# Incredibly accurate recursions for the Riemann Zeta function

Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here.

During some statistical research about iterated convolutions and moving averages, I came up by chance with surprising results for the Dirichlet Eta function related to $$\zeta(s)$$, and this spiked again my interest about its complex roots in the critical strip.

It started like this. Let $$X(t)$$ be a time-discrete time series. Apply the following transformation: $$Y(t)=\sum_{k=-N}^N h(k)X(t-k), \mbox{ with } \sum_{k=-N}^N h(k)=1$$

so that $$Y = h*X$$, where $$*$$ denotes the discrete convolution operator. Iterate $$n$$ times: $$Y_{n+1} = h * Y_n$$ with $$Y_0=X$$ and $$Y_1=Y$$. Denote as $$h_n$$ the $$n$$-fold self-convolution of $$h$$, with $$h_1=h$$, $$h_2=h * h$$, and so on. In short, $$Y_n(t)=\sum_{k=-N_n}^{N_n} h_n(k)X(t-k), \mbox{ with } \sum_{k=-N_n}^{N_n} h_n(k)=1.$$ Start with $$N=N_1=1$$ and $$h_1(-1) = h_1(0) = h_1(1) = \frac{1}{3}$$. Clearly, the properly scaled $$h_n(k)$$ coefficients have a bell shape, just like the Binomial coefficients. Indeed, they are known as trinomial coefficients, see here. The scaling factor is $$O(\sqrt{n})$$: this is a direct application of the central limit theorem. Note that $$N_n=n$$. I wanted to use business data to illustrate the concept, but instead ended up using the real and imaginary part of $$\eta(s)$$, with $$s=0.75 + ti$$ a complex number. So in what follows, $$X(t)$$ represents the real or imaginary part of the complex Dirichlet Eta function $$\eta(s)$$.

Applying $$h_n$$ with $$n=60$$ to $$X(t)$$ yields the result illustrated in the picture below, after proper scaling.

The blue curve at the bottom is $$X(t)$$, in this case the real part of $$\eta(s)$$ for integer values $$60\leq t \leq 240$$. The red curve is $$h_{60} * X$$ (convolution) on the same domain. It is very well approximated by a cosine function, so well that you can not visually tell the difference between $$h_{60} * X$$ and its approximation in the upper part of the picture: the two curves are almost perfectly super-imposed. This is caused by the choice of $$h_n$$ and its connection to the central limit theorem (my guess); ordinary weighted moving averages don't work. If instead you consider the imaginary part of $$\eta(s)$$, you get the same kind of graph with same scaling factor, but this time the approximation is with a sine function.

Also, the real part of $$\eta(s)$$ investigated here is

$$X(t)=\sum_{k=1}^\infty (-1)^{k+1}\frac{\cos(\frac{1}{2} t\log k)}{k^{0.75}}.$$ For the imaginary part, replace cosines by sines. You can not replace $$\log k$$ by (say) $$\sqrt{k}$$ or will lose the near-perfect fit. If you replace $$\log k$$ by a function growing too fast, the behavior of the approximation becomes chaotic. This is illustrated below if you replace $$\log k$$ by $$\sqrt{k}$$ in the definition of $$X(t)$$:

The dips are not a glitch in my computations. I also checked values of $$t$$ up to $$t=600$$, with still a perfect fit for the real and imaginary parts of $$\eta(s)$$, but with additional dips and bumps if you replace $$\log k$$ by (say) $$\sqrt{k}$$.

Question

Could this type of formula be of any use to study the zeroes of the Riemann Zeta function? How do you explain such a coincidence (the excellent approximation)? Is my formula also true for $$t>600$$? Is it peculiar to $$\eta(s)$$? How to generalize to continuous convolutions?

• The coefficients $h_n(k)$ are known as triangular coefficients, see oeis.org/… Jan 24, 2021 at 10:54