On a theorem of Bombieri–Vinogradov type for prime numbers from a thin set

LXXXI.1 (1997)

On a theorem of Bombieri–Vinogradov type for prime numbers from a thin set

by

D. I. Tolev (Plovdiv) In 1940 I. M. Vinogradov considered the set

S

= {p prime | { √

p} < p

},

where λ > 0 is a fixed number and {t} denotes the fractional part of t.

Vinogradov proved ([17], Chapter 4) that if 0 < λ < 1/10 then

(1) X

p≤x, p∈Sλ

1 ∼ x

(1 − λ) log x as x → ∞.

A different approach to this problem was developed by Linnik [11] in 1945. In 1979 Kaufman [10] used the method of Linnik and proved the asymptotic formula (1) for λ < 0.1631 . . . He also proved that if the Riemann Hypothesis is assumed then (1) holds for λ < 1/4.

In 1983 Balog [1] and Harman [8] used Vaughan’s identity and mean value estimates for Dirichlet polynomials and independently proved without assuming the Riemann Hypothesis that the formula (1) is true for λ <

1/4. Later Balog [2] generalized his result to prime numbers in arithmetic progressions. We should also mention the works of Schoissengeier [14], [15], Gritsenko [7] and Rivat [13].

In the present paper we use the method of Balog and Harman and we prove a theorem of Bombieri–Vinogradov type for prime numbers from the set S

.

Let λ, θ be real numbers such that

(2) 0 < λ < 1/4, 0 < θ < 1/4 − λ.

Let x be a sufficiently large real number, L = log x; y, u, v, t, α, ν, τ, V, Y, K, M, N, D real numbers; a, k, l, m, n integers; A an arbitrarily large positive

1991 Mathematics Subject Classification: Primary 11N13.

Research done with the financial support of Bulgarian Ministry of Education and Science, grant MM-430.

[57]

number; ε an arbitrarily small positive number. In formulas which do not involve ε the constants in O-terms and -symbols are absolute or depend only on A, λ, θ. In formulas which involve ε the constants also depend on ε.

As usual, [t] denotes the integer part of t, e(t) = e

; µ(n), Λ(n), ϕ(n), τ (n) denote M¨obius’ function, von Mangoldt’s function, Euler’s function and the number of positive divisors of n, respectively. P

χ mod k

denotes the sum over all characters (mod k), and P

χ mod k

the sum over all primitive characters (mod k); finally, ψ

(y; k, a) = X

n≤y n≡a (mod k)

{√ n}<n^−λ

Λ(n).

We prove the following theorem:

Theorem. If λ and θ satisfy (2) and A > 0 is arbitrarily large then E = X

k≤x^θ

max

max

(a,k)=1

 ψ

(y; k, a) − y

ϕ(k)(1 − λ)

   x

L

.

P r o o f. We may suppose that A > 10. Let B = 10A. If k ≤ x

then using only a simple counting argument we find

(3) ψ

(xL

; k, a)  L X

n≤xL^−B n≡a (mod k)

{√ n}<n^−λ

1  k

x

L

.

Note that to prove the last estimate the upper bound for θ need not be so tight as in (2). The same happens in other places as well. We use the strong restriction θ < 1/4 − λ only at the end of the proof to obtain (38) and (39) from (36) and (37).

From (3) we get

(4) E  E

+ x

L

,

where

E

= X

k≤x^θ

xL^−B

max

max

(a,k)=1

 ψ

(y; k, a) − y

ϕ(k)(1 − λ)

   .

Define u

= v(1 − (log v)

), and S

(v; k, a) = X

u_v<n≤v n≡a (mod k)

{√ n}<n^−λ

Λ(n), S

(v; k, a) = X

u_v<n≤v n≡a (mod k) {√

n}<√

nv^−1/2−λ

Λ(n),

P

(v; k, a) = X

u_v<n≤v n≡a (mod k)

Λ(n)n

, P

(v; k, a) = v

X

u_v<n≤v n≡a (mod k)

Λ(n)n

.

It is not difficult to see that if xL

≤ v ≤ x and k ≤ x

then (5) S

(v; k, a) − S

(v; k, a)  k

x

L

,

P

(v; k, a) − P

(v; k, a)  k

x

L

.

Let us prove, for example, the first of the inequalities above. We have S

(v; k, a) − S

(v; k, a)  L X

uv<n≤v n≡a (mod k)

√n v^−1/2−λ≤{√ n}<√

n u^−1/2−λ_v

1.

