OCTOGON MATHEMATICAL MAGAZINE
Vol. 17, No.2, October 2009, pp 727-731
ISSN 1222-5657, ISBN 978-973-88255-5-0,
www.hetfalu.ro/octogon
727
On certain conjectures in prime number
theory
József Sándor25
ABSTRACT. Some remarks between connections of certain famous conjectures on
prime differences are pointed out.
Let pn denote the nth prime number. Put dn = pn+1 − pn . As
pn+1 − 2
d1 + d2 + ... + dn
=
∼ log n ∼ log pn as n → ∞
n
n
(by the prime number theorem pn ∼ n log n); we can say that the average
order of dn is log pn .
MAIN RESULTS
The study of gaps between primes has a long history.
In 1920 H. Crámer ([1]) proved that the famous Riemann hypothesis implies
that
³
´
dn = O p1/2
·
log
p
(1)
n
n
We note that the Riemann hypothesis implies a weaker hypothesis, namely,
the so-called Lindelöf hypothesis, to the effect that for any ε > 0 one has
µ
¶
1
ζ
+ it = O (tε ) as t → ∞
2
A.E. Ingham showed (see e.g. [11]) that by assuming only the Lindelöf
hypothesis, one has for any ε > 0
25
Received: 04.09.2009
2000 Mathematics Subject Classification. 11A41; 11N05.
Key words and phrases. Primes, difference of consecutive primes, Riemmans
conjecture.
728
József Sándor
dn < pn1/2+ε , for n ≥ n0
(2)
(i.e. if n is sufficiently large).
Clearly, relation (1) is stronger than (2) as c log pn < pεn for any fixed
constants c, ε > 0; if n is sufficiently large.
In 1937, based on probability arguments, Crámer conjectured ([5]) even a
stronger relation than (1), namely that
´
³
(3)
dn = O (log pn )2
The strong inequality (3) perhaps is not true; and indeed, recently A.
Granville ([4], [6]) conjectured that for infinitely many n one has
dn > k (log pn )2 ,
(4)
where k = 2 · e−γ ≈ 1, 12292... (e and γ being the two Euler constants). See
also H. Maier [7].
Crámer conjectured in fact a more precise form of (3), namely that
lim sup
n→∞
dn
=c
(log pn )2
(5)
where c is finite constant (perhaps; c = 1).
As noted by K. Soundararajan [9], (5) is for beyond what ”reasonable”
conjectures such as the Riemann hypothesis would imply.
An old conjecture says that there is always a prime between two consecutive
squares (see e.g. [8], [11]). Even this lies slightly beyond the reach of the
Riemann hypothesis, and all it would imply is that
dn
lim sup √ ≤ 4,
pn
n→∞
(6)
which is weaker than (5) with a finite value of c.
A. Schinzel conjectured (see [8]) that between x and x + (log x)2 there is a
prime, if x ≥ x0 (x0 ≈ 7, 1374035).
Thus pn < pn+1 < pn + (log pn )2 if pn ≥ 8 (i.e. n ≥ 4) , so this would imply
lim sup
n→∞
pn+1 − pn
≤ 1,
(log pn )2
i.e. c of (5) is ≤ 1. This would imply also the conjecture (3) of Crámer.
Schinzel‘s conjecture gives also
(7)
On certain conjectures in prime number theory
729
pn+1 − pn
(log pn )2
0<
< √
→ 0 as n → ∞,
√
pn
pn
so
dn
√ → 0 as n → ∞
pn
(8)
As
pn+1 − pn
pn+1 − pn
√
√
´
q
pn+1 − pn = √
=√ ³
√
pn + pn+1
pn 1 + pn+1
pn
and since
pn+1
pn
→ 1, we get that (8) is equivalent with
√
√
pn+1 − pn → 0 as n → ∞
(9)
In our paper [10] it is proved that (and even stronger results)
lim inf
n→∞
√
√
√
4
pn ( pn+1 − pn ) = 0
(10)
Motivated by our paper, D. Andrica ([8]) conjectured that
√
√
pn+1 − pn < 1 for all n ≥ 1
√
As for x ≥ 121 one has (log x)2 < 2 x + 1, we get that
(11)
√
(log pn )2 < 2 pn + 1
if pn ≥ 121, so
√
pn+1 − pn < (log pn )2 < 2 pn + 1, i.e.
(11) holds true, which means that Schinzel‘s conjecture implies the Andrica
conjecture. Another conjecture, due to Piltz is that for any ε > 0
dn = O (pεn )
As (log pn )2 /pεn → 0 as n → ∞, clearly (3) is stronger than (12).
On the other hand, (12) is stronger than (11), for sufficiently large n, as
letting e.g. ε = 13 in (12) we get
1
1
pn+1 − pn < M pn3 < 2pn2 + 1 if n ≥ n0
(12)
730
József Sándor
1
But (12) implies even relation (2), as from pn+1 − pn < M · pn3 it follows
−pn
√
immediately that pn+1
→ 0 as n → ∞, i.e. (8), which is equivalent to (9).
pn
Inequality (11)√seems√to be true; at least for n = 4 we get
√
√
p5 − p4 = 11 − 7 ≈ 0, 670873 which is the largest value among the
first 105 primes.
In 2001 Baker, Harman and Pintz ([3]) obtained an unconditional result on
the maximum value of dn . They proved that, without any hypothess one has
dn < p0,525
for n ≥ n0
n
(13)
and perhaps there are hopes to prove unconditionally inequality (2) for not
only ε = 0, 025 but any ε > 0.
Recently, a major advance was made by Goldston, Pintz and Yildirim (see
e.g. [9]); who proved without any assumption that
lim inf
n→∞
dn
=0
log pn
Clearly, this implies also that the lim inf of expression (8) is true; but
remains open the proof of the similar fact for the lim sup .
As noted by K. Soundarajan [9], theorem (14) shows that given ε > 0, for
infinitely many n, the interval [n, n + ε · log n] contains at least two primes.
REFERENCES
[1] Crámer, H., Some theorems concerning prime numbers, Arkiv fur
Mathematik, Astr., Fysk, 5(1920), pp. 1-33.
[2] Crámer, H., On the Distribution of primes, Proc. Cambridge Phil. Soc.
20, (1921), pp. 272-28.
[3] Baker, R.C., Harman,G. and Pintz, J., The difference between consecutive
primes, II, Proc. London, M.S. 83(2001), No. 3, pp. 532-562.
[4] Granville, A., Unexpected inequalities in the distribution of prime
numbers, in: Proc. of the Int. Conf., Zurich, 1994, Vol. I(1994), pp. 388-399.
[5] Crámer, H., On the order of magnitude of the difference between
consecutive prime numbers, Acta Arith. 2(1936), pp. 23-46.
[6] Granville, A., Harold Crámer and the distribution of prime numbers,
Scand. Act. J. 1(1995),pp. 12-28.
[7] Maier, H., Primes in short intervals, Michigan Math. J., 32(1985), pp.
221-225.
[8] Guy, R.K., Unsolved problems in number theory, 3rd ed., Springer Verlag,
2004.
On certain conjectures in prime number theory
731
[9] Soundarajan, K., Small gaps between prime numbers: the work of
Goldston- Pintz- Yildirim, Bull. Amer. Math. Soc. 44(2007), No. 1, pp. 1-18
[10] Sandor, J., On sequences series and applications in prime number theory
(Romanian), Gaz. Mat. Perf. Mat 6(1985), pp. 38-48.
[11] Ribenboim, P., The new book of prime number records, Springer Verlag,
1996.
Babeş-Bolyai University,
Cluj and Miercurea Ciuc,
Romania