u/rhackbar

For better or for worse, I have become a somewhat regularly contributor to r/numbertheory. This time I am back with what I think is a pretty amazing result, described in the linked paper, and I wanted to also provide a tool for you to explore the result as well.

Sage Cell Server is a web-based math system that lets you run python scripts without needing to login or install anything - https://sagecell.sagemath.org/ . Thanks to PeakMath on YouTube for introducing me to SageMath.

You can try my function and plot the results by copying and pasting the below code:

####################################################################

import pylab as plt


def ps_euler_product(b,u):
#function takes imaginary input 'b', and upper limit on euler product 'u'
    
    r = (1/(2-2^(.5-b*i))) * prod([1/(1-(1/(j^(.5+b*i)))) for j in list(Primes(modulus=0, classes=range(u)))])
    
    return r
 
b_values = numpy.arange(10, 35, .1).tolist() 
#Range of b values to iterate over

#Separate into real and imaginary parts for easier plotting
result_real = [ps_euler_product(b,500).real() for b in b_values]

result_imag = [ps_euler_product(b,500).imag() for b in b_values]

#plotting results
plt.plot(b_values,result_real, color = 'blue', linestyle = '-')
plt.plot(b_values,result_imag, color = 'red', linestyle = '--')

major_ticks = numpy.arange(10, 36, 5)
minor_ticks = numpy.arange(10, 36, 1)
plt.xticks(major_ticks)
plt.xticks(minor_ticks, minor=True)

#Customizing plot colors and style
plt.axvline(x=14.134, color = 'green', linestyle = 'dotted', linewidth = '1')
plt.axvline(x=21.022, color = 'green', linestyle = 'dotted', linewidth = '1')
plt.axvline(x=25.01, color = 'green', linestyle = 'dotted', linewidth = '1')
plt.axvline(x=30.424, color = 'green', linestyle = 'dotted', linewidth = '1')
plt.axvline(x=32.935, color = 'green', linestyle = 'dotted', linewidth = '1')

plt.grid(which='minor', alpha=0.2)
plt.grid(which='major', alpha=0.5)

plt.show()

####################################################################

Hello Numbertheory,

I'm here to present an equation I derived from the Dirichlet Eta function. It is a Taylor Series times a Euler Product which converges to zero at the Zeroes of the Zeta function. The exact equation is:

https://preview.redd.it/gfuqjchm3oug1.png?width=253&format=png&auto=webp&s=bf75a2df274655cc397e543e3dada1d7d0dd2abb

I also posted a video explaining the equation: https://www.youtube.com/watch?v=izpTymRUFO8.

I have had a suspicion that a a converging Euler Product would be needed, but I had to go back to fundamentals to find a path forward.