If l

≤ n < (l + 1)

and √

n v

≤ { √

n} < √

n u

then we have l

(1 − v

)

≤ n < l

(1 − u

)

.

We use the definition of u

and the restriction imposed on k and after some calculations we find that the expression being estimated is

 L X

√uv−1<l≤√ v

(1 + k

l

((1 − u

)

− (1 − v

)

))

 k

x

L

.

For each y ∈ [xL

, x] we define the sequence y

, 0 ≤ i ≤ i

, in the following way:

(6) y

= y, y

= y

(1 − (log y

)

), y

< y(log y)

≤ y

. Clearly

(7) i

 L

.

If y ∈ [xL

, x] then using (3), (5)–(7) we get ψ

(y; k, a) − X

n≤y n≡a (mod k)

Λ(n)n

 X

0≤i≤i0

(|S

(y

; k, a) − S

(y

; k, a)| + |S

(y

; k, a) − P

(y

; k, a)|

+ |P

(y

; k, a) − P

(y

; k, a)|) + k

x

L

 X

0≤i≤i₀

|S

(y

; k, a) − P

(y

; k, a)| + k

x

L

.

The last inequality and (7) imply

(8) E

 E

+ L

E

+ x

L

, where

E

= X

k≤x^θ

xL^−B

max

max

(a,k)=1

X

n≤y n≡a (mod k)

Λ(n)n

− y

ϕ(k)(1 − λ)

   ,

E

= X

k≤x^θ

xL^−2B

max

max

(a,k)=1

|S

(v; k, a) − P

(v; k, a)|.

We use the Bombieri–Vinogradov theorem [4] to obtain

(9) E

 x

L

.

Obviously

(10) E

 L max

xL^−2B≤V ≤x

E

, where

E

= E

(V ) = X

k≤x^θ

V /2≤v≤V

max max

(a,k)=1

|S

(v; k, a) − P

(v; k, a)|.

Hence, using (4) and (8)–(10) we have

(11) E  L

max

xL^−2B≤V ≤x

E

+ x

L

. Suppose that

(12) xL

≤ V ≤ x, V /2 ≤ v ≤ V, T = V

L

, T

= V

/10, k ≤ x

, (a, k) = 1.

Let χ be a character (mod k). We define F (s) = X

uv<n≤v n≡a (mod k)

Λ(n)n

, F

(s) = X

uv<n≤v

χ(n)Λ(n)n

,

L(s) = X

V^1/2/10<l≤10V^1/2

l

, H(s) = 1

s (1 − (1 − v

)

),

I = 1 2πi

1/2+iT

\

1/2−iT

F (s/2)L(s)H(s) ds, I

= 1 2πi

1/2+iT

\

1/2−iT₀

F (s/2)L(s)H(s) ds.

We use Perron’s formula ([6], §17) to get S

(v; k, a) = X

uv<n≤v n≡a (mod k)

Λ(n)([ √

n ] − [ √

n(1 − v

)])

(13)

= X

u_v<n≤v n≡a (mod k)

Λ(n)

× X

V^1/2/10<l≤10V^1/2

 1 2πi

1/2+iT

\

1/2−iT

n

l

H(s) ds

+ O

 min

 1, T

    log

√ n l

−1



+ min

 1, T

    log

√ n

l (1 − v

)    

−1



= I + O(L(∆

+ ∆

)), where

∆

= X

V /4≤n≤V n≡a (mod k)

X

V^1/2/10<l≤10V^1/2

min

 1, T

    log

√ n l

−1

 ,

∆

= X

V /4≤n≤V n≡a (mod k)

X

V^1/2/10<l≤10V^1/2

min

 1, T

    log

√ n

l (1 − v

)    

−1

 .

We use (12) and after some standard calculations we obtain (14) ∆

, ∆

 k

x

L

.

If s = 1/2 + it, |t| ≤ T

then we may approximate the exponential sum L(s) by an integral ([9], Chapter III, §1, Corollary 1) to get

L(s) = (10V

)

− (V

/10)

1 − s + O(V

) = O

 V

1 + |t|

 . We also have

(15) H(s)  V

, H(s) = v

+ O(|s − 1|V

).

Hence

I

= v

2πi

1/2+iT

\

1/2−iT0

F (s/2) (10V

)

− (V

/10)

1 − s ds

(16)

+ O



V

T

\

−T0

 F

+

it
  dt 

.

