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

Ilija Tanackov; Željko Stević; Ilija Tanackov; Željko Stević

doi:10.3934/math.2023688

AIMS Mathematics

2023, Volume 8, Issue 6: 13556-13571. doi: 10.3934/math.2023688

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Research article

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

Ilija Tanackov ^{1
,
,},
Željko Stević ^{2
,}

1.
Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Novi Sad, Serbia
2.
Faculty of Transport and Traffic Engineering Doboj, University of East Sarajevo, Vojvode Mišića 52, Doboj 74000, Bosnia and Herzegovina

Received: 11 December 2022 Revised: 22 March 2023 Accepted: 29 March 2023 Published: 07 April 2023
MSC : 11M26

Newton's identities of an infinite polynomial with complex-conjugate roots n^−(σ+it) and n^−(σ−it) are multiple zeta functions for n∈[1, ∞), σ∈R and t∈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+it)+ζ(σ−it) = 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 (n!)⁻¹. The general value of the infinite multiple zeta function is calculated based on Vieta's rules and is equal to (n!)^−2σ. The identity based on the relation of the special case and the general case (n!)⁻¹ = (n!)^−2σ is reduced to the equation −1 = −2σ. The value of the critical line for all non-trivial zeros is singular, σ = ½.
- non trivial zeros,
- factorial,
- imaginary unit,
- 3.141592…,
- 2.718281…
Citation: Ilija Tanackov, Željko Stević. Calculation of the value of the critical line using multiple zeta functions[J]. AIMS Mathematics, 2023, 8(6): 13556-13571. doi: 10.3934/math.2023688

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Abstract

Newton's identities of an infinite polynomial with complex-conjugate roots n^−(σ+it) and n^−(σ−it) are multiple zeta functions for n∈[1, ∞), σ∈R and t∈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+it)+ζ(σ−it) = 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 (n!)⁻¹. The general value of the infinite multiple zeta function is calculated based on Vieta's rules and is equal to (n!)^−2σ. The identity based on the relation of the special case and the general case (n!)⁻¹ = (n!)^−2σ is reduced to the equation −1 = −2σ. The value of the critical line for all non-trivial zeros is singular, σ = ½.

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