On the difference between consecutive primes

  title={On the difference between consecutive primes},
  author={Martin N. Huxley},
  journal={Inventiones mathematicae},
  • M. Huxley
  • Published 1 June 1971
  • Chemistry
  • Inventiones mathematicae
This disclosure relates to prostaglandins of the PG3 series including PGE3, PGF3 alpha , PGF3 beta , and PGB3, to various analogs of those in racemic form, and to novel processes for making those. This disclosure also relates to certain fluorine and alkyl substituted analogs and certain acetylenic analogs of PGE3, PGF3 alpha , PGF3 beta , PGA3, and PGB3 in both racemic and optically active form, and to processes for making those. These various analogs are useful for the same pharmacological… 
On the difference between consecutive primes
Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new
On polynomial-factorial diophantine equations
We study equations of the form P(x) = n! and show that for some classes of polynomials P the equation has only finitely many solutions. This is the case, say, if P is irreducible (of degree greater
Pair Correlation of Zeros, Primes in Short Intervals and Exponential Sums over Primes
)and#, #$ run over the imaginary part of the non-trivial zeros of the Riemann zeta function. In view of the above results, wemay therefore expect that the quantities R(X, !), J(X, h) and F(X, T)
On the Möbius function in all short intervals
We show that, for the Mobius function $\mu(n)$, we have $$ \sum_{x 0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's result corresponded to
Note On Prime Gaps And Very Short Intervals
Assuming the Riemann hypothesis, this article discusses a new elementary argument that seems to prove that the maximal prime gap of a finite sequence of primes p_1, p_2, ..., p_n 0 constant.
Irreducible polynomials over F2r with three prescribed coefficients
Bounded gaps between primes in short intervals
Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $$[x-x^{0.525},x]$$[x-x0.525,x] for large x. In this paper, we extend a result of Maynard
Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function
Abstract We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in “almost all” short intervals. It is well known that a relation
Prime numbers in logarithmic intervals
Let X be a large parameter. We will first give a new estimate for the i ntegral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will


On the difference between consecutive primes
[Received 3 August 1937] 1. LET pn denote the nth. prime, and 77(2;) the number of primes p not exceeding x. The existence of an absolute constant 6 < 1 such that a (1) n(x) ~ . log a; when x -> 00,
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value
Zeros ofL-functions
for some fixed A and B. Both (2) and (3) are proved by the use of mean value theorems, the large sieve in the case of (3). The lack of a bound for (1) which is good with respect to both Q and Tmay be