Written: April-May 2018

This paper presents a collection of experimental results regarding the truncated Riemann Zeta function conducted by students in Dr. Z.'s Spring 2018 Experimental Mathematics Class.

ToDo List

• Write efficient programs that find as many as possible minimum points and minimum values for abs(Zeta_N(1/2+I*t))^2, for as many N as possible

• Since abs(Zeta_N(1/2+I*t))^2 (called Ze(N,t) in C23.txt) is "almost" a trigonometic polynomial, but with irrational (in fact transcendental) frequencies, (involving log(n) for n small positive integers), use continued fractions to approximate the log by rational numbers, and make approximations fo Ze(N,t) (for a given N) that is a linear combination of cos(rational*t). These are periodic (with large, but finite, period), so its absolute minimum should be calculable. Then by bounding the "error" possibly establish rigorously the absolute minimum of Ze(N,t), and in particular prove that is strictly positive.

  Added April 29, 2018: Anthony Zaleski pointed out that this approach, taken naively, is doomed to failure (the "approximations" will get worse and worse as t gets larger). Perhaps a more clever variation will work?

• By using naive numerical integration (Maple's evalf(Int(...)) is way too slow), study the asymoptotic behavior as both N and T become large (and they may grow large together, but not necessarily at the same pace) of the k-th moment of abs(Zeta_N(1/2+I*t))^2 for as many k as possible. Do you see any trends? Can you say something about the rate of growth of the k-th moment of abs(Zeta(1/2_I*t))^2 herself?

• You are welcome to think of other, related, problems, e.g. try to implement Ch. 29 of Iwaniec's book

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