Visualization of the Eta and Zeta Functions from 0 to 100 on the Critical Line скачать в хорошем качестве

Visualization of the Eta and Zeta Functions from 0 to 100 on the Critical Line

Hi everyone! In this video, I've overlaid the famous Zeta function and its cousin, the Eta function! You can see some really great interactions between the two functions! The Eta function is the same as the Zeta function with the exception that the 1 in the numerator is replaced by an alternating sign, (-1)^(n+1). However, to be equivalent to the Zeta function, the Eta function is divided by the denominator 1-2^(1-s). This is the analytical continuation of the Zeta function. This particular plot shows the value of the two functions overlaid where x=1/2 and t steadily increases from 0 to around 100. As many of you may already know, this value of x is of great importance as it is the value of the real part of s where all values of t form a vertical line in the complex plane through x=1/2 and is known as the Critical Line. As the plot moves through the origin of the plot, these are known as zeros of the functions. Note the imaginary value as the yellow dots lie closest to the origin. You can correlate these approximate values with an absolutely wonderful resource provided online by Andrew Odlyzko from his work at the University of Minnesota. http://www.dtc.umn.edu/~odlyzko/zeta_... I state approximate because these values are irrational and go on forever, also, I may be off due to the step value in which I increase t, at around 0.0192. I fade the lines as the plots continues to reduce the amount of clutter from the plots. Also, note that the zeros of the Zeta function are identical to the zeros of the Eta function. This is due to the analytically continued form of the Zeta function in which Zeta(s) = Eta(s)/1-2^(1-s). If you write this as simple fractions, it makes sense why the zeros would match. If a = b/c, a = 0 when b = 0. Coincidentally, the denominator here never equals zero and so is continuous. For details on the individual functions, you can click over to my other videos on just these functions: Riemann Zeta Function -    • The Riemann Zeta Function from 0 to 100 on...   Dirichlet Eta Function -    • The Dirichlet Eta Function from 0 to 100 o...   Enjoy! Andrew T Gonzalez