The Haros-Farey sequence at two hundred years. A survey.
The Hooley-Huxley contour method for problems in number fields III : frobenian functions
In this paper we study finite valued multiplicative functions defined on ideals of a number field and whose values on the prime ideals depend only on the Frobenius class of the primes in some Galois extension. In particular we give asymptotic results when the ideals are restricted to “small regions”. Special cases concern Ramanujan's tau function in small intervals and relative norms in “small regions” of elements from a full module of the Galois extension.
The hyperbola
We include several results providing bounds for an interval on the hyperbola containing lattice points.
The iterated Carmichael λ-function and the number of cycles of the power generator
The joint distribution of -additive functions on polynomials over finite fields
Let be a finite field and a polynomial of positive degree. A function on is called (completely) -additive if , where and . We prove that the values are asymptotically equidistributed on the (finite) image set
The mean square of the error term in a generalization of Dirichlet's divisor problem
The multiplicative group of rationals generated by the shifted primes, I.
The -th prime asymptotically
A new derivation of the classic asymptotic expansion of the -th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994).Realistic bounds for the error with , after having retained the first terms, for , are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible such that, for , we have where is the sum of the first four terms of the asymptotic expansion.
The number of cube-full numbers in an interval
The number of k-ary divisors of a generalized integer
The number of k-ary, (k + 1)-free divisors of an integer.
The number of k-free and k-ary divisors of m which are prime to n.
The number of k-free divisors of an integer
The number of k-free divisors of an integer. II.
The number of solutions of .
The number of squarefull numbers in an interval
The number of unsieved integers up to x
The Piltz divisor problem in number fields: An improved lower bound by Soundararajan's method
The propinquity of divisors
The range of the sum-of-proper-divisors function
Answering a question of Erdős, we show that a positive proportion of even numbers are in the form s(n), where s(n) = σ(n) - n, the sum of proper divisors of n.