |S(α + \frac{h}{k})|\}

= mes\{α ∈ [\frac{s}{N} + \frac{h}{k}, \frac{s + 1}{N} + \frac{h}{k}] | L_{1} ≤ |S(α)|\} = ∑_{t=0}^{2N−1} mes(E_{8}) = ∑_{t∈𝔑_{1}} mes(E_{8}),(48)

where

E_{8} = [\frac{t}{N} , \frac{t + 1}{N}] ∩ [\frac{s}{N} + \frac{h}{k}, \frac{s + 1}{N} + \frac{h}{k}] ∩ \{α | L_{1} ≤ |S(α)|\}

and where 𝔑_{1} is the set of all integers t ∈ {0, 1, …, 2N − 1} such that |S(α_{t})| ≥ L_{1} for some α_{t} ∈ [t/N, (t + 1)/N]. For the number of elements of 𝔑_{1} we find, as above, that

#𝔑_{1} ≪ \frac{N^{2} log N}{L_{1}^{2}} .  (49)

Clearly, the measure of the set E_{8} is at most 1/N and E_{8} is empty unless |s − t + Nh/k| ≤ 1. Therefore we get, by (46) and (48),

mes(E_{6}) ≪ \frac{1}{N} \begin{matrix}∑_{t∈𝔑_{1}} & ∑_{s∈𝔑_{2}} \\ |s−t+Nh/k|≤1 \end{matrix} 1 ≪ \frac{1}{N} ∑_{t∈𝔑_{1}} ∑_{s∈𝔑_{2}} e^{−π(s−t+Nh/k)^{2}}.

We use the well-known identity

e^{−πx^{2}} = ∫_{−∞}^{∞} e^{−πα^{2}}e(αx) dα  (50)

and find that

mes(E_{6}) ≪ \frac{1}{N} ∑_{t∈𝔑_{1}} ∑_{s∈𝔑_{2}} ∫_{−∞}^{∞} e^{−πα^{2}}e(α(s − t + N \frac{h}{k})) dα.

ON THE NUMBER OF REPRESENTATIONS OF AN ODD INTEGER 425

We substitute the last estimate into (45) and change the order of summation and integration. As a result we get

E_{8} ≪ \frac{1}{N} ∫_{−∞}^{∞} e^{−πα^{2}}|F_{1}(α)F_{2}(α)K^{∗}(α)| dα,  (51)

where

F_{1}(α) = ∑_{t∈𝔑_{1}} e(αt),  F_{2}(α) = ∑_{t∈𝔑_{2}} e(αt),  K^{∗}(α) = ∑_{G<k≤2G} ∑_{\begin{matrix}h=1 \\ (h,k)=1 \end{matrix}}^{k} e(αN \frac{h}{k}).  (52)

Suppose that r > 2, u > 2 are real numbers such that

\frac{1}{2} + \frac{1}{r} + \frac{1}{u} = 1.  (53)

To estimate the right-hand side of (51), we apply Hölder’s inequality and get

E_{8} ≪ \frac{1}{N} (I_{1})^{1/2}(I_{2})^{1/r}(ξ_{u})^{1/u},  (54)

where

I_{1} = ∫_{−∞}^{∞} e^{−πα^{2}}|F_{1}(α)|^{2} dα,  I_{2} = ∫_{−∞}^{∞} e^{−πα^{2}}|F_{2}(α)|^{r} dα,  ξ_{u} = ∫_{−∞}^{∞} e^{−πα^{2}}|K^{∗}(α)|^{u} dα. (55)

It is clear that if we substitute

K(α) = ∑_{G<k≤2G} ∑_{h=1}^{k} e(αN \frac{h}{k})  (56)

for K^{∗}(α) in (51), then the inequality remains valid, and therefore we also have

E_{8} ≪ \frac{1}{N} (I_{1})^{1/2}(I_{2})^{1/r}(η_{u})^{1/u},  (57)

where

η_{u} = ∫_{−∞}^{∞} e^{−πα^{2}}|K(α)|^{u} dα.  (58)

