TY - JOUR
T1 - The distribution of spacings between small powers of a primitive root
AU - Rudnick, ZeéV
AU - Zaharescu, Alexandru
PY - 2000
Y1 - 2000
N2 - We examine the distribution of spacings between the first N powers of a primitive root modulo a large prime p, where N ≪ p1-ε. We aim to show that the distribution of these spacings follows the random (or Poisson) model. We show that the m-level correlation functions are Poissonian if JV ≫ p1-1/2m+ε. For the pair correlation function (m = 2) we can do better: the exponent 3/4 may be replaced by 5/7 unconditionally, and by 2/3 on GRH. Moreover, we show unconditionally that the exponent 2/3 is valid for almost all choices of primitive roots.
AB - We examine the distribution of spacings between the first N powers of a primitive root modulo a large prime p, where N ≪ p1-ε. We aim to show that the distribution of these spacings follows the random (or Poisson) model. We show that the m-level correlation functions are Poissonian if JV ≫ p1-1/2m+ε. For the pair correlation function (m = 2) we can do better: the exponent 3/4 may be replaced by 5/7 unconditionally, and by 2/3 on GRH. Moreover, we show unconditionally that the exponent 2/3 is valid for almost all choices of primitive roots.
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U2 - 10.1007/s11856-000-1280-z
DO - 10.1007/s11856-000-1280-z
M3 - Article
AN - SCOPUS:0040487257
SN - 0021-2172
VL - 120
SP - 271
EP - 287
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
ER -