Using the orthogonality of characters (mod k), Cauchy’s inequality and

Theorem 6.1 of [12] we get

T

\

−T₀

 F

+

it


 dt  1 ϕ(k)

X

χ mod k T

\

−T₀

 F

4

+

it
  dt

 1

ϕ(k) X

χ mod k

T



\

−T0

 F

+

it
 

dt



 x

L.

We substitute the last estimate in (16) and apply Perron’s formula again.

We get

(17) I

= P

(v; k, a) + O(k

x

L

).

From (13)–(17) and from the orthogonality of the characters (mod k) we obtain

S

(v; k, a) − P

(v; k, a)

 k

x

L

+ V

ϕ(k)

X

χ mod k T

\

T0

 F

4

+

it
  · 

L

+ it
  dt.

The last estimate and (11) imply

(18) E  L

max

xL^−2B≤V ≤x

(V

E

) + x

L

, where

E

= E

(V ) = X

k≤x^θ

1 k

X

χ mod k

V /2≤v≤V

max

T

\

T0

 F

4

+

it
  · 

L

+ it
  dt.

We approximate the exponential sum L in the last expression by a shorter one ([9], Chapter III, §1, Theorem 1) and we obtain

 L

+ it


  1 + 

  X

tV^−1/2/(20π)<l≤5tV^−1/2/π

l

 = 1 + |L

(t)|, say. Hence we have

E

 X

k≤x^θ

1 k

X

χ mod k

V /2≤v≤V

max

T

\

T₀

 F

+

it


(1 + |L

(t)|) dt (19)

 L max

x^1/2L^−1−B≤Y ≤x^1/2+λL^2B

E

,

where

E

= E

(V, Y ) = X

k≤x^θ

1 k

X

χ mod k

V /2≤v≤V

max

Y

\

Y /2

 F

+

it


 (1 + |L

(t)|) dt.

The interval of summation in L

(t) depends on t. To get rid of this depen- dence we apply, for example, Lemma 2.2 of [5] to get

|L

(t)| 

∞

\

−∞

K(α)|L

(t, α)| dα, where

L

(t, α) = X

Y V^−1/2/200<l≤2Y V^−1/2

e(αl)l

and where the kernel K(α) depends only on α, Y, V and satisfies the in- equalities

K(α) ≥ 0, 1 

∞

\

−∞

K(α) dα  L.

Therefore

(20) E

 L max

0≤α≤1

E

, where

E

= E

(V, Y, α)

= X

k≤x^θ

1 k

X

χ mod k

V /2≤v≤V

max

Y

\

Y /2

 F

4

+

it


(1 + |L

(t, α)|) dt.

We use the properties of primitive characters and the inequality (21)

Y

\

Y /2

|L

(t, α)| dt  Y,

which is a consequence of Cauchy’s inequality and Theorem 6.1 of [12]. After some calculations we get

(22) E

 L(E

+ E

) + x, where

E

= E

(V, Y, α) = max

V /2≤v≤V Y

\

Y /2

 X

uv<n≤v

Λ(n)n

(1 + |L

(t, α)|) dt, E

= E

(V, Y, α)

= X

k≤x^θ

1 k

X

χ mod k

V /2≤v≤V

max

Y

\

Y /2

 F

4

+

it


(1 + |L

(t, α)|) dt.

It remains to prove that if V and Y satisfy the conditions imposed in (18), (19) then we have

(23) E

, E

 x

for some ν > 0. The proof of the theorem follows from (18)–(20), (22), (23).

Let us consider E

. The estimation of E

is similar and, in fact, it was done in [1]. Clearly

(24) E

 L max

K≤x^θ

(K

E

), where

E

= E

(V, Y, α, K)

= X

k≤K

X

χ mod k

V /2≤v≤V

max

Y

\

Y /2

 F

+

it


(1 + |L

(t, α)|) dt.

Let

(25) D = x

.

We apply Vaughan’s identity [16] to get F

+

it

= F

− F

− F

− F

, where

F

= X

m≤D

X

u_v/m<n≤v/m

µ(m)(log n)χ(mn)(mn)

,

F

= X

m≤D

X

uv/m<n≤v/m

c(m)χ(mn)(mn)

,

F

= X

D<m≤D²

X

u_v/m<n≤v/m

c(m)χ(mn)(mn)

,

F

= X X

uv<mn≤v, m,n>D

a(m)Λ(n)χ(mn)(mn)

,

|c(m)| ≤ log m, |a(m)| ≤ τ (m).

