The transcendental zeta function is a zeta function that is made from two divergence sub-functions. The fascinating part of the transcendental zeta function is the difference of two infinite values gives a finite value. (Re(s) > 0, s≠1) .
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 Riemann’s last Theorem article for detail).
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In the video below , you will see the connection among Riemann’s, Transcendental and ABC zeta functions. These functions give us the last key, which we need it to prove the Riemann Hypothesis.