Let us consider I_{2}. We use (50) and the trivial estimate |F_{2}(α)| ≪ #𝔑_{2} to get

I_{2} ≪ (#𝔑_{2})^{r−2} ∫_{−∞}^{∞} e^{−πα^{2}}|F_{2}(α)|^{2} dα = (#𝔑_{2})^{r−2} ∫_{−∞}^{∞} e^{−πα^{2}} ∑_{t_{1},t_{2}∈𝔑_{2}} e(α(t_{1} − t_{2})) dα
= (#𝔑_{2})^{r−2} ∑_{t_{1},t_{2}∈𝔑_{2}} e^{−π(t_{1}−t_{2})^{2}} ≪ (#𝔑_{2})^{r−1}.  (59)

Similarly,

I_{1} ≪ #𝔑_{1}.  (60)

From (47), (49), (54), (59), (60) we obtain

E_{8} ≪ \frac{N^{2−2/r} log^{2} N}{L_{1}L_{2}^{2−2/r}} (ξ_{u})^{1/u}.

We substitute the last estimate for E_{8} in (44) and use (43) to obtain

E_{7} ≪ N^{2}(ξ_{u})^{1/u}(log N)^{4−(B−10)/r}.  (61)

If we consider estimate (57) instead of (54), we also find

E_{7} ≪ N^{2}(η_{u})^{1/u}(log N)^{4−(B−10)/r}.  (62)

Let us state now the conjectures that we mentioned at the end of Section 2.

Conjecture 1. Suppose that u > 0 is a real number and ξ_{u} is defined by (55). There exist c_{1}(u), c_{2}(u) ≥ 0 such that if

G ≤ \frac{N^{1/u}}{(log N)^{c_{1}(u)}} ,

then

ξ_{u} ≤ G^{u}(log N)^{c_{2}(u)}.

Conjecture 2. Suppose that u > 0 is a real number and η_{u} is defined by (58). There exist c_{1}(u), c_{2}(u) ≥ 0 such that if

G ≤ \frac{N^{1/u}}{(log N)^{c_{1}(u)}} ,

then

η_{u} ≤ G^{u}(log N)^{c_{2}(u)}.

Let us assume that one of the conjectures stated above is true for a particular u > 2. We define the real number r by (53) and substitute the estimate for ξ_{u} (or η_{u}) into (61) (or (62)). We obtain

E_{7} ≪ N^{2}G(log N)^{4+c_{2}(u)−(B−10)/r}  (63)

provided that

G ≤ \frac{N^{1/u}}{(log N)^{c_{1}(u)}} .

Suppose that θ < 1/u. We apply (33)–(35), (38), (41), and (63) and find that

E_{2} ≪ N^{2}((log N)^{−A} + (log N)^{6+c_{2}(u)−(B−10)/r}).

We choose

B ≥ r(10A + c_{2}(u))  (64)

ON THE NUMBER OF REPRESENTATIONS OF AN ODD INTEGER 427

and get

E_{2} ≪ \frac{N^{2}}{(log N)^{A}} for each θ < \frac{1}{u}.

At the end of this section we prove Conjecture 1 for u = 4. Suppose that G ≤ N^{1/4}. By virtue of (52) and (55), we obtain

ξ_{4} = ∫_{−∞}^{∞} e^{−πα^{2}}K^{∗}(α)^{2}K^{∗}(−α)^{2} dα = ∑_{k_{i},h_{i} (65)} ∫_{−∞}^{∞} e^{−πα^{2}}e(αN(\frac{h_{1}}{k_{1}} + \frac{h_{2}}{k_{2}} − \frac{h_{3}}{k_{3}} − \frac{h_{4}}{k_{4}} )) dα.

