Historically was the Zeta function something that was just "arbitrarily defined" or did it come about through ring theory in the sense of the video? I've seen the Zeta function presented in several ways. The usual way is simply to poof it out of thin air of course. One simply defines it as the typical sum or product. The second way I've seen it presented is through the generalization of the Gamma function typical of Riemann. The third way is through Dirichlet series which obviously begs the question. The fourth way is given in the video but likely not the origin of the Zeta function. Obviously it is very likely people "discovered" the zeta function by simply toying around long before anyone knew what to call it but I'm curious if the "ring theory version" was known long before or if it is relatively recent. My guess is that it is 20th century and that, in some sense, some of the ring theory mathematics were constructed "around" the zeta function in trying to formulate a solution to the RH. Euler in the 1700's obviously was toying around with the Zeta function as were some of those before him but these were generally for specific values and without any real understanding of any deeper connection to number theory. From best I can tell is that the "ring theory" version simply fits ring theory to the zeta function to connect it. Since the zeta function is defined in terms of primes there is the obvious connection to finite fields. That is the link that brings the Zeta function under the umbrella of ring theory which enables one to view the Zeta function in a much greater light.
I believe the zeta functions for curves at least was first studied by Artin and I'm sure he would have been well aware of the connections with Riemann's zeta function. The latter I believe was first studied by Euler who loved to play with series in general. His representation of the function as an infinite product was no doubt a strong motivating factor for studying it.
@@DanielChanMaths of course is 100% right, and to add a couple specifics on the early days: Euler first considered the specific series representation of zeta(2) and calculation of its value in his solution to the Basel problem, and proceeded to generalize the foundations of analytic number theory including the computation of the Euler-Mascheroni constant and concepts like his totient function. Riemann would expand greatly on this later in his 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" or "On the Number of Primes Less than a Given Magnitude" in which he defined his zeta function and specified the relationship with prime numbers and complex analysis in detail. Apologies for any errors. I am not a university graduate and simply love math and math history.
thanks for uploading
Just found your channel, time to binge all your videos 😁😁
Historically was the Zeta function something that was just "arbitrarily defined" or did it come about through ring theory in the sense of the video? I've seen the Zeta function presented in several ways. The usual way is simply to poof it out of thin air of course. One simply defines it as the typical sum or product. The second way I've seen it presented is through the generalization of the Gamma function typical of Riemann. The third way is through Dirichlet series which obviously begs the question. The fourth way is given in the video but likely not the origin of the Zeta function. Obviously it is very likely people "discovered" the zeta function by simply toying around long before anyone knew what to call it but I'm curious if the "ring theory version" was known long before or if it is relatively recent. My guess is that it is 20th century and that, in some sense, some of the ring theory mathematics were constructed "around" the zeta function in trying to formulate a solution to the RH. Euler in the 1700's obviously was toying around with the Zeta function as were some of those before him but these were generally for specific values and without any real understanding of any deeper connection to number theory.
From best I can tell is that the "ring theory" version simply fits ring theory to the zeta function to connect it. Since the zeta function is defined in terms of primes there is the obvious connection to finite fields. That is the link that brings the Zeta function under the umbrella of ring theory which enables one to view the Zeta function in a much greater light.
I believe the zeta functions for curves at least was first studied by Artin and I'm sure he would have been well aware of the connections with Riemann's zeta function. The latter I believe was first studied by Euler who loved to play with series in general. His representation of the function as an infinite product was no doubt a strong motivating factor for studying it.
@@DanielChanMaths of course is 100% right, and to add a couple specifics on the early days: Euler first considered the specific series representation of zeta(2) and calculation of its value in his solution to the Basel problem, and proceeded to generalize the foundations of analytic number theory including the computation of the Euler-Mascheroni constant and concepts like his totient function. Riemann would expand greatly on this later in his 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" or "On the Number of Primes Less than a Given Magnitude" in which he defined his zeta function and specified the relationship with prime numbers and complex analysis in detail.
Apologies for any errors. I am not a university graduate and simply love math and math history.