J. Nig. Soc. Phys. Sci. 5 (2023) 967 Journal of the Nigerian Society of Physical Sciences The Derivation of the Riemann Analytic Continuation Formula from the Euler’s Quadratic Equation Opeyemi O. Enocha,∗, Adejimi A. Adenijib, Lukman O. Salaudeena aDepartment of Mathematics, Federal University Oye-Ekiti, Ekiti State, Nigeria bDepartment of Mathematics and Statistics, Tshwane University of Technology, South Africa Abstract The analysis of the derivation of the Riemann Analytic Continuation Formula from Euler’s Quadratic Equation is presented in this paper. The connections between the roots of Euler’s quadratic equation and the Analytic Continuation Formula of the Riemann Zeta equation are also considered. The method of partial summation is applied twice on the resulting series, thus leading to the Riemann Analytic Continuation Formula. A polynomial approach is anticipated to prove the Riemann hypothesis; thus, a general equation for the zeros of the Analytic Continuation Formula of the Riemann Zeta equation based on a polynomial function is also obtained. An expression in Terms of Prime numbers and their products is considered and obtained. A quadratic function, G(tn), that is required for Euler’s quadratic equation (EQE) to give the Analytic Continuation Formula of the Riemann Zeta equation (ACF) is presented. This function thus allows a new way of defining the Analytic Continuation Formula of the Riemann Zeta equation (ACF) via this equivalent equation. By and large, the Riemann Zeta function is shown to be a type of L function whose solutions are connected to some algebraic functions. These algebraic functions are shown and presented to be connected to some polynomials. These Polynomials are also shown to be some of the algebraic functions’ solutions. Conclusively, ς(z) is redefined as the product of a new function which is called H(tn, z) and this new function is shown to be dependent on the polynomial function, G(tn). DOI:10.46481/jnsps.2023.967 Keywords: Riemann zeta function, Euler’s equation, Meromorphic function, Non-trivial zeros Article History : Received: 31 July 2022 Received in revised form: 20 November 2022 Accepted for publication: 25 November 2022 Published: 20 December 2022 c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: J. Ndam 1. Introduction Many authors have recently presented some polynomial ap- proaches to prove the Riemann Hypothesis [1-9]. Some of the proofs were based on Jensen polynomials, Laguerre polynomi- als as Jensen polynomials of Laguerre–Pólyaentire functions, and some worked on a general theorem which models such polynomials by Hermite polynomials. The authors presented ∗Corresponding author tel. no: +234 7066466859 Email address: opeyemi.enoch@fuoye.edu.ng (Opeyemi O. Enoch) an Approximation to Zeros of the Riemann Zeta Function us- ing Fractional Calculus [10]. This allowed the authors to con- struct a fractional iterative method to nd the zeros of functions in which it is possible to avoid expressions that involve hyper- geometric functions, Mittag-Leffler functions or infinite series [10], to mention a few. As good as their works are, seeking a clearer insight into these possible polynomials is expedient, knowing fully that so- lutions to most difficult problems may not necessarily be com- plex. Significantly, it is known that the Riemann Zeta function 1 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 2 is a type of L function whose solutions are connected to some algebraic