Lyapunov exponents for the map that passes through the non-trivial zeros of Riemann zeta-function
LLyapunov exponents for the map that passes through the non-trivial zeros of Riemann zeta-function
J. L. E. da Silva [email protected]
Lab. of Quantum Information Technology, Department of Teleinformatic Engineering – Federal University of Ceara - DETI/UFC, C.P. 6007 – Campus do Pici - 60455-970 Fortaleza-Ce, Brazil.
Abstract
The Riemann Hypothesis is the main open problem of Number Theory and several scientists are trying to solve this problem. In this regard, in a recent work [8], a difference equation has been proposed that calculates the nth non-trivial zero in the critical range. In this work, we seek to optimize this estimation by calculating Lyapunov numbers for this non-linear map in order to seek the best value for the bifurcation parameter. Analytical results are presented.
Key words – Riemann zeta function. Lambert W function. Lyapunov exponents. The Riemann ( ) z function is a central point in the study of the Riemann Hypothesis [1]. This function has a close relationship with the prime numbers P ( ) (1 ) z zn p P z n p − − −= = = − . (1) Its analytical continuity can be obtained through ( ) ( ) ( ) 1 2 1 nz zn z n − −− −= = − − , (2) for 0 ≤ Re ( z )<1 in the critical range. . Statistics about the non-trivial zeros of the function ( ) z Statistics in the critical range of zeros of the Riemann function is one of the possibilities for current research on the Riemann Hypothesis [2]. All non-trivial zeros of ( ) z are known to be complex numbers with positive imaginary part in the form 𝜌 = 1/2 + 𝑖𝑡 𝑛 with 𝑛 ≤𝑡 . The number of non-trivial zeros with multiplicity of the function up to a range t can be calculated using [3] ( ) ( )
7( ) log2 2 2 8 t t tN t S t O t − = − + + + , (3) where S ( t ) represents the argument function [4] S t it = + . (4)
In [5], França and Leclair investigated the non-trivial zeros statistic through the solution of transcendental equation n n n t t it ne + → + + + = − , (5) for (𝑛 = 1,2, . . . , 𝑡) . They obtained the following estimate for t n ~ 10
112 8118 n nt W e n − − = − , (6) here W is the main branch of the Lambert W function [6,7]. Based on this proposal, Ramos and Silva [8] proposed a difference equation that calculates the n -th non-trivial zero of the function ( ) z in the critical line
11 12 arg8 21 11 1arg8 2 knkn kn n itt W n ite −− − − + = − − + . (7) ( ) z on the critical line One way to search for the best value of 𝛿 de (7), considering that 𝛿 = 1 is not always the best value for the bifurcation parameter of the map, in this perspective, we seek to calculate the Lyapunov numbers for the map of (7) [9]. Considering 𝑡 𝑛𝑘 = 𝑡 and deriving the map as a function of t we obtain as a result ( ) ( ) W n iteN t eO t W n it ne eW n iteN − − − + −→ − − − + − − + + − − + =
20 0 Nn itW n ite = − − + (8) whose numerical calculation must consider that O ( t -2 ) is equal to zero.
4. Conclusions sing (7) to numerically estimate the numbers of Lyapunov (8), we can find the bifurcation points that indicate the best 𝛿 values to be used to obtain with better precision the complex part of the non-trivial zeros of the function ( ) z . Appendix
11 12 arg8 21 11 1arg8 2 n itddt W n ite − − + = − − +
Let
S t it = + , we have to
112 ( )81 11 ( )8 n S tddt W n S te − − = − −
11 1 11 1 11 112 ( ) ( ) ( ) 2 ( )8 8 8 81 11 ( )8 d dn S t W n S t W n S t n S tdt e dt eW n S te − − − − − − − − − − − = by [4,6], we have to ( ) ( )
1( ) log2 2 d tS t O tdt − = − + , e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 W u t d e dW u t u tdt dtW u t − = + , inding ( ) W n S te d e dS t W n S t n S t n S tdt e dt eW n S teW n S te − − − − − − − − − − − + − − − − = ( ) W n S te d e dS t W n S t S t n S tdt e e dtW n S teW n S te − − − − − − − − − − + − − = − − ( ) W n S te d e dS t W n S t S t n S tdt e e dtW n S teW n S te − − − − − − − − − − + − − = − − ( ) W n S te d eS t W n S t n S tdt e eW n S teW n S te − − − − − − − − − + − − = − − ( ) W n S te t eO t W n S t n S te eW n S teW n S te − − − − − − + − − − − − + − − − − = ( ) W n S te t eO t W n S t n S te eW n S teW n S te − − − − − − − − − − + − − − − = ) ( ) W n S teN t eO t W n S t n S te eW n S teN W n S te − − − −→ − − − − − − + − − = − − Nn = = ( ) ( ) W n iteN t eO t W n it ne eW n iteN − − − + −→ − − − + − − + + − − + =
20 0 Nn itW n ite = − − + . References [1]
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