By Komaravolu Chandrasekharan
ISBN-10: 0387152954
ISBN-13: 9780387152950
ISBN-10: 3540152954
ISBN-13: 9783540152958
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Extra info for Elliptic Functions
Sample text
Removing the brackets [ ] in the first of the above representations for T(x) introduces an error which is, from what we have already proved, O(x). 2 P- log p to bP -z V p(P - t), - pox which is bounded for all x >_ 2, we obtain xj E loge+0(1)}=xlogx+O(x). 2). 1. Variants of Well-Known Arithmetic Inequalities 26 The third desired estimate now follows with an integration by parts. The fourth is essentially an exponentiated form of the third, since log fl (1-p-')-' = Y_ -log(1-p ') = Y p` +c4+O((log x)-').
With certain constants the estimates x '7T(X) << log X, loge=logx+O(l), Y_ psx P psx _ px\1 hold for x ? 2. P = log log x +c, + O (log x)' logx(l+O( P1 logx)/ 24 1. Variants of Well-Known Arithmetic Inequalities These results have mirror proofs. 1). By means of the identity log m = Y_ log p", m > 0, P'11- we have S(x) = I g(m) log m = Y_ log p° t-xp-"I msx g(p°t). 5x0 Y_ g(t)t'. 2). 2). 1), now using the representation log m = Y- log p, P'Im with g(n) identically one. This gives T(x) _ Y- log p a J= mix log m Lp P°-x = x log x+O(x), the last estimate being obtained with an integration by parts.
In the notation of that lemma S= Y 17(p)log p`i Y logpElliptic Functions by Komaravolu Chandrasekharan
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