functions, and polynomials are also types of solutions of algebraic functions. A polynomial approach can be antici- pated to prove the Riemann Hypothesis. This study aims to present the link that connects Euler’s Quadratic Equation (EQE) and the Analytic Continuation For- mula of the Riemann Zeta equation (ACF) by using the par- tial summation on a generalized polynomial function of Euler’s Quadratic Equation. (EQE). The remaining part of this paper is organized as follows: Section Two is the Materials and Methods. The Derivation of the Analytic Continuation Formula of the Riemann Zeta equa- tion (ACF) from Euler’s Quadratic Equation (EQE) is presented in Section Three. Section Four considers the obtained Expres- sion in Terms of Prime Numbers and Their Products. While Section Five is for obtaining a quadratic function; G (tn) , in terms of the Analytic Continuation Formula of the Riemann Zeta equation (ACF). Section six presents a new way of defin- ing the Analytic Continuation Formula of the Riemann Zeta equation (ACF) via an equivalent equation. The Concluding remark is presented in section seven. 2. Materials and Methods The following equations and method shall be used in the derivation of the Analytic Continuation Formula of the Rie- mann Zeta equation (ACF) from the Euler’s quadratic equation (EQE) [11]: ∏ ( z 2 ) = − z 2 Γ ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) (1) nz = m∏ i=1 P∝i zi (2) From the expression [14-15], ( 1 pz − 1 ) ∞∑ n=0 1 pnz = −1 (3) The method of partial summation as found in literature [14-15] is given as: ∑ P≤x log2 p + ∑ pq≤x log p log q = 2x log x + ϑ (x) (4) By partial summation, we get from Equation (5) ∑ p≤x log p + ∑ pq≤x log p log q log pq = 2x + ϑ ( x log x ) , (5) the above Equation (5) gives Equation (6) ∑ pq≤x log p log q = ∑ p≤x log p ∑ q≤x/p log q (6) Or =2x ∑ p≤x log p p − ∑ p≤x log p ∑ qr≤x/p log q log r log qr + ϑ x ∑ p≤x log p p ( 1 + log xp )  (7) = 2x log x − ∑ qr≤x/p log q log r log qr ϑ ( x pq ) + o ( x log log x ) (8) These equations above shall be used to derive the analytic continuation formula of the Riemann Zeta Function from Eu- ler’s Quadratic Equation. By this derivation, it will be readily obvious to see through into the generation of the zeros of the Riemann Zeta Function. 3. Derivation of the ACF from EQE The polynomial function in (1) was always discovered to be prime for 1 ≤ x ≤ 40 [12]. It was from this quadratic equation that Euler obtained the first few prime numbers [13]. 1 ≤ x ≤ 40 : x2 + x + 41 (9) Another known fact is that Equation 10 is also a prime for x = 1 . . . , 39. x2 − x + 41 (10) Interestingly, the roots of (9, 10) are x = −0.5±6.3836i and x = 0.5 ± 6.3836i respectively, which are the similitude of the non-trivial zeros of the Riemann zeta function. With these sim- ilarities [16], the question that readily comes to mind is; what is the possibility of using the structure of this Euler’s equation to obtain ACF if the coefficients of x2 and x are taken as k and for the constant integral, 41, is replaced with G (tn)? To seek an an- swer to this question, a transformation of the Euler’s equation, when it is multiplied by another linear equation whose roots will are −2n : n = 1, 2, 3, . . ., will be ζE (z) = ( kz2 − kz + G(tn) ) (z + 2n) (11) The roots of this polynomial will be the same as the trivial and the non-trivial zeros of the Riemann Zeta function under certain conditions that G(tn) is known. It has been shown that there are