ON THE RESIDUE CLASSES OF π(n) MODULO t
نویسندگان
چکیده
The prime number theorem is one of the most fundamental theorems of analytic number theory, stating that the prime counting function, π(x), is asymptotic to x/ log x. However, it says little about the parity of π(n) as an arithmetic function. Using Selberg’s sieve, we prove a positive lower bound for the proportion of positive integers n such that π(n) is r mod t for any fixed integers r and t. Moreover, we generalize this to the counting function of any set of primes with positive density.
منابع مشابه
On equitable zero sums
There is a rich literature on conditions guaranteeing that certain sums of residue classes cover certain other residue classes modulo some integer N . Often it is of particular importance for applications to know that the class 0 mod N can be represented as a sum of the studied residue classes, and the zero class is often the most difficult case. A well-known result along these lines is the fam...
متن کاملSimple Groups Generated by Involutions Interchanging Residue Classes modulo Lattices in Z
We present a series of countable simple groups, whose generators are involutions which interchange disjoint residue classes modulo lattices in Zd (d ∈ N). This work is motivated by the famous 3n + 1 conjecture.
متن کاملOn Multiplicative Semigroups of Residue Classes
The set of residue classes, modulo any positive integer, is commutative and associative under the operation of multiplication. The author made the conjecture: For each finite commutative semigroup, S, there exists a positive integer, n, such that S is isomorphic with a subsemigroup of the multiplicative semigroup of residue classes (mod n). (A semigroup is a set closed with respect to a single-...
متن کاملCounting primes in residue classes
We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x) can be used for computing efficiently π(x, k, l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n ± 1 less than x for several values of x up to 1020 and found a new region where π(x, 4, 3) is less than π(x, 4, 1) near x = 1018.
متن کاملDistribution of Farey Fractions in Residue Classes and Lang–Trotter Conjectures on Average
We prove that the set of Farey fractions of order T , that is, the set {α/β ∈ Q : gcd(α, β) = 1, 1 ≤ α, β ≤ T}, is uniformly distributed in residue classes modulo a prime p provided T ≥ p1/2+ε for any fixed ε > 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013