Monday, 26 March 2007

nt.number theory - Question concerning the arithmetic average of the Euler phi function:

There is information on page 68 of Montgomery and Vaughan's book, and also on page 51 of "Introduction to analytic and probabilistic number theory" by GĂ©rald Tenenbaum. Briefly, Montgomery has established that



$$
limsup_{x rightarrow +infty}frac{R(x)}{xsqrt{loglog(x)}} > 0
$$



and similarly with the limit inferior. So there is only modest room for improvement. Unfortunately I cannot find any reference to an upper bound conditional on RH. On page 40 Tenenbaum has a reference to page 144 of Walfisz' book on exponential sums. Walfisz uses Vinogradov's method to show that



$$
R(x) = Oleft(xlog^{2/3}(x)(loglog(x))^{4/3}right).
$$



I don't own a copy of Walfisz' book, so I have no further details.

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