The summation is over k_{1}, …, k_{4}, h_{1}, …, h_{4} such that

G < k_{i} ≤ 2G, 1 ≤ h_{i} ≤ k_{i}, (h_{i}, k_{i}) = 1, 1 ≤ i ≤ 4. (65)

We denote by Γ the number of solutions of the inequality

|\frac{h_{1}}{k_{1}} + \frac{h_{2}}{k_{2}} − \frac{h_{3}}{k_{3}} − \frac{h_{4}}{k_{4}} | ≤ \frac{log N}{N}

in the integers k_{i}, h_{i} satisfying (65). We apply (50) and get

ξ_{4} = ∑_{k_{i},h_{i} (65)} e^{−πN^{2}(h_{1}/k_{1}+h_{2}/k_{2}−h_{3}/k_{3}−h_{4}/k_{4})^{2}} ≪ Γ.

Now we use Lemma 3 of [3] and obtain

ξ_{4} ≪ G^{4}(log N)^{17}.

Thus inequality (5) holds for each θ < 1/4.

5. PROOF OF CONJECTURE 2 FOR u = 3

We can assume that G ≥ 10. The sum K(α) defined by (56) satisfies

|K(α)| ≤ ∑_{G<k≤2G}|∑_{h=1}^{k} e(αN \frac{h}{k})| ≪ ∑_{G<k≤2G} min(G, \frac{1}{∥αN/k∥}).  (66)

It is well known that

min(G, \frac{1}{∥x∥}) = ∑_{m=−∞}^{∞} c(m)e(mx),  (67)

where the coefficients c(m) satisfy

|c(m)| ≪ \{ \begin{matrix}log G & for all m \\ G^{2}/m^{2} & for m ≠ 0. \end{matrix}  (68)

We use (66) and (67) and find that

|K(α)| ≪ ∑_{G<k≤2G} ∑_{m=−∞}^{∞} c(m)e(αN \frac{m}{k})  (69)
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS  Vol. 218  1997

uniformly with respect to α. On the other hand,

|K(α)|^{2} = K(α)K(−α) = ∑_{G<k_{1},k_{2}≤2G} ∑_{h_{1}=1}^{k_{1}} ∑_{h_{2}=1}^{k_{2}} e(αN(\frac{h_{1}}{k_{1}} − \frac{h_{2}}{k_{2}} )).  (70)

Formulas (69) and (70) imply that uniformly with respect to α we have

|K(α)|^{3} ≪ ∑_{G<k,k_{1},k_{2}≤2G} ∑_{h_{1}=1}^{k_{1}} ∑_{h_{2}=1}^{k_{2}} ∑_{m=−∞}^{∞} c(m)e(αN(\frac{h_{1}}{k_{1}} − \frac{h_{2}}{k_{2}} + \frac{m}{k})).

We substitute the last estimate for |K(α)|^{3} into the expression of η_{3} (defined by (58)), change the order of summation and integration, and apply (50) to obtain

η_{3} ≪ ∑_{G<k,k_{1},k_{2}≤2G} ∑_{h_{1}=1}^{k_{1}} ∑_{h_{2}=1}^{k_{2}} ∑_{m=−∞}^{∞} |c(m)|e^{−πN^{2}(h_{1}/k_{1}−h_{2}/k_{2}+m/k)^{2}}.  (71)

We denote by η′_{3} the contribution of the summands in (71) satisfying

|\frac{h_{1}}{k_{1}} − \frac{h_{2}}{k_{2}} + \frac{m}{k}| ≤ \frac{log N}{N} .  (72)

If condition (72) does not hold, then the corresponding summand on the right-hand side of (71) is very small. The numbers |c(m)| are not large either (see (68)). Thus we have

η_{3} ≪ 1 + η′_{3}.  (73)

Suppose that

G ≤ \frac{N^{1/3}}{log N} .  (74)

It is easy to see that if G < k, k_{1}, k_{2} ≤ 2G, then conditions (72), (74) imply

\frac{h_{1}}{k_{1}} − \frac{h_{2}}{k_{2}} + \frac{m}{k} = 0.  (75)

