A note on primes between consecutive powers

A Note on Primes Between Consecutive Powers

DANILOBAZZANELLA(*)

ABSTRACT- In this paper we carry on the study of the distribution of prime numbers between two consecutive powers of integers.

1. Introduction.

A well known conjecture about the distribution of primes asserts that all intervals of type [n²;(n‡1)²] contain at least one prime. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. To get a conditional proof of the conjecture we need to assume a stronger hypothesis about the behaviour of Selberg's integral in short intervals, see D. Bazzanella [3]. This paper concerns with the distribution of prime numbers between two consecutive powers of in- tegers, as a natural generalization of the above problem. The well known result of M. N. Huxley [8] about the distribution of prime in short intervals implies that all intervals [n^a;(n‡1)^a] contain the expected number of primes fora>12

5 andn! 1. This was slightly improved by D. R. Heath- Brown [7] toa12

5.

Assuming some heuristic hypotheses we can obtain the expected distribution of primes for smaller values of a. In particular under the assumption of the LindeloÈf hypothesis, which states that the Riemann Zeta-function satisfies

z(s‡it)t^h s1 2;t2

;

(*) Indirizzo dell'A.: Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy.

E-mail: danilo.bazzanella@polito.it

1991Mathematics Subject Classification. 11NO5.

for any h>0, the classical result of A. E. Ingham [9] implies that all in- tervals [n^a;(n‡1)^a] contain the expected number of primes fora>2.

In a previous paper, see [2], the author proved that all intervals [n^a;(n‡1)^a][N;2N], withat most O(B(N;a)) exceptions, contain the expected number of primes, for suitable functionB(N;a). More precisely the author proved that we can choose

B(N;a)ˆ

(N^1=a)^{85 a2‡e} 6

55a6 5‡c (N^1=a)^{52 a‡e} 27

165a53 26 (N^1=a)^{72 9a3(a‡12)8a2‡e} 53

26a512 5 8>

>>

>>

><

>>

>>

>>

: (1)

fore>0 and ca suitable positive constant. The author proved also that, under the assumption of the LindeloÈf Hypothesis, we can choose

B(N;a)ˆ(N^1=a)^{2 a‡e} for 15a2 (2)

and, under the assumption of the Riemann Hypothesis, we can choose B(N;a)ˆ(N^1=a)^{2 a}log²N g(N) for 15a2;

(3)

withg(N)! 1arbitrarily slowly.

In this paper we establish the upper bounds for the exceptional set of the distribution of primes between two consecutive powers of integers under the assumption of some other heuristic hypotheses.

The first hypothesis regards the counting functions N(s;T) and N(s;T). The former is defined as the number of zeros rˆb‡ig of Riemann zeta function which satisfy sb1 and jgj T, while N(s;T) is defined as the number of ordered sets of zerosr_j ˆb_j‡ig_j (1j4), eachcounted byN(s;T), for whichjg₁‡g₂ g₃ g₄j 1. If we make the heuristic assumption that there exists a constant T₀ suchthat

N(s;T)N(s;T)⁴ T

1

2s1;TT0

; (4)

as in D. Bazzanella and A. Perelli [4], then we can obtain the following re- sult.

THEOREM1. Assume(4)and lete>0. Then all intervals[n^a;(n‡1)^a] [N;2N], with at most O((N^1=a)^h(a)‡e) exceptions, contain the expected

number of primes , where h(a)ˆ

4a‡12 8 

p3a 27

16a48 25 12

5 a 48

25a512 5 8>

><

>>

: :

Foranear 6=5 the assumption of (4) is not helpful to obtain a stronger result than the unconditional result (1) proved in [2]. A corollary of this theorem is Theorem 3 of D. Bazzanella [1], which states that, under the assumption of (4), all intervals [n²;(n‡1)²][N;2N], withat most O(N^1=5‡e) exceptions, contain the expected number of primes. Recalling that fora12=5 there are not exceptions, we expect to have

a!12=5lim h(a)ˆ0:

We note that the above condition is implies by the Theorem 1, but not by the unconditional result (1).

Moreover we assume the Density Hypothesis, which states that for everyh>0 the counting functionN(s;T) satisfies

N(s;T)T^{2(1 s)‡h} 1

2s1

;

obtaining our last result.

THEOREM 2. Assume the Density Hypothesis and (4), let e>0 and 15a2. Then all intervals [n^a;(n‡1)^a][N;2N], with at most O((N^1=a)^h(a)‡e)exceptions, contain the expected number of primes, where

h(a)ˆ2(2 a):

If we assume the Riemann Hypothesis, it is known that fora>2 there are not exceptions and then we expect to haveh(2)ˆ0. Indeed, although the assumptions of the Theorem 2 are weaker than the Riemann Hy- pothesis, we obtainh(2)ˆ0 again.

2. The basic lemma.

Throughout the paper we always assume thatn,x,XandN are suffi- ciently large as prescribed by the various statements, and e>0 is arbi- trarily small and not necessary the same at each occurrence. The basic

lemma is a result about the structure of the exceptional set for the asymptotic formula

c(x‡h(x)) c(x)h(x) as x! 1:

(5)

Let j j denote the modulus of a complex number or the Lebesgue measure of an infinite set of real numbers or the cardinality of a finite set.

