Calculation of the value of the critical line using multiple zeta functions

Abstract

<abstract> <p>Newton's identities of an infinite polynomial with complex-conjugate roots <italic>n</italic>^{<italic>−(</italic>σ+<italic>it</italic>)} and <italic>n</italic>^{<italic>−(</italic>σ<italic>−it</italic>)} are multiple zeta functions for <italic>n</italic>∈[1, ∞), σ∈R and <italic>t</italic>∈R. All Newton's identities can be represented by Macdonald determinants. In a special case of the Riemann hypothesis, the multiple zeta function of the first order is equal to zero, ζ(σ+<italic>it</italic>)+ζ(σ−<italic>it</italic>) = 0. The special case includes all non-trivial zeros. The value of the last, infinite multiple zeta function, in the special case, changes the structure of the determinant that can be calculated. The result is the reciprocal of the factorial value (<italic>n</italic>!)⁻¹. The general value of the infinite multiple zeta function is calculated based on Vieta's rules and is equal to (<italic>n</italic>!)^−2σ. The identity based on the relation of the special case and the general case (<italic>n</italic>!)⁻¹ = (<italic>n</italic>!)^−2σ is reduced to the equation −1 = −2σ. The value of the critical line for all non-trivial zeros is singular, σ = ½.</p> </abstract>