We have

(26) E

 E

+ E

+ E

+ E

, where E

denotes the contribution to E

arising from F

.

Let us consider E

. We have F

= X

m≤D

µ(m)χ(m)m

W

,

where

W

= X

uv/m<n≤v/m

(log n)χ(n)n

= X

1≤l≤k

χ(l) X

u_v/m<n≤v/m n≡l (mod k)

(log n)n

= X

1≤l≤k

χ(l)Γ

,

say. We use (2) and (25) to conclude that the exponential sum Γ

may be approximated by an integral ([9], Chapter III, §1, Corollary 1). More precisely, we have

Γ

= 1 k

v/m

\

uv/m

(log y)y

dy + O

 x m



L

 .

Since the character χ is primitive we have P

1≤l≤k

χ(l) = 0. Hence W

 K

 x m



L, F

 DKx

L.

The last estimate and (21) imply

(27) E

 DK

x

Y L.

Using the bounds for Y and K imposed in (19) and (24) and also (2), (25), (27) we get

(28) E

 Kx

.

We estimate E

analogously and we obtain

(29) E

 Kx

.

Consider now E

. We have E

 X

k≤K

X

χ mod k

V /4≤v≤V

max

Y

\

Y /2

 X X

m,n>Dmn≤v

a(m)Λ(n)χ(mn)(mn)

  (30) 

×(1 + |L

(t, α)|) dt

 L

max

D≤M,N ≤x/D M N ≤x

E

,

where

E

= E

(V, Y, α, K, M, N ) (31)

= X

k≤K

X

χ mod k

V /4≤v≤V

max

Y

\

Y /2

|F

(t)|(1 + |L

(t, α)|) dt,

F

(t) = X X

M <m≤2M N <n≤2N

mn≤v

a(m)Λ(n)χ(mn)(mn)

.

We may suppose that the maximum in (31) is taken over v of the form 1/2 + l where l is an integer. Applying again Perron’s formula we obtain

F

(t) = X

M <m≤2M

X

N <n≤2N

a(m)Λ(n)χ(mn)(mn)

×

 1 2πi

L⁻¹

\

 v mn



ds

s + O

 x

    log v

mn    

−1



. Hence

(32) F

(t)  L

x

\

−x

|M| · |N | dτ

1 + |τ | + x

, where

M = X

M <m≤2M

a(m)χ(m)m

,

N = X

N <n≤2N

Λ(n)χ(n)n

. Formulas (21), (31) and (32) imply

(33) E

 L

x

\

−x

E

dτ

1 + |τ | + Kx

where

E

= E

(Y, α, K, M, N, τ ) = X

k≤K

X

χ mod k Y

\

Y /2

|M| · |N |(1 + |L

(t, α)|) dt.

Suppose, for example, that M ≤ N . Then M, N satisfy (34) D ≤ M ≤ x

, D ≤ N ≤ x/D, M N ≤ x.

By the Cauchy inequality we have

(35) E

 (E

)

(E

)

, where

E

= E

(Y, α, K, N, τ ) = X

k≤K

X

χ mod k Y

\

Y /2

|N |

dt,

E

= E

(Y, α, K, M, τ ) = X

k≤K

X

χ mod k Y

\

Y /2

|M |

(1 + |L

(t, α)|

) dt.

We estimate E

by Theorem 7.1 of [12] to get (36) E

 L(K

Y + N )N

.

To estimate the integral in the expression for E

we use Theorem 6.1 of [12] and also Theorem 1 of [3]. We obtain

Y

\

Y /2

|M|

(1 + |L

(t, α)|

) dt  x

M

Y, hence

(37) E

 x

M

K

Y.

We use (2), (24), (25), (30), (33)–(37) to get

(38) E

 Kx

for some ν > 0. Let us point out that only in this place do we need the tight restriction θ < 1/4 − λ.

We proceed with E

analogously to obtain

(39) E

 Kx

for some ν > 0.

We use (24), (26), (28), (29), (38), (39) to find that E

 x

for some ν > 0. The estimation of E

is similar, so we have proved (23) and the proof of the theorem is complete.

Finally, the author would like to thank the referee for his useful remarks.

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Department of Mathematics Plovdiv University “P. Hilendarski”

Plovdiv 4000, Bulgaria

E-mail: dtolev@ulcc.uni-plovdiv.bg

Received on 30.8.1996

and in revised form on 20.1.1997 (3041)