On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant H 3 X ^[7/24]+7dH\ge X^{\,{7\over24}+7\delta} we have |E(X,H) | << H^1-d/600\vert E(X,H) \vert \ll H^{1-\delta/600} .     

Explicit relations between pair correlation of zeros and primes in short intervals

In this paper we obtain a quantitative version of the well known theorem by Goldston and Montgomery about the equivalence between the asymptotic behaviors of the mean-square of primes in short intervals, and of the pair-correlation function of the zeros of the Riemann zeta function.     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

Short intervals asymptotic formulae for binary problems with primes and powers,I: density 3/2

We investigate two kinds of q-analogues of generalized Stirling numbers. One is a q-analogue of Hsu-Shiue’s generalized Stirling numbers and the other is a q-analogue of Comtet’s numbers. In particular, we derive a q-analogue of an equality of Dobinski-type. Moreover, a determinant of the matrix consisting of the q-analogue of Comtet’s numbers is evaluated.     

On an average ternary problem with prime powers

The Ramanujan Journal - We continue our work on averages for ternary additive problems with powers of prime numbers in Languasco and Zaccagnini (J Number Theory 159:45–58, 2016; Rocky...     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

On the sum of two primes and k powers of two

Let X be a large integer. We prove that, for any fixed positiveinteger k, a suitable asymptotic formula for the number of representationsof an even integer N [1,X] as the sum of two primes and k powersof 2 holds with at most exceptions.