My formal education includes a Bachelor of Science in Electrical Engineering, and I worked for over three decades in the fields of hardware, software, and systems engineering. My formal education exposed me to a good bit of calculus, but I didn’t really become fully aware of the fascinating number theory branch of mathematics until some time later. Consequently explorations into number theory became one of my hobbies for the past two to three decades, and I’d describe myself as a recreational mathematician. I first became aware of the Riemann hypothesis a decade or two ago at which time I read a few popular books on the topic. At the beginning of October of 2015 I returned my attention to the topic and began a fairly intense investigation into prime counting functions and the Riemann hypothesis. The primary result of my investigation at this point in time is the definition of three general methods for derivation of Fourier series for prime counting functions and successful application of these methods to several prime counting functions as illustrated on this website.
I developed the general methods for derivation of Fourier series for prime counting functions in the fall of 2015, and initially applied these methods to the base prime-number counting function, Riemann’s prime-power counting function, and an additional related prime-power counting function. I became aware of the Chebyshev functions at the beginning of 2016, and consequently added the second Chebyshev function to my investigation at that point in time. I more recently (late September 2016) added the first Chebyshev function and an additional related function to my investigation.
After successful derivation of Fourier series for these prime counting functions, my primary objective has been the illustration of how the terms of Riemann’ s formula for his prime-power counting function and von Mangoldt’s formula for the second Chebyshev function evolve from their corresponding Fourier series. I’ve been particularly interested in determining how each of the individual zeta zero terms evolves from these Fourier series as this could potentially lead to a proof of the Riemann hypothesis. I have not yet achieved this goal, but I’ve decided to illustrate some of the results of my investigation at this point in time because I believe these results are potentially of significant importance to the theory of prime numbers.