Asymptotically Exact Heuristics for (Near) Primitive Roots
✍ Scribed by Pieter Moree
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 211 KB
- Volume
- 83
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let g # Q"[ &1, 0, 1]. Let p be a prime. Let ord p (g) denote the exponent of p in the canonical factorization of g. If ord p ( g)=0, we define r g ( p)=[(ZÂp Z)* : (g mod p) ], that is, r g ( p) is the residual index mod p of g. For an arbitrary natural number t we consider the set N g, t of primes p with ord p ( g)=0 and r g ( p)=t. Write g=\g h 0 , where g 0 is positive and not an exact power of a rational. We introduce a function w g, t ( p) # [0, 1, 2], for which it is proved, under the Generalized Riemann Hypothesis (GRH), that N g, t (x)=(h, t) :
.(( p&1)Ât) p&1
+O \
x log log x log 2 x + , where N g, t (x) is the counting function for N g, t . This modifies the naive and, under GRH, false heuristic in which one takes w g, t ( p)=1.