In 1982, Glyn Harman [2] proved that for almost
all n, the interval [n,
n+n(1/10)+ε]
contains a prime number. By this we mean that the set of n[les ]N
for which the
interval does not contain a prime has measure o(N)
as n→+∞. It follows from
Huxley's work [6] that if θ>1/6
then there will almost always be asymptotically
nθ(log n)−1
primes in the interval [n,
n+nθ]. In 1983, Glyn Harman
[3] pointed
that for almost all n, the interval [n,
n+n(1/12)+ε] contains a prime
number, and
meantime Heath-Brown gave the outline of this result in
[5]. The exponent was reduced to
1/13 by Jia [10], 2/27 by Li [12]
and 1/14 by Jia [11], and meantime N. Watt
[16]
got the same result. In this paper we shall prove the following result.THEOREM. For almost all n, the intervalformula herecontains a prime number.
The prime number theorem in short intervals for automorphic L-functions