Multiplicative Properties of Integers in Short Intervals

In this thesis we consider multiplicative properties of integers in short intervalsusing techniques involving exponential sums, sieve methods and a wide variety of other principles from analytic number theory. The existence of products of three pairwise coprime integers are investigated in short intervals of the form (x, x + x 1/2 ]. A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. The author’s result has been published in a Journal of the London Mathematical Society [22]. A particular case of relevance to elliptic curve cryptography, when all three integers are of order x 1/3 , is then presented as a corollary to this result. The techniques used in the proof include Fourier series for fractional parts and bounds for an exponential sum. We investigate the sum of differences between consecutive primes where the gap between these consecutive primes is greater than x 1/2−∆ for some fixed number 0 < ∆ < 1/48 and show by using Dirichlet polynomials and the sieve of Harman that Σ pn+1−pn>x1/2−∆ x≤pn≤2x pn+1 − pn x 2/3+5∆and thereby generalise an existing result corresponding to ∆ = 0. We showthis bound provides significant improvements to several existing results forconstant 0 < ∆ ≤ −3 + 16√327 = 0.01385...We establish a corollary which further improves the currently establishedbound on the sum of squared differences between consecutive primes in certain intervals.By applying the result on sums of differences between consecutive primeswe prove the existence of a significantly improved form of a prime representingfunction. We show that there exists α > 2 and β = 1/(12 + ∆) for 0 < ∆ ≤−3 + 16√327 such that the sequence[αβn] for α > 2 is prime for all n ∈ Nthereby reducing best known present result β = 2 in the exponent to 1/(12 +∆) = 1.946067... .We also establish the existence of a prime representing function whichonly takes values which are primes in Beatty sequences [mξ +η] for irrationalξ > 1 and η ∈ R.