Meromorphic functions that are equivalent to the Rie- mann zeta function [6-7], and they are given as: 2 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 3 ζE (z) = (z + 2n) (z − 1) ( kz2 − kz + G(n) ) ; k = 4 (12) Or ζE (z) = (z + 2n) (ez−1 − 1) ( kz2 − kz + G(n) ) ; k = 4 (13) Provided that G(n) is also known. The Equation (12) and Equation (13) are transformed into matrices whose Eigenvalues are the trivial and non-trivial spec- tral points of the Riemann zeta function [5-7, 13] provided that; G (n) = 800.162 + 968.548J (n) (14) Or G (n) = 800.162 + 968.548nv(n) (15) Or G (tn) = 1 + kt 2 n where k = 4 (16) Such that G (tn) = G(n). From 11, let ζE (z) = ( kz2 − kz + G (tn) ) (z + 2n) (17) Riemann gave the following expression in his work [11]: − ∏ ( z 2 ) = ∞∑ n≥1 ( 1 + z 2n ) (18) which is the same as z2 Γ ( z 2 ) , By taking ∏ ( z 2 ) = − z 2 Γ ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) , (19) Equation (17) can written as ζE (z) = 2n ( 1 + z 2n ) ( kz2 − kz + G (tn) ) (20) Using the method of discretization, Equation (20) becomes; γ (z) = ∞∑ n≥1 (ζE (z)) (21) = ∞∑ n≥1 [ 2n ( 1 + z 2n ) ( kz2 − kz + G (tn) )] (22) Thus applying the method of partial summation in [15],as in Equation (6) the resulting equation from Equation (22) shall be ∑ dq≤n [ 2n ( 1 + z2n ) ( kz2 − kz + G(tn) )] = ∑ d≤n 2n [ kz2 − kz + G(tn) ] ∑ q≤n/d (1 + z 2n ), (23) Where d = 2n [ kz2 − kz + G(tn) ] and q = 1 + z2n Eventually, Equation (23) can be written as: γ (z) = ∞∑ n≥1 (ζE (z)) (24) = ∑ d≤n [ 2n ( kz2 − kz + G(tn) )] ∑ q≤n/d ( 1 + z 2n ) (25) Interestingly, since − z 2 Γ( z 2 ) = ∞∑ n≥1 ( 2n + z 2n ) (26) 25 becomes: γ (z) = ∞∑ n≥1 (ζE (z)) = − z 2 Γ ( z 2 ) (z − 1) ∑ d≤n 2n ( kz + G (tn) (z − 1) ) (27) With a little rearrangement and the introduction of π −z 2 π z 2 = 1, Equation (27) is the same as: γ (z) = ∅(z) ∑ d≤n 2n [( kz + G(tn) (z − 1) ) π z 2 ] (28) where ∅(z) = − z2 (z − 1)π −z 2 Γ( z2 ) If the principle of partial summation is again applied on the series in Equation (28), Equation (29) we will obtain: ∑ rb≤n 2n [( kz + G(tn) (z − 1) ) π z 2 ] = π z2 ∑ r≤n ( kz + G (tn) (z − 1) ) ∑ b≤ nr 2n  ; rb = d (29) By this, (28) becomes: γ (z) = ∅(z) π z2 ∑ r≤n ( kz + G (tn) (z − 1) ) ∑ b≤ nr 2n  (30) where r = (kz + G(tn(z−1) )π z 2 and b = 2n. (30) shall be used subsequently in this paper. 3 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 4 4. An Expression in Terms of Prime Numbers and Their Products A non-conventional expression for −1 [12-13] is;( 1 pz − 1 ) ∞∑ n=0 1 pnz = −1, (31) (31) allows us to write (28) as: γ (z) = − ∅ (z) ( 1 pz − 1 ) ∞∑ n=0 1 pnz ∑ d≤n 2nF (t, z) (32) F (t, z) = [( kz + G (tn) (z − 1) ) π z 2 ] Pleasantly (32) gives: γ (z)  ( 1 pz − 1 ) π z 2 ∑ d≤n 2n [F(t, z)]  −1 = z 2 (z − 1) π −z 2 Γ ( z 2 ) ∞∑ n=0 1 pnz . (33) Intuitively, if (33) is multiplied over prime numbers one comes to; γ (z) ∏ p  ( 1 pz − 1 ) ∑ d≤n [2nF(t, z)]  −1 = z 2 (z − 1)π −z 2 Γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz . (34) Conclusively, the RHS of (34) is the same as the Analytic Continuation formula of the Riemann Zeta function, while the LHS is an equivalence of the RHS. 5. Obtaining G (tn) in Terms of the ACF Riemann defines ε (z) [11] as ; ε (z) = z 2 (z − 1) π −z 2 Γ ( z 2 ) ζ (z) (35) By this, ζ (z) can be represented as: ζ (z) = 2ε (z) π z 2 z (z − 1) Γ ( z 2 ) (36) For the LHS of (34) to be equal to (34), we represent εe in Equation (37) as εe = γ (z) ∏ p ( 1 pz − 1 ) ∞∑ d≤n 2nF(t, z)  −1 (37) Using (29) on (34), we obtain: εe = γ (z) J(n) ∏ p ( 1 pz − 1 ) ∑ r≤n ∏ p F (t, z)  −1 (38) such that J (n) = ∑ b≤ nr ∏ p 2n and εe = γ (z) π − z 2 B (t, z) ∏ p ( 1 pz − 1 )−1 (39) where B (t, z) = ∏ p ∑ r≤n ( kz + G (tn) (z − 1) ) ∑ b≤nr 2n  −1 Gives: εe = γ (z) π − z 2 ζ (z) B (t, z) . (40) By making the series containing G (tn) the subject of the expression in (38), we obtain: ∑ r≤n ∏ p ( kz + G (tn) (z − 1) ) = γ (z) εe π z2 ∑ b≤ nr ∏ p 2n  −1 ∏ p ( 1 pz − 1 ) −1 (41) followed by Further simplification shown that (41) becomes: ∑ r≤n ∏ p ( kz + G (tn) (z − 1) ) = γ (z) εe ζ (z) π z2 ∑ b≤ nr ∏ p 2n  −1 (42) Since ζ (z) = ∏ p ( 1 pz − 1 ) −1 . (43) It can be shown in existing works [1-2] that; nz = m∏ i=1 P∝i zi (44) 4 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 5 implies that ∑ d≤n 2n = 2 ∑ d≤n m∏ i=1 P∝ii (45) With this, (42) can be expressed as (45), in terms of prime numbers such that: ∑ r≤n ∏ p ( kz + G (tn) (z − 1) ) = γ (z) εe ζ (z) π− z 2 ∑ b≤ nr ∏ p  12 m∏ i=1 P−∝ii  (46) 6. Defining the ACF via (34) If the LHS of (34) is written as: εe = γ (z) ∏ p [( 1 pz − 1 ) H (t, z) ]−1 (47) where H (t, z) = π z 2 ∑ d≤n 2n ( kz + G (tn) (z − 1) ) Then we can write (35) as the RHS of (34) such that; εe = z 2 (z − 1) π −z 2 Γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (48) Where � (z) = εe and From the Nachlass of Riemann [15], ε (z) is also defined as; ε (z) = 1 2 + z 2 (z − 1) D (x, z) (49) where D (x, z) = ∫ ∞ 1 ψ (x) ( x z 2 −1 + x− (z+1) 2 ) d x (50) and ψ (x) = ∞∑ n=1 e−n 2πx (51) Such that if ε (z) = εe then; 1 2 + z 2 (z − 1) D (x, z) = z 2 (z − 1) π −z 2 Γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (52) Or as; 1 2 + z 2 (z − 1) D (x, z) = γ (z) ∏ p [( 1 pz − 1 ) H (t, z) ]−1 (53) Since H (t, z) = π z 2 ∑ d≤n [ 2n ( kz + G (tn) (z − 1) )] The evaluation of the intergrades in (52 and 53) will give; D (x, z) = 2N(z) (54) Where N (z) = ∞∑ n=1 e−n2π  1( z2 −1)( 2n2π + z − 2 ) + 1( z+12 )( 2n2π− z − 1 )  (55) With (52), we can write (52) and (53) as follows; 1 2 + z (z − 1) N(z) = z 2 (z − 1) π −z 2 Γ ( z 2 ) ∏ p ∞∑ n≥1 1 pnz (56) and 1 2 + z (z − 1) N(z) = γ (z) ∏ p [( 1 pz − 1 ) H(t, z) ]−1 (57) (53) and (54) now give new definitions of the ζ (z) as: [ z 2 (z − 1) π −z 2 Γ ( z 2 )]−1 [ 1 2 + z (z − 1) N (z) ] = ∏ p ∞∑ n≥1 1 pnz (58) and γ (z)−1 ∏ p H(t, z) [ 1 2 + z (z − 1) N(z) ] = ∏ p [( 1 pz − 1 )]−1 (59) 5 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 6 Equating (56) and (57), (60) is obtained: [ z 2 (z − 1) π −z 2 Γ ( z 2 )]−1 [ 1 2 + z (z − 1) N (z) ] = γ (z)−1 ∏ p H (t, z)  [ 1 2 + z (z − 1) N (z) ] (60) From (60), γ (z) is obtained (61) : γ (z) = [ z 2 (z − 1) π −z 2 Γ ( z 2 )] ∏ p H (t, z) (61) such that H (t, z) is represented in (62 H (t, z) = π z 2 ∑ d≤n [ 2n ( kz + G (tn) (z − 1) )] (62) It can be written as Equation (29). 7. Conclusion With the contents of G (tn),in (13 and 14): G (tn) = 1 + kt2n where k = 4, (61) and (62) can be implemented to give the same results and zeros of the Analytic Continuation formula and that of the Riemann Zeta function. It has been shown that the Analytic Continuation formula of the Riemann Zeta func- tion can be obtained from Euler’s Quadratic Equation. Riemann Zeta Function can be written as (63) provided G (tn) holds as defined in (58) and (59). ζ (z) = ∏ p H (t, z) = ∏ p ∑ d≤n [ 2nπ z 2 ( kz + G (tn) (z − 1) )] (63) Enoch obtained the following for the generation of the ze- ros of the Analytic Continuation formula of the Riemann Zeta function [5-7]: G (tn) = kz (z − 1) σ τ−ϑ (64) Where σ =  z2π−z2 Γ ( z 2 ) ( 1 pz − 1 ) ∞∑ n=0 1 pnz − 1  (65) Such that; ϑ = z 2 π −z 2 Γ ( z 2 ) ( 1 pz − 1 ) ∞∑ n=0 1 pnz (66) and τ = [ 1 2 (z − 1) + zN (z) ] ∏ p ( 1 pz − 1 ) (67) He pointed out that (58) and (59) definition of G (tn) is the same as obtained in Equations (14), (15) and (16). The func- tions; Jn and vn are written as polynomials of order two or three for this to be possible by the authors [8-9]. He was able to obtain a general equation for the zeros of the Analytic Contin- uation formula from Equation (16) as; G (tn) = 1 + kt 2 n ; k = 4 (68) By which Equation (69) holds as : tn = ( G (tn) − 1 k )1/2 (69) Again from: 1 + kt2n = kz (z − 1) σ τ−ϑ ; k = 4 (70) Such that; tn = ± ( z (z − 1) σ (τ−ϑ) − 1 k ) 1 2 ; k = 4 (71) The k value can hold for any integer, depending on the pat- tern of choice. Acknowledgments The authors wish to thank Tshwane University of Technol- ogy for their financial support and the Department of Higher Education and Training, South Africa. References [1] M. Griffin, K. Ono, L. Rolen, & D. Zagier, “Jensen polynomials for the riemann zeta function and other sequences”, Proceedings of the National Academy of Sciences 116 (2019) 11103. [2] D. K. Dimitrov & Y. B. Cheikh, “Laguerre polynomials as jensen polyno- mials of laguerre–pólya entire functions”, Journal of computational and applied mathematics 233 (2009) 703. [3] S. DeSalvo & I. Pak, “Log-concavity of the partition function”, The Ra- manujan Journal 38 (2015) 61. [4] J. Derbyshire, Prime obsession: Bernhard Riemann and the greatest un- solved problem in mathematics, Joseph Henry Press (2003). [5] O. O. Enoch, “The eigenvalues (energy levels) of the riemann zeta func- tion”, International Conference on Pure and Applied Mathematics (2015) 81. [6] O. O. Enoch, “The derivation of the riemann zeta function from euler’s quadratic equation and the proof of the Riemann hypothesis”, Interna- tional Scientific Journal. Journal of Mathematics (2016). 6 Enoch et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 967 7 [7] O. O. Enoch, “From the zeros of the Riemann zeta function to its analyt- ical continuation formula”, Global Journal of Pure and Applied Mathe- matics 13 (2017) 8423. [8] M. Chudnovsky & P. Seymour, “The roots of the independence polyno- mial of a claw free graph”, Journal of Combinatorial Theory, Series B 97 (2007) 350. [9] H. Larson & I. Wagner, “Hyperbolicity of the partition jensen polynomi- als”, Research in Number Theory 5 (2019) 1. [10] A. Torres-Hernandez & F. Brambila-Paz, “An approximation to zeros of the riemann zeta function using fractional calculus”, arXiv preprint arXiv:2006 (2020) 14963. [11] A. Selberg, “Harmonic analysis and discontinuous groups in weakly sym- metric riemannian spaces with applications to dirichlet series”, The Jour- nal of the Indian Mathematical Society 20 (1956) 47. [12] S. Andreas, Riemann’s second proof of the analytic continuation of the riemann zeta function (1987). [13] R. R. Ben, The zeta function and its relation to the prime number theorem (2000). [14] S. J. Patterson, “An introduction to the theory of the Riemann zeta- functio”, Cambridge University Press 14 (1995). [15] A. Selberg, “An elementary proof of the prime number theorem”, Annals of Mathematics (1949) 305. [16] O. O. Enoch, T. O. Ewumi, & Y. Skwame, “On the turning point, critical line and the zeros of riemann zeta function”, Australian Journal of Basic and Applied Sciences 6 (2012) 279. 7