In turn, (75) and the conditions imposed on k, k_{1}, k_{2}, h_{1}, h_{2} in (71) imply that |m| ≤ k. We also use (68) and get

η′_{3} ≪ log G × \begin{matrix}∑_{G<k,k_{1},k_{2}≤2G} & ∑_{h_{1}=1}^{k_{1}} & ∑_{h_{2}=1}^{k_{2}} & ∑_{|m|≤k} \\ h_{1}/k_{1}−h_{2}/k_{2}+m/k=0 \end{matrix} 1 ≪ log G × (Gη′′_{3} + η′′′_{3}),  (76)

where

η′′_{3} = \begin{matrix}∑_{G<k_{1},k_{2}≤2G} & ∑_{h_{1}=1}^{k_{1}} & ∑_{h_{2}=1}^{k_{2}} \\ h_{1}/k_{1}=h_{2}/k_{2} \end{matrix} 1, η′′′_{3} = \begin{matrix}∑_{G<k,k_{1},k_{2}≤2G} & ∑_{h_{1}=1}^{k_{1}} & ∑_{h_{2}=1}^{k_{2}} & ∑_{h=1}^{k} \\ h_{1}/k_{1}+h_{2}/k_{2}=h/k \end{matrix} 1.

(For convenience, we change the notation in the expression for η′′′_{3}.) It is clear that

η′′_{3} ≪ ∑_{n≤4G^{2}}( ∑_{\begin{matrix}G<k≤2G \\ h≤2G \\ hk=n \end{matrix}} 1)^{2} ≪ ∑_{n≤4G^{2}} τ^{2}(n) ≪ G^{2} log^{3} G.  (77)

It is more difficult to estimate η′′′_{3}. We have

η′′′_{3} = ∑_{q≤2G} ∑_{\begin{matrix}a=1 \\ (a,q)=1 \end{matrix}}^{q} L_{1}(a, q)L_{2}(a, q),  (78)

where

L_{1}(a, q) = \begin{matrix}∑_{G<k≤2G} & ∑_{h=1}^{k} \\ h/k=a/q \end{matrix} 1, L_{2}(a, q) = \begin{matrix}∑_{G<k_{1},k_{2}≤2G} & ∑_{h_{1}=1}^{k_{1}} & ∑_{h_{2}=1}^{k_{2}} \\ h_{1}/k_{1}+h_{2}/k_{2}=a/q \end{matrix} 1.

Since (a, q) = 1, we have

L_{1}(a, q) = \begin{matrix}∑_{G<k≤2G} & ∑_{h=1}^{k} \\ k≡0 (q), h≡0 (a), hq=ak \end{matrix} 1 = ∑_{G/q<k≤2G/q} ∑_{\begin{matrix}1≤h≤kq/a \\ h=k \end{matrix}} 1 ≪ \frac{G}{q}.  (79)

To estimate L_{2}(a, q), we proceed as in the proof of Lemma 3 from [3]. We have

L_{2}(a, q) = ∑_{d_{1},d_{2}≤2G} ∑_{G<k_{1},k_{2}≤2G} ∑_{h_{1}=1}^{k_{1}} ∑_{h_{2}=1}^{k_{2}} ; \begin{matrix}(h_{1},k_{1})=d_{1}, (h_{2},k_{2})=d_{2} \\ h_{1}/k_{1}+h_{2}/k_{2}=a/q \end{matrix} 1 = ∑_{d_{1},d_{2}≤2G} ∑_{G/d_{1}<k_{1}≤2G/d_{1}} ∑_{G/d_{2}<k_{2}≤2G/d_{2}} L_{3},  (80)

where L_{3} = L_{3}(a, q, k_{1}, k_{2}) is the number of solutions of the equation

\frac{h_{1}}{k_{1}} + \frac{h_{2}}{k_{2}} = \frac{a}{q}  (81)

in the integers h_{1}, h_{2} such that

1 ≤ h_{1} ≤ k_{1}, 1 ≤ h_{2} ≤ k_{2}, (h_{1}, k_{1}) = (h_{2}, k_{2}) = 1. (82)

