Abstract: dvi (3K), pdf (84K), ps (29K).
Paper: pdf (932K).
Review by Daniel Shanks in Mathematics of Computation 30 (1976), 379: pdf (144K).
Earlier review by Daniel Shanks in Mathematics of Computation 29 (1975), 331: pdf (83K).
2. Page 45 mentions an empirical standard deviation (say C) of about 0.21.
s2 (corresponding to Riemann's approximation
R(x)) appears to have mean 0 and standard
s1 (corresponding to Gauss's approximation L(x)) appears to have mean 1 and standard deviation C.
Thus on average Riemann's approximation R(x) is better than Gauss's approximation L(x).
This contrasts with the fact that, from Littlewood's "Omega" result (page 44), the two approximations are about equally accurate in the worst case.
3. Assuming the Riemann Hypothesis,
where gamma is Euler's constant, so
This constant is related to a constant occurring in the Hadamard product formula for the Riemann zeta function zeta(s): the exponential factor in the Hadamard product is exp((-C2/2)s).
An equivalent expression for C2/2 is the sum of the reciprocals of the complex zeros of the Riemann zeta function (summed in the natural order to ensure convergence).
Thanks to Andrey Kulsha, David Broadhurst and Mike Oakes
(November 2006) for these
observations on the constant C.
For more see http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0611&L=nmbrthry&T=0&P=1333
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