Number Theory


On the Riemann Hypothesis, Complex Scalings and Logarithmic Time Reversal

Authors: Carlos Castro

An approach to solving the Riemann Hypothesis is revisited within the framework of the special properties of $\Theta$ (theta) functions, and the notion of $ {\cal C } { \cal T} $ invariance. The conjugation operation $ {\cal C }$ amounts to complex scaling transformations, and the $ {\cal T } $ operation $ t \rightarrow ( 1/ t ) $ amounts to the reversal $ log (t) \rightarrow - log ( t ) $. A judicious scaling-like operator is constructed whose spectrum $E_s = s ( 1 - s ) $ is real-valued, leading to $ s = {1\over 2} + i \rho$, and/or $ s $ = real. These values are the location of the non-trivial and trivial zeta zeros, respectively. A thorough analysis of the one-to-one correspondence among the zeta zeros, and the orthogonality conditions among pairs of eigenfunctions, reveals that $no$ zeros exist off the critical line. The role of the $ {\cal C }, {\cal T } $ transformations, and the properties of the Mellin transform of $ \Theta$ functions were essential in our construction.

Comments: 13 Pages. Submitted to Mod. Phys. Letts A

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Submission history

[v1] 2017-05-09 06:38:25
[v2] 2017-05-09 20:04:55
[v3] 2017-05-13 03:50:38

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