Abstract
An equivalent, but variant form of Riemann’s functional equation is explored, and several discoveries are made. Properties of Riemann’s zeta function ζ(s), from which a necessary and sufficient condition for the existence of zeros in the critical strip, are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for arg(ζ(s)) on the critical line s=1/2+iρ, the consequences of which are explored, and the “LogZeta" function is introduced. A singular linear transform between the real and imaginary components of ζ and ζ′ on the critical line is derived, and an implicit relationship for locating a zero (ρ=ρ0) on the critical line is found between the arguments of ζ(1/2+iρ) and ζ′(1/2+iρ). Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent, is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of anomalous zeros, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative ζ′(1/2+iρ) will never vanish on the perforated critical line (ρ≠ρ0). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of ζ(1/2+iρ) as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.
Highlights
The course of other work has led to an exploration of a not-so-well-known variant of Riemann’s functional equation
)(−s), in which case Equation (3.2) can be written. Since it is known (Spira, 1973) that (s) ≠ 0 in the open critical half-strip 0 ≤ R(s) < 1∕2, the following requires that s be constrained to that region, the known symmetry imposed by the functional equation, viz. (1 − s0) = 0 implies (s0) = 0 and the reverse, means that the following can be generalized to R(s) > 1∕2
The imaginary counterpart of Equation (3.9) vanishes. Because this point does not coincide with a zero of (1∕2 + i ), on the critical line, Equation (3.6), or equivalently Equation (3.7), is a sufficient condition for Equation (3.8) to be true, unless 0 = 6.2898..., a constant quoted to many significant figures in de Reyna and Van de Lune (2014, Corollary 9)
Summary
The course of other work has led to an exploration of a not-so-well-known variant of Riemann’s functional equation. In equivalent form, define the normalized right- and left-hand sides of Equation (3.2) by (s) ≡ (ln(2 ) − (s) + tan( s∕2)) (1 − s)∕ (s) 2 Since it is known (Spira, 1973) that (s) ≠ 0 in the open critical half-strip 0 ≤ R(s) < 1∕2, the following requires that s be constrained to that region, the known symmetry imposed by the functional equation, viz. The imaginary counterpart of Equation (3.9) vanishes Because this point does not coincide with a zero of (1∕2 + i ), on the critical line, Equation (3.6), or equivalently Equation (3.7), is a sufficient condition for Equation (3.8) to be true, unless 0 = 6.2898..., a constant quoted to many significant figures in de Reyna and Van de Lune (2014, Corollary 9). This proves necessity subject to the simplicity of a zero
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