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introducing a wave function $W$ in the series, to replace the cosine function (itself a particular case). But we still keep $\lambda(n)=n$. The goal is to show that RH can be generalized in a new way, different (I think) from the Generalized Riemann Hypothesis (GRH). The classic GRH involves L-functions while my version does not. This offers more hopes to solve Riemann's conjecture, by first trying to prove it for the easiest $W$, and understand what those $W$'s having a RH attached to them (as opposed to those that do not) have in common. In the three examples below, $\alpha=0, \beta=1$, while $\gamma=0$ for the $\phi_1$ series attached to the real part, and $\gamma=-\frac {\pi}{2}$ for the shifted $\phi_2$ series attached to the imaginary part. The classic RH corresponds to the cosine wave: $W(x)=\cos x$.

introducing a wave function $W$ in the series, to replace the cosine function (itself a particular case). But we still keep $\lambda(n)=\log n$. The goal is to show that RH can be generalized in a new way, different (I think) from the Generalized Riemann Hypothesis (GRH). The classic GRH involves L-functions while my version does not. This offers more hopes to solve Riemann's conjecture, by first trying to prove it for the easiest $W$, and understand what those $W$'s having a RH attached to them (as opposed to those that do not) have in common. In the three examples below, $\alpha=0, \beta=1$, while $\gamma=0$ for the $\phi_1$ series attached to the real part, and $\gamma=-\frac {\pi}{2}$ for the shifted $\phi_2$ series attached to the imaginary part. The classic RH corresponds to the cosine wave: $W(x)=\cos x$.

Final update on 1/9/2021: As I make more progress and to avoid too many edits on my question, I posted my most recent findings, generalizing the material below, as a new question, here.

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.

Update on 1/5/2020: I added the section "more interesting results" featuring a generalization of RH very different from the current versions (not involving L-function), and spectacular plots for the orbit as well as the error when using only the first 200 terms in the series representing $\phi$

The orbit consists, for a fixed $\sigma$, of the points $(X(t),Y(t))$ with $X(t)=\phi(\sigma,t;0,1,0)$ and $Y(t)=\phi(\sigma,t,0;1,-\frac{\pi}{2})$. The lower three plots represent the error between the true value $(X(t),Y(t))$ and its approximation based on using only the first 200 terms in the series that defines $\phi$. The error distribution is very surpring; I was expecting the points to be radially but randomly distributed around the origin; instead, they are located on a ring. Note that for $t>600$ (and for the triangular wave, for $t>80$) you need to use more than 200 terms for the pattern to remain strong.

The left part of the plot corresponds to the cosine wave (that is, classical RH), the middle part corresponds to the triangular wave, and the right part corresponds to the alternating quadratic wave.

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.

Update on 1/5/2020: I added the section "more interesting results" featuring a generalization of RH very different from the current versions (not involving L-function), and spectacular plots for the orbit as well as the error when using only the first 200 terms in the series representing $\phi$

The orbit consists, for a fixed $\sigma$, of the points $(X(t),Y(t))$ with $X(t)=\phi(\sigma,t;0,1,0)$ and $Y(t)=\phi(\sigma,t,0;1,-\frac{\pi}{2})$. The lower three plots represent the error between the true value $(X(t),Y(t))$ and its approximation based on using only the first 200 terms in the series that defines $\phi$. The error distribution is very surpisring; I was expecting the points to be radially but randomly distributed around the origin; instead, they are located on a ring. Note that for $t>600$ (and for the triangular wave, for $t>80$) you need to use more than 200 terms for the pattern to remain strong.

The left part of the plot corresponds to the cosine wave (that is, classical RH), the middle part corresponds to the triangular wave, and the right part corresponds to the alternating quadratic wave. Interestingly, when $\sigma=\frac{1}{2}$ the orbit does not have a hole anymore as predicted, yet the error points are still distributed on a similar ring.