On Sieve method and Goldbach problem
On sums over primes
On the Brun-Titchmarsh theorem.
On the composition of the arithmetic functions σ and φ
On the constant β(k) in Rosser's sieve
On the coprimality of values of Euler's function at consecutive integers.
On the difference between consecutive prime numbers
On the Difference Between Consecutive Primes.
On the error term in the linear sieve
On the least almost-prime in arithmetic progression
Let denote an almost-prime with at most prime factors, counted according to multiplicity. Suppose that and are positive integers satisfying . Denote by the least almost-prime which satisfies . It is proved that for sufficiently large , there holds This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range in place of .
On the least almost-prime in arithmetic progressions
Let denote a positive integer with at most prime factors, counted according to multiplicity. For integers , such that , let denote the least in the arithmetic progression . It is proved that for sufficiently large , we have This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained
On the Lengths of the Periods of the Continued Fractions of Square-Roots of Integers.
On the non-linear sieve
On the Number of Solutions of p + h = Pr.
On the numbers that are representable as the sum of two cubes.
On the range of Carmichael's universal-exponent function
Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds for all large x, while for φ it is equal to , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.
On the representation of a number in the form x^2+y^2+p^2+q^2 where p, q are odd primes
On the switching principle in sieve theory.
On the third iterates of the φ- and σ-functions
On twin almost primes