Letd>0 and leth(x) be an increasing function suchthatx^eh(x)xfor somee>0,

D(x;h)ˆc(x‡h(x)) c(x) h(x) and

Ed(X;h)ˆ fXx2X:jD(x;h)j dh(x)g:

It is clear that (5) holds if and only if for everyd>0 there existsX₀(d) such thatE_d(X;h)ˆ ;forXX₀(d). Hence for smalld>0,Xtending to1and h(x) suitably small withrespect tox, the setE_d(X;h) contains the excep- tions, if any, to the expected asymptotic formula for the number of primes in short intervals. Moreover, we observe that

E_d(X;h)E_d⁰(X;h) if 05d⁰5d:

We will consider increasing functionsh(x) of the formh(x)ˆx^u‡e(x), with some 05u51 and a functione(x) suchthatje(x)jis decreasing,

e(x)ˆo(1) and e(x‡y)ˆe(x)‡O jyj xlogx

;

for everyjyj5x. A function satisfying these requirements will be called of typeu.

The basic lemma provides the structure of the exceptional setEd(X;h).

LEMMA. Let 05u51, h(x) be of type u, X be sufficiently large de- pending on the function h(x)and05d⁰5dwithd d⁰exp ( 

logX

p ).

If x02Ed(X;h) then E_d⁰(X;h) contains the interval [x0 ch(X);

x0‡ch(X)]\[X;2X], where cˆ(d d⁰)u=5. In particular, if E_d(X;h)6ˆ ; then jE_d⁰(X;h)j u (d d⁰)h(X):

The above Lemma is part (i) of Theorem 1 of D. Bazzanella and A.

Perelli, see [4], and it essentially says that if we have a single exception in Ed(X;h), witha fixedd, then we necessarily have an interval of exceptions inE_d⁰(X;h), withd⁰a little smaller thand.

3. Proof of the Theorems.

We defineHˆ(n‡1)^a n^aand

Ad(N;a)ˆ fN^1=an(2N)^1=a :jc((n‡1)^a) c(n^a) Hj dHg:

This set contains the exceptions, if any, to the expected asymptotic formula for the number of primes in intervals of the type [n^a;(n‡1)^a][N;2N].

The main step of the proof is to connect the exceptional set A_d(N;a) with the exceptional set for the distribution of primes in short intervals and to show that

jA_d(N;a)j jE_d=2(N;h)j N^{1 1=a} ‡1;

(6)

for everyd>0,a>1 andh(x)ˆ(x^1=a‡1)^a x.

In order to prove (6) we choosen2A_d(N;a) and letxˆn^a2[N;2N].

From the definition ofAd(N;a) we get

jc((n‡1)^a) c(n^a) Hj dH;

and then

jc(x‡h(x)) c(x) h(x)j dh(x);

which implies that x2E_d(N;h). Using the Lemma, with d⁰ˆd=2, we obtain that there exists an effective constantcsuchthat

[x;x‡ch(x)]\[N;2N]E_d=2(N;h):

Letm2A_d(N;a),m>n. As before we can defineyˆm^a2[N;2N] such that [y;y‡ch(y)]\[N;2N]E_d=2(N;h):

Choosingc51 we find

y xˆm^a n^a(n‡1)^a n^a>ch(x);

and then

[x;x‡ch(x)]\[y;y‡ch(y)]ˆ ;:

Hence (6) is proved, since for everyn2A_d(N;a) andxˆn^a, withat most one exception, we have

[x;x‡ch(x)][N;2N]:

Now we can conclude the proof of the theorems providing a suitable bounds for the measure of the exceptional setE_d=2(N;h).

If we considerx2E_d=2(N;h) we get

jc(x‡h(x)) c(x) h(x)j N^{1 1=a} and then

jE_d=2(N;h)jN^{4 4=a}

Z

jc(x‡h(x)) c(x) h(x)j⁴dx

Ed=2(N;h)

Z^2N

N

jc(x‡h(x)) c(x) h(x)‡S(x)j⁴dx;

(7)

for everyS(x) suchthat

S(x)N^{1 1=a} logN :

Now we use the classical explicit formula, see H. Davenport [5, Chapter 17], to write

c(x‡h(x)) c(x) h(x)ˆ X

jgjT

x^rcr(x)‡O Nlog²N T

!

