On Primes in Arithmetic Progression Having a Prescribed Primitive Root

✍ Scribed by Pieter Moree

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 152 KB
• Volume: 78
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2409

✦ Synopsis

Let g # Z"[ &1, 0, 1] and let h be the largest integer such that g is an hth power. Let p be a prime. Put w g ( p)=2.( p&1)Â( p&1) if (g p)=&1 (Legendre symbol) and ( p&1, h)=1 and w g ( p)=0 otherwise, with . Euler's totient. Let a (mod f ) be a primitive residue class. Let ? g (x; f, a) denote the number of odd primes p x such that p#a (mod f ) and g is a primitive root mod p. It is shown, under the GRH, that

Thus the function w g ( p) behaves as if it were some kind of ``probability'' that g is a primitive root mod p.