On the power-free parts of consecutive integers
On the powerful part of
We show that is powerfull for integers at most, thus answering a question of P. Ribenboim.
On the prime number theorem for the coefficients of certain modular forms
On the probability that k generalized integers are relatively H-prime
On the probability that n and f (n) are relatively prime II
On the probability that n and f(n) are relatively prime
On the probability that n and f(n) are relatively prime III
On the problem of divisors
On the problem of Goldbach's type.
On the product of divisors of and of .
On the products of Hecke L-functions of holomorphic cusp forms
On the proof of the prime number theorem
On the properties of oscillation and almost periodicity of certain convolutions
On the quantitative unit sum number problem-an application of the subspace theorem
On the quotient sequence of sequences of integers
On the -free values of the polynomial
Let be a fixed integer. We study the asymptotic formula of , which is the number of positive integer solutions such that the polynomial is -free. We obtained the asymptotic formula of for all . Our result is new even in the case . We proved that , where is a constant depending on . This improves upon the error term obtained by G.-L. Zhou, Y. Ding (2022).
On the radical of a perfect number.
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 remainder term of the prime number formula I. On a problem of Littlewood
On the remainder term of the prime number formula II; On a theorem of Ingham