Note on a Paper by G. J. Rieger

✍ Scribed by Doug Hensley

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 103 KB
• Volume: 179
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19961790107

✦ Synopsis

Let c E IN. For d E Z with gcd (d, c) = 1 let 6(d, c) be defined by d . 6 ( d , c) I 1 mod c, 1 5 6(d, c) 5 c. Let s, t E IN with 1 5 s 5 c, 1 5 t 5 c. The main result is that for arbitrary fixed E > 0, but uniformly over c, s and t, # { d E I N : ( ~, d ) = l , I 5 d S S and 1 < 6 ( d , c ) < t } = -s t + O e ~( c ) ( c t + e ) . C2 Equivalently, the points (d, 6(c, d ) ) are approximately uniformly distributed in (0, c] x [0, c]: The twodimensional discrepancy of { ( d , 6(c, d )): 1 < d < c and (c, d ) = 1) in (0, c] x [O, c] is 0, (ce-ll2).

= c € 1 (d,c)=l, l s d l s l s j < t , d j = l m o d c 1 1991 Mathematics Subject Classification. Primary llK36, llL05. Keywords and phrases. uniform distribution, multiplicative inverse of amod b.