Equation (81) is equivalent to

q(h_{1} \frac{k_{2}}{(k_{1}, k_{2})} + h_{2} \frac{k_{1}}{(k_{1}, k_{2})}) = a\frac{k_{1}}{(k_{1}, k_{2})} \frac{k_{2}}{(k_{1}, k_{2})}(k_{1}, k_{2}).  (83)

The integers

\frac{k_{1}}{(k_{1}, k_{2})},  \frac{k_{2}}{(k_{1}, k_{2})},  h_{1} \frac{k_{2}}{(k_{1}, k_{2})} + h_{2} \frac{k_{1}}{(k_{1}, k_{2})}
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430 TOLEV

are, obviously, pairwise coprime. Hence, we find from (83) that

\frac{k_{1}k_{2}}{(k_{1}, k_{2})^{2}} | q,  (84)

i.e., q(k_{1}, k_{2})^{2}/(k_{1}k_{2}) is an integer dividing q. Equation (83) implies

\frac{q(k_{1}, k_{2})^{2}}{k_{1}k_{2}} (h_{1} \frac{k_{2}}{(k_{1}, k_{2})} + h_{2} \frac{k_{1}}{(k_{1}, k_{2})}) = a(k_{1}, k_{2}).  (85)

Since (a, q) = 1, we find that

\frac{q(k_{1}, k_{2})^{2}}{k_{1}k_{2}} | (k_{1}, k_{2}),  i.e.,  q | \frac{k_{1}k_{2}}{(k_{1}, k_{2})}.  (86)

Using (85), we arrive at

h_{1} \frac{k_{2}}{(k_{1}, k_{2})} + h_{2} \frac{k_{1}}{(k_{1}, k_{2})} = a\frac{k_{1}k_{2}}{q(k_{1}, k_{2})}.  (87)

If h_{1} = H_{1}, h_{2} = H_{2} is a special solution of (87), then all solutions are given by

h_{1} = H_{1} + \frac{k_{1}}{(k_{1}, k_{2})}t, h_{2} = H_{2} − \frac{k_{2}}{(k_{1}, k_{2})}t, t ∈ ℤ.

Using (82), we infer that the integer t takes ≪ (k_{1}, k_{2}) values. Hence, we have L_{3} ≪ (k_{1}, k_{2}). We have found that L_{3} = 0 unless conditions (84), (86) hold. Therefore we get from (80)

L_{2}(a, q) ≪ ∑_{d_{1},d_{2}≤2G} ∑_{\begin{matrix}k_{1}≤2G/d_{1}, k_{2}≤2G/d_{2} \\ k_{1}k_{2}/(k_{1},k_{2})≡0 (q), k_{1}k_{2}/(k_{1},k_{2})^{2}|q \end{matrix}}(k_{1}, k_{2})
= ∑_{d_{1},d_{2}≤2G} ∑_{d≤min(2G/d_{1},2G/d_{2})} d ∑_{\begin{matrix}k_{1}≤2G/d_{1}, k_{2}≤2G/d_{2} \\ (k_{1},k_{2})=d, k_{1}k_{2}/d≡0 (q), k_{1}k_{2}/d^{2}|q \end{matrix}} 1
≪ ∑_{d_{1},d_{2}≤2G} ∑_{d≤min(2G/d_{1},2G/d_{2})} d ∑_{\begin{matrix}k_{1}≤2G/(d_{1}d), k_{2}≤2G/(d_{2}d) \\ k_{1}k_{2}d≡0 (q), k_{1}k_{2}|q \end{matrix}} 1.  (88)

