New Erdös-Kac type theorems

Abstract.

Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the type $Q(\zeta _q ,\sqrt[q]{a}),$ we prove the following prime analogue of a conjecture of Erdös & Pomerance (1985) concerning the exponent function f _a (p) (defined to be the minimal exponent e for which a^e ≡ 1 modulo p):

$$ \mathop {\lim }\limits_{x \to \infty } \frac{{F_a (x;A,B)}} {{^{\pi (x)} }} = \frac{1} {{\sqrt {2\pi } }}\int\limits_A^B {e^{ - \frac{1} {2}t^2 } dt,} $$

((‡))

where

$$ F_a (x;A,B): = \# \left\{ {p\,\underline{\underline < } \,x\;:\;A\,\underline{\underline < } \frac{{\omega (f_a (p)) - \log \log p}} {{\sqrt {\log \log p} }}\underline{\underline < } \;B} \right\}. $$

The main result is obtained by computing all the higher moments corresponding to ω(f _a (p)), and by comparing them, via the Fréchet-Shohat theorem, with estimates due to Halberstam concerning the moments of ω(p − 1).