Abstract
To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.
Highlights
In his article [1] from 1859 Bernhard Riemann expressed the conjecture
To the Riemann hypothesis, we investigate first the approximation by stepwise Omega functions Ω (u ) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ ( z)
The Omega function is supposed to be a monotonically decreasing function up to zero in infinity for u → ∞ and it is stated that the Xi function possesses zeros only on the imaginary axis y if the Omega function is not a step-wise constant function with commensurable step lengths
Summary
With pn ,(n = 1, 2, ) the (ordered) sequence of prime numbers and extended by him to complex variable s= σ + it possesses nontrivial zeros only on the imaginary axis s=. 1 ) that remained unproved up to now This function was already known to Euler (Euler product) and using the uniqueness of the prime-number decompositions of natural numbers n = 1, 2,3, Euler established in about 1737 the connection to the sum form. Its main importance is as Riemann showed in [1] that one can derive from it approximations for the prime-number distribution. With respect to the nontrivial zeros it is fully equivalent to the zeta function ζ (s) and, in addition, it excludes the only pole (a simple one) of the last at s = 1. There are many known functions with zeros only on the imaginary axis, in particular, the entire modified Bessel functions Iν ( z )
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