Formulas (78), (79), (88) give us

η′′′_{3} ≪ ∑_{q≤2G} ∑_{\begin{matrix}a=1 \\ (a,q)=1 \end{matrix}}^{q} \frac{G}{q} ∑_{d_{1},d_{2}≤2G} ∑_{d≤min(2G/d_{1},2G/d_{2})} d ∑_{\begin{matrix}k_{1}≤2G/(d_{1}d), k_{2}≤2G/(d_{2}d) \\ k_{1}k_{2}d≡0 (q), k_{1}k_{2}|q \end{matrix}} 1
≪ G ∑_{d_{1},d_{2}≤2G} ∑_{d≤min(2G/d_{1},2G/d_{2})} d ∑_{k_{1}≤2G/(d_{1}d), k_{2}≤2G/(d_{2}d)} ∑_{\begin{matrix}q≤2G \\ q≡0 (k_{1}k_{2}) \\ q|k_{1}k_{2}d \end{matrix}} 1.

The sum over q is ≪ τ(q), and, hence

η′′′_{3} ≪ G ∑_{d_{1},d_{2}≤2G} ∑_{d≤min(2G/d_{1},2G/d_{2})} dτ(d) ∑_{k_{1}≤2G/(d_{1}d), k_{2}≤2G/(d_{2}d)} 1

≪ G^{3} ∑_{d,d_{1},d_{2}≤2G} \frac{τ(d)}{dd_{1}d_{2}} ≪ G^{3} log^{4} G.  (89)

Estimates (73), (76), (77), (89) imply η_{3} ≤ G^{3} log^{6} N provided that

G ≤ \frac{N^{1/3}}{log N} .

Conjecture 2 for u = 3 is proved, and therefore the Theorem is proved too.

Consider an approach that is used to prove Conjecture 1 or Conjecture 2 for u > 2 and u which is close to 2.

Let u = 2 + 2/n, where n is a large integer. We consider, for example, the quantities K(α), η_{u} defined by (56) and (58). Assume that uniformly with respect to α we have

|K(α)| ≪ |H(α)|^{n} + R_{0},  (90)

where

H(α) = ∑_{s=1}^{∞} ∑_{l=1}^{s} c(s, l)e(αN \frac{l}{s})

and where R_{0} and |c(s, l)| are small in a certain sense. Then, by (90), we get

η_{2+2/n} = ∫_{−∞}^{∞} e^{−πα^{2}}|K(α)|^{2+2/n} dα ≪ η^{∗} + R_{1},

where R_{1} is “small” and

η^{∗} = ∫_{−∞}^{∞} e^{−πα^{2}}|K(α)|^{2}|H(α)|^{2} dα.

According to (50) we have

η^{∗} ≪ ∑_{\begin{matrix}G<k_{1}, k_{2}≤2G \\ s_{1},s_{2}≥1 \end{matrix}} ∑_{h_{1}=1}^{k_{1}} ∑_{h_{2}=1}^{k_{2}} ∑_{l_{1}=1}^{s_{1}} ∑_{l_{2}=1}^{s_{2}} |c(s_{1}, l_{1})c(s_{2}, l_{2})|e^{−πN^{2}(h_{1}/k_{1}−h_{2}/k_{2}+l_{1}/s_{1}−l_{2}/s_{2})^{2}}.

We would be able to estimate the last expression if we had information about the size of |c(s, l)| and if we could calculate the number of solutions of the inequality

|\frac{h_{1}}{k_{1}} − \frac{h_{2}}{k_{2}} + \frac{l_{1}}{s_{1}} − \frac{l_{2}}{s_{2}} | ≤ \frac{log N}{N}

in integers satisfying some conditions.

Other versions of the same approach are also possible. However, the author cannot realize this idea at present.

ACKNOWLEDGMENTS

This research was begun during the author’s visit to Budapest in May 1996 and was sponsored by the Institute of Mathematics of the Hungarian Academy of Sciences. The author would like to thank Professor A. Balog for the interesting and valuable discussions.

This research was carried out under the financial support of the Ministry of Science and Education of Bulgaria (grant MM-430).

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The text was submitted by the author in English