; (8)

uniformly for Nx2N, where 10TN, rˆb‡ig runs over the non-trivial zeros ofz(s) and

cr(x)ˆ(1‡h(x)=x)^r 1

r :

Let

TˆN^1=alog³N;

(9) and then

cr(x)min N ^1=a; 1 jgj

: (10)

Follow the method of D. R. Heath-Brown, see [6], we find a constant 05u51 suchthat

X

jgjT;b>u

x^rc_r(x)N^{1 1=a} logN ;

obtaining

c(x‡h(x)) c(x) h(x)ˆ X

jgjT;bu

x^rc_r(x)‡O N^{1 1=a} logN

;

and then, from (7), we have

jEd(N;h)jN^{4 4=a} Z^2N

N

X

jgjT;bu

x^rcr(x)

4

dx:

(11)

To estimate the fourth power integral we divide the interval [0;u] into O(lnN) subintervalsIkof the form

I_kˆ k

logN; k‡1 logN

;

and by HoÈlder inequality we obtain X

jgjT;bu

x^rcr(x)

4

ln³NX

k

X

jgjT;b2Ik

x^rcr(x)

4

and then Z^2N

N

X

jgjT;bu

x^rc_r(x)

4

dxN^{1 4=a‡e}max

su N^4sM(s;T);

(12)

where

M(s;T)ˆ X

b1;...;b4s jg1jT;...;jg4jT

1

1‡ jg₁‡g₂ g₃ g₄j: It is not difficult to prove that

M(s;T)N(s;T)logN;

see [6], and then from (11) and (12) this yields jE_d=2(N;h)j N ^3‡emax

su N^4sN(s;T):

The assumption of (4) then implies

jEd=2(N;h)j N ^{3 1=a‡e} max

su N^sN(s;T)

₄

: (13)

Using the Ingham-Huxley density estimate, asserting that for everyn>0 we have

N(s;T) T^{3(1 s)=(2 s)‡n} 1

2s3 4 T^{3(1 s)=(3s 1)‡n} 3

4s1

; 8>

><

>>

(14) :

see [10, Theorem 11.1], we obtain an upper bound that fora48=25 attains its maximum atsˆ3=4, and so we get

jE_d=2(N;h)j N ^{3 1=a‡e}N^{4 3=4‡5a3}

N^5a7 ‡e: (15)

From (6) and (15) we can conclude jA_d(N;a)j jE_d=2(N;h)j

N^{1 1=a} ‡1N^5a7 ‡e N^{1 1=a} N^125a 1‡e(N^1=a) ¹²⁵ a

‡e; for everyd>0 anda48=25.

For 27=16a48=25 the above bound attains its maximum at sˆ2 

p3=a

and then we have

jE_d=2(N;h)j N ^{3 1=a‡e} N^sN(s;T)₄

N^{5‡11a 8} 

p3=a

‡e: (16)

Thus, from (6) and (16), we deduce jA_d(N;a)j jE_d=2(N;h)j

N^{1 1=a} ‡1N^{5‡11a 8} 

p3=a

‡e

N^{1 1=a} N^{4‡12a 8} 

p3=a

‡e(N^1=a) ^{4a‡12 8} 

p3a

… †^‡e:

for everyd>0 and 27=16a48=25, and then Theorem 1 follows.

In order to prove Theorem 2 we imitate the proof of Theorem 1 up to equation (13) and then we write

jE_d=2(N;h)j N ^{3 1=a‡e} max

su N^sN(s;T)

₄

:

Recalling that under the assumption of the Density Hypothesis we have

N(s;T) T^{2(1 s)‡n} 1

2s11 14

T^{9(1 s)=(7s 1)‡n} 11

14s1

; 8>

><

>>

: (17)

for everyn>0, we thus obtain an upper bound for the exceptional set. For every 15a2 sucha bound attains its maximum atsˆ1=2, and so we obtain

jE_d=2(N;h)j N ^{3 1=a‡e} N²N(1=2;T)₄

N^{3=a 1‡e} The above bound and (6) imply that

jAd(N;a)j jE_d=2(N;h)j

N^{1 1=a} ‡1N^{3=a 1‡e} N^{1 1=a} N^{4=a 2‡e}(N^1=a)^{(4 2a)‡e}: This concludes the proof of Theorem 2.

REFERENCES

[1] D. BAZZANELLA,The exceptional set for the distribution of primes between consecutive squares, Preprint 2008.

[2] D. BAZZANELLA,The exceptional set for the distribution of primes between consecutive powers, Acta Math. Hungar,116(3) (2007), pp. 197-207.

[3] D. BAZZANELLA,Primes between consecutive square, Arch. Math.,75(2000), pp. 29-34.

[4] D. BAZZANELLA- A. PERELLI,The exceptional set for the number of primes in short intervals, J. Number Theory,80(2000), pp. 109-124 .

[5] H. DAVENPORT,Multiplicative Number Theory, volume GTM 74 (Springer - Verlag, 1980), second edition.

[6] D. R. HEATH-BROWN,The difference between consecutive primes II, J. London Math. Soc.,19(2) (1979), pp. 207-220.

[7] D. R. HEATH-BROWN, The number of primes in a short interval. J. Reine Angew. Math.,389(1988), pp. 22-63.

[8] M. N. HUXLEY,On the difference between consecutive primes, Invent. Math., 15(1972), pp. 164-170.

[9] A. E. INGHAM, On the difference between consecutive primes, Quart. J. of Math. (Oxford),8(1937), pp. 255-266.

[10] A. IVICÂ,The Riemann Zeta-Function, John Wiley and Sons, New York, 1985.

Manoscritto pervenuto in redazione l'11 febbraio 2008.