The first "few" partial sums of this series are plotted
in the complex plane. This is a somewhat more visually
interesting version of a picture in the third of
H.L.Montgomery's * Ten Lectures on the Interface
between Analytic Number Theory and Harmonic Analysis *
(cited in the Cebysev notes).
Count the whorls and estimate the length of the sum.
Or cheat by consulting the
PostScript source. **Warning:** Real numbers
in PostScript have limited precision, which will eventually
foil attempts to extend this picture to longer sums or
adapt it to faster-growing functions. To generate similar
pictures of the sums of e(f(n)) for much larger values of f(n),
compute the fractional parts of f(n) recursively using power-series
approximations to the first or (if necessary) higher finite
differences of f - which after all is also a key ingredient
in our analytic estimates on such exponential sums.

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