Transcendental Zeta Function
The transcendental zeta function is a unique type of zeta function that is composed of two divergent sub-functions. What makes the transcendental zeta function particularly intriguing is that the difference between these two infinite values yields a finite result. This remarkable property holds true for any complex number "s" as long as its real part is greater than zero and "s" is not equal to one. In essence, the transcendental zeta function is an excellent example of how seemingly divergent and infinite series can be manipulated to produce fascinating and unexpected outcomes. This function has found applications in many areas of mathematics, physics, and engineering, and has been the subject of intense study by researchers across various fields.
To avoid trivial mistakes, it is critical to understand that no specific symbol exists that distinguishes the analytic continuation of the Riemann zeta function and the zeta function itself. Therefore, we rely on the reader to distinguish and recognize the difference between the classical definition of the Riemann zeta function as the infinite sum 1/1^s + 1/2^s + 1/3^s + ... = ∑(n=1 to infinity) 1/n^s , and its analytic continuation to the critical strip, which we denoted as ζ(s) and left the Riemann zeta function in its original form before analytic continuation, written in the summation format.
Notice the symmetry of this function (Re(S)=Re(1-S) => Re(S) =1/2). This new function helps us understand Riemann’s thought process and why he proposed his famous hypothesis. We believe this new zeta function proves the Riemann Hypothesis (See Riemannarticle’s last Theorem article for detail).
We offer a $10,000 prize if you can disprove this function by providing a numeric counterexample. Also, we offer a $10,000 referral bonus if you can help us find a person who can give us a numeric counterexample.
See below for proof of the above function. Also, see the ABC Zeta Function for a second proof of this amazing function. (See Riemann's Last Theorem article for detail).
The proof for the function mentioned above is presented below. Moreover, there is a second proof for this function utilizing the ABC Zeta Function, which is a generalized form of the Riemann Zeta Function. Further information on Transcendental Zeta Function can be found in the Riemann's Last Theorem Article.The function discussed here is a significant result in the field of mathematics, specifically in number theory. The proof offered involves a rigorous mathematical argument and simple mathematical techniques. The ABC Zeta Function implies that this zeta function is linked to other zeta functions, which are essential objects in number theory. The existence of multiple proofs for this zeta function demonstrates its importance and significance in mathematics.This indicates that this function and its proof are valuable contributions to the field of mathematics, with potential implications for various areas of research.
The video below provides valuable insights into the connections between three important zeta functions in mathematics: the Riemann zeta function, the Transcendental zeta function, and the ABC zeta function. These functions are known to have a significant role in number theory . The Transcendental zeta function and the ABC zeta function have been shown to be intimately related to the Riemann's zeta function and have played a significant role in advancing our understanding of the Riemann Hypothesis. By highlighting the connections among these zeta functions, the video provides a valuable insight into the complexity of the Riemann Hypothesis and the various tools that mathematicians have developed to tackle this challenging problem. Overall, the video is a great resource for anyone interested in the fascinating world of number theory and the search for solutions to some of the most important mathematical problems of our time.
How can we eliminate irrelevant information and focus on the core essence?
The solution lies in the Transcendental Zeta function, which enables us to eliminate infinite elements and uncover the golden nugget.
You can see major benefits of the Transcendental Zeta Function in explaining why we believe that we can remove infinity from infinity and obtain a finite number. Please note that this function wasn't available at the time of this video. This remarkable new zeta function provides compelling evidence supporting the claim made by Professor Edward Frenkel regarding the removal of infinite dirt and the attainment of the gold nugget within the Critical Strip.
The Transcendental Zeta Function is Analytic
Author: Riemann's Last Theorem
Date:
For simplicity, let the ABC zeta function \(\zeta_{abc}(s)_b\) be defined as:
where \( b \) is a parameter depending on \( k \).
Since \(\zeta_{abc}(s)_1\) is equal to the Riemann zeta function, which is holomorphic and thus analytic (for one variable only), for \( b=1 \) it follows that:
where \(\zeta(s)\) is known to be analytic.
Now, assume that \(\zeta_{abc}(s)_b\) is analytic for some integer \( b = k \). That is, we assume the function is analytic for \( b = k \), i.e.,
which is holomorphic and thus analytic.
Now, consider the case when \( b = k + 1 \). We have:
Expanding the sum:
This can be rewritten as:
Since all functions involved are holomorphic, they are analytic. By the principle of mathematical induction, if \(\zeta_{abc}(s)_k\) is analytic, then \(\zeta_{abc}(s)_{k+1}\) is also analytic. Therefore, by induction, the transcendental zeta function \(\zeta_{abc}(s)_b\) is analytic since the ABC zeta function is analytical for any arbitrarily large \( b \in \mathbb{N} \).
Note: For a holomorphic function of a single complex variable, the function is differentiable at every point in a region. A fundamental result in complex analysis is that holomorphic functions of one variable are also analytic. This means that if a function is holomorphic in a region, it can be locally expressed as a convergent power series around any point in that region. The reason for this is that holomorphic functions satisfy the Cauchy-Riemann equations, which not only guarantee differentiability but also ensure the function is representable by a power series. Thus, for functions of a single complex variable, being holomorphic is equivalent to being analytic.