Unformatted text preview: x â‰¡ a k (mod m k ). A system of covering congruences is called exact when for every value of x there is exactly one value of k such that x â‰¡ a k (mod m k ). Show that if the system is exact, then K Â° k =1 z a k 1 âˆ’ z m k = 1 1 âˆ’ z . When z = re (1 /m K ) (where e ( z ) denotes e 2 Ï€iz ) with r âˆˆ R > and r â†’ 1 âˆ’ , show that the left hand side above is âˆ¼ e ( a K /m K ) m K (1 âˆ’ r ) whereas the right hand side is bounded for z in a neighbourhood of e (1 /m K )....
View
Full Document
 Spring '08
 Vaughan,R
 Math, Number Theory, Prime number, Multiplicative function, congruences, Arithmetic function

Click to edit the document details