The Shape of the Differences of the Complex Zeros of Three-Term Exponential Polynomials In my last blog, I looked at the distribution of the distances of the real zeros of functions of the form with incommensurate , . And after analyzing the real case, I now want to have a look at the differences of the zeros of three-term exponential polynomials of the form for real , , . (While we could rescale to set and for the zero set , keeping and will make the resulting formulas look more symmetric.) Looking at the zeros in the complex plane, one does not see any obvious pattern. But by forming differences of pairs of zeros, regularities and patterns emerge, which often give some deeper insight into a problem. We do not make any special assumptions about the incommensurability of , , . The differences of the zeros of this type of function are all located on oval-shaped curves. We will find a closed form for these ovals. Using experimental mathematics techniques, we will show that ovals are described by the solutions of the following equation: ... where:

Identifying Peaks in Distributions of Zeros and Extrema of Almost-Periodic Functions: Inspired by Answering a MathOverflow Question

One of the Holy Grails of mathematics is the Riemann zeta function, especially its zeros. One representation of is the infinite sum . In the last few years, the interest in partial sums of such infinite sums and their zeros has grown. A single cosine or sine function is periodic, and the distribution of its zeros is straightforward to describe. A sum of two cosine functions can be written as a product of two cosines, . Similarly, a sum of two sine functions can be written as a product of . This reduces the zero-finding of a sum of two cosines or sines to the case of a single one. A sum of three cosine or sine functions, , is already much more interesting.

Fifteen years ago, in the notes to chapter 4 of Stephen Wolfram’s A New Kind of Science, a log plot of the distribution of the zero distances... ... of the zero distribution of ---showing characteristic peaks---was shown.