Criterion for the Reality of the Logarithmic Derivative of the Zeta Function on the Critical Line and Its Relation to the Riemann Hypothesis and the Law of Distribution of Prime Numbers

This work investigates the relationship between the gamma function, the Riemann zeta function, and the distribution of prime numbers. From the functional equation of the zeta function, an explicit expression is derived connecting the ratio of gamma functions with the product over primes. Differentiation of this relation leads to an equation linking the logarithmic derivatives of the gamma and zeta functions. On the critical line Re s = 1/2, this equation takes the form of a reality condition for ζ′/ζ (1/2 + it).Furthermore, from the gamma function relation, an explicit law for the distribution of prime numbers on the complex plane is derived which represents an exact identity connecting all prime numbers with the argument of the gamma function. Numerical verification of the law is conducted, and the dependence of convergence speed on the parameter t is investigated.The equivalence of the obtained law to the Riemann hypothesis is established, and it is proved that the critical line is the only vertical line on which the logarithmic derivative of the zeta function can be real. A comparison with the classical explicit Riemann formula is made, and a sketch of its derivation from our relation is outlined.Applied aspects are investigated: vulnerabilities of deterministic prime generation schemes in cryptography, connection with Mills’ formula, and relation to random matrix theory. Finally, a spectral operator of prime numbers is constructed, whose trace yields the obtained law, and its connection with the generalized Dirac operator from noncommutative geometry is established.