An ~2-Theorem for Ramanujan's z-Function
نویسندگان
چکیده
This function was first studied by Ramanujan [6]. He wrote, for every prime p, z(p) = 2p I */2 COS Op and conjectured that 0v is real. This was proved by Deligne [2]. it is known that r(p~) =p11~/2 sin(a+ 1) 0p sin 0p If d(n) denotes the number of divisors of n, then it follows that It(n)[ _-< n 11/2 d(n), as r is a multiplicative function. Therefore, for some constant cl >0, { (CI 1ON F/.~ ~ "r(n) O ~nl 1/2 exp \log log n! 1" It is conjectured that [ ( c2 log n_.~ "~(H) O ~nl 1/2 exp \log log n] ] (1)
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