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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