Primes in short intervals

1997 ◽  
Vol 122 (2) ◽  
pp. 193-205 ◽  
Author(s):  
HONGZE LI
Keyword(s):  
Prime Number ◽  
Short Intervals ◽  
Almost All

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.

scholarly journals Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

2017 ◽  
Vol 15 (1) ◽  
pp. 1517-1529
Author(s):  
Zhao Feng
Keyword(s):  
Short Intervals ◽  
Exceptional Sets ◽  
Goldbach Problem ◽  
Almost All

Abstract In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

scholarly journals Vertex-primitive half-transitive graphs

1994 ◽  
Vol 57 (1) ◽  
pp. 113-124 ◽  
Author(s):  
D. E. Taylor ◽  
Ming-Yao Xu
Keyword(s):  
Prime Number ◽  
Infinite Family ◽  
Primitive Groups ◽  
Almost All ◽  
Transitive Graphs ◽  
Vertex Primitive

AbstractGiven an infinite family of finite primitive groups, conditions are found which ensure that almost all the orbitals are not self-paired. If p is a prime number congruent to ±1(mod 10), these conditions apply to the groups P S L (2, p) acting on the cosets of a subgroup isomorphic to A5. In this way, infinitely many vertex-primitive ½-transitive graphs which are not metacirculants are obtained.

Quadratic approximation in ℚp

2014 ◽  
Vol 11 (01) ◽  
pp. 193-209 ◽  
Author(s):  
Yann Bugeaud ◽  
Tomislav Pejković
Keyword(s):  
Prime Number ◽  
Continued Fractions ◽  
Quadratic Approximation ◽  
Almost All

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

scholarly journals Primes in almost all short intervals

1998 ◽  
Vol 84 (3) ◽  
pp. 225-244 ◽  
Author(s):  
Alessandro Zaccagnini
Keyword(s):  
Short Intervals ◽  
Almost All

scholarly journals ON THE DISTRIBUTION OF -FREE NUMBERS AND NON-VANISHING FOURIER COEFFICIENTS OF CUSP FORMS

2012 ◽  
Vol 54 (2) ◽  
pp. 381-397 ◽  
Author(s):  
KAISA MATOMÄKI
Keyword(s):  
Fourier Coefficients ◽  
Cusp Forms ◽  
Short Intervals ◽  
Almost All

AbstractWe study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.

scholarly journals Almost primes in almost all short intervals

2016 ◽  
Vol 161 (2) ◽  
pp. 247-281 ◽  
Author(s):  
JONI TERÄVÄINEN
Keyword(s):  
Short Intervals ◽  
Almost Primes ◽  
Prime Factors ◽  
Almost All ◽  
Positive Integers

AbstractLet Ek be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log1+ϵx] contain E3 numbers, and almost all intervals [x,x + log3.51x] contain E2 numbers. By this we mean that there are only o(X) integers 1 ⩽ x ⩽ X for which the mentioned intervals do not contain such numbers. The result for E3 numbers is optimal up to the ϵ in the exponent. The theorem on E2 numbers improves a result of Harman, which had the exponent 7 + ϵ in place of 3.51. We also consider general Ek numbers, and find them on intervals whose lengths approach log x as k → ∞.

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