tag:blogger.com,1999:blog-264226589944705290.post2980002185374579008..comments2021-12-14T05:53:12.175-08:00Comments on God Plays Dice: Conditioned to rationalityMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-264226589944705290.post-62684714082935912462009-03-08T16:04:00.000-07:002009-03-08T16:04:00.000-07:00I have always been fascinated whenever numbers gre...I have always been fascinated whenever numbers greater than 1 show up (which, in my current fields of interest, is infrequent). One of the memorable instances I recall was in the proof in Rudin's book of the polynomial approximation theorem for continuous functions on compact sets which are holomorphic in the interior, where even 10,000 came up.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-22903307067163791672009-03-07T12:03:00.000-08:002009-03-07T12:03:00.000-08:00If you sum a multiplicative function f(n) over int...If you sum a multiplicative function f(n) over integers up to N, then assuming it is such that f(n) doesn't grow faster than a power of n (e.g., typically, if f is bounded on primes, and polynomially bounded on prime powers in terms of the exponent of the prime), then the typical asymptotic will be <BR/><BR/>cN(log N)^{r-1}<BR/><BR/>where r is the average value of f restricted to primes. This can be quite arbitrary (in my recent post <BR/><BR/>http://blogs.ethz.ch/kowalski/2009/02/28/a-beautiful-analogy-2/<BR/><BR/>there is a case where it is is used with r ranging over the whole unit circle...)Anonymousnoreply@blogger.com