Uniform distribution of primes having a prescribed primitive root
نویسندگان
چکیده
منابع مشابه
On the density of primes in arithmetic progression having a prescribed primitive root
Let g ∈ Q be not −1 or a square. Let Pg denote the set of primes p such that g is a primitive root mod p. Let 1 ≤ a ≤ f, (a, f) = 1. Under the Generalized Riemann Hypothesis (GRH) it can be shown that the set of primes p ∈ Pg with p ≡ a(mod f) has a natural density. In this note this density is explicitly evaluated. This generalizes a classical result of Hooley.
متن کاملOn primes in arithmetic progression having a prescribed primitive root. II
Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p ≡ a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.
متن کاملBounded gaps between primes with a given primitive root, II
Let m be a natural number, and let Q be a set containing at least exp(Cm) primes. We show that one can find infinitely many strings of m consecutive primes each of which has some q ∈ Q as a primitive root, all lying in an interval of length OQ(exp(C ′m)). This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin’s conjecture. Let...
متن کاملBounded Gaps between Primes with a given Primitive Root
Fix an integer g 6= −1 that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which g is a primitive root. Forty years later, Hooley showed that Artin’s conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley’s analysis into the Maynard–Tao work on bounded gaps between primes. This leads to the following GRH-conditional resu...
متن کاملUniform Bounds for the Least Almost-prime Primitive Root
A recurring theme in number theory is that multiplicative and additive properties of integers are more or less independent of each other, the classical result in this vein being Dirichlet’s theorem on primes in arithmetic progressions. Since the set of primitive roots to a given modulus is a union of arithmetic progressions, it is natural to study the distribution of prime primitive roots. Resu...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1999
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-89-1-9-21