Noname manuscript No. (will be inserted by the editor) An investigation of the non-trivial zeros of the Riemann zeta function Yuri Heymann Received: date / Accepted: date Abstract The zeros of the Riemann zeta function outside the critical strip are the so-called trivial zeros. While many zeros of the Riemann zeta function are located on the critical line ℜpsq “ 1{2, the non-existence of zeros in the remaining part of the critical strip ℜpsqPs0, 1r remains to be proven. The Rie- mann zeta functional leads to a relationship between the zeros of the Riemann zeta function on either sides of the critical line. Given s a complex number ands ¯ its complex conjugate, if s is a zero of the Riemann zeta function in the critical strip ℜpsq Ps0, 1r, then we have ζpsq “ ζp1 ´ s¯q. As the Riemann hypothesis states that all non-trivial zeros lie on the critical line ℜpsq“ 1{2, it is enough to show there are no zeros on either sides of the critical line within the critical strip ℜpsqPs0, 1r, to say the Riemann hypothesis is true. Keywords Riemann zeta function, Riemann hypothesis 1 Introduction The Riemann zeta function named after Bernhard Riemann (1826 - 1866) is an extension of the zeta function to complex numbers, which primary purpose is the study of the distribution of prime numbers [14]. Although the zeta function which is an infinite series of the inverse power function was studied long time ago and is referred to in a formula known as the Euler product establishing the connection between the zeta function and primes, Riemann’s approach is linked with more recent developments in complex analysis at the time of Cauchy. The Riemann hypothesis stating that all non-trivial zeros lie on the critical line ℜpsq“ 1{2 is crucial for the prime-number theory. Not only the Riemann-von Mangoldt explicit formula for the asymptotic expansion of arXiv:1804.04700v15 [math.GM] 15 Jun 2020 Yuri Heymann Georgia Institute of Technology, Atlanta, GA 30332, USA E-mail: [email protected] Present address: 3 rue Chandieu, 1202 Geneva, Switzerland 2 Yuri Heymann the prime-counting function involves a sum over the non-trivial zeros of the Riemann zeta function, but also the Riemann hypothesis has implications for the accurate estimate of the error involved in the prime-number theorem, which is the description of the asymptotic behavior of the number of primes less than or equal to a real variable. The Riemann zeta function is a holomorphic function defined as follows: 8 1 ζpsq“ , (1) ns n“1 ÿ where s is a complex number and domain of congruence ℜpsqą 1. The domain of convergence of the Riemann zeta function is extended to the left of ℜpsq“ 1 by analytic continuation. The Dirichlet eta function, which is the product of 2 ℜ the factor 1 ´ 2s with the Riemann zeta function is convergent for psqą 0. This is an alternating series, expressed as follows: ` ˘ 2 8 p´1qn`1 ηpsq“ 1 ´ ζpsq“ , (2) 2s ns n“1 ˆ ˙ ÿ where ℜpsqą 0, and ηp1q“ lnp2q by continuity. 2 ℜ The function 1 ´ 2s has an infinity of zeros on the line psq “ 1 given by 2kπi Z˚ 2 α´1 iβ ln 2 α iβ ln 2 sk “ 1 ` ln 2 where k P . As 1 ´ 2s “ 2 ˆ 2 e ´ 1 { 2 e , ` 2 ˘ the factor 1 ´ s has no poles nor zeros in the strip ℜpsq Ps0, 1r. It follows 2 ` ˘ ` ˘ that by analytic continuation, the Dirichlet eta function can be used as a proxy ` ˘ for zero finding of the Riemann zeta function in the critical strip ℜpsqPs0, 1r. The Riemann zeta function can be further extended to the left, i.e. ℜpsqď 0 by analytic continuation with the Riemann zeta functional. Whenever the Riemann zeta function is invoked for ℜpsqď 1, it is implicitly referring to its analytic continuation. 2 Elementary propositions Before delving into core propositions, a prerequisite is the Riemann zeta func- tional, which is used explicitly in propositions 2, 4 and 9 and implicitly in 5, 6, 7 and 8. The Riemann zeta functional is expressed as follows: sπ ζpsq“ Πp´sqp2πqs´12 sin ζp1 ´ sq . (3) 2 ´ ¯ This equation was established by Riemann in 1859. The details of the deriva- tion are provided in [2], p. 13. The standard formulation of (3) is as follows: sπ ζp1 ´ sq“ Γ psqp2πq´s2cos ζpsq , (4) 2 ´ ¯ An investigation of the non-trivial zeros of the Riemann zeta function 3 involves the substitution s Ñ 1 ´ s and the relationship Πps ´ 1q“ Γ psq. Let us proceed with key propositions. Proposition 1 A formula is introduced here for calculations further down, which is expressed as follows: a2 ´ b2 ´2 ab ` i ra ` ibs“ a ´ i b , (5) a2 ` b2 a2 ` b2 „ ˆ ˙ where a and b are reals. Proof By identifying the real and imaginary parts of px`iyqpa`biq“ a´ib, we obtain two equations a x ´ by “ a and b x ` ay “ ´b. Eq. (5) is the resolution of these two equations. Proposition 2 Given s a complex number ands ¯ its complex conjugate, we have: ζps¯q“ ζpsq , (6) where ζpsq is the complex conjugate of ζpsq. Proof Let us say s “ α ` iβ where α and β are real numbers and i2 “ ´1. We have: 8 rcospβ ln nq´ i sinpβ ln nqs ζpsq“ , (7) nα n“1 ÿ where ℜpsqą 1. 1 1 1 We have ns “ nα exppβi ln nq “ nαpcospβ ln nq`i sinpβ ln nqq . We then multiply both the numerator and denominator by cospβ lnpnqq ´ i sinpβ lnpnqq. After sev- eral simplifications, we get (7). Note that in (7) when β changes its sign, the real part of ζpsq remains unchanged, while imaginary part changes sign. This means, the Riemann zeta function has mirror symmetry with respect to the real axis of the complex plane and ζps¯q“ ζpsq when ℜpsqą 1. By the same process, the Dirichlet eta function is expressed as follows: 8 rcospβ ln nq´ i sinpβ ln nqs ηpsq“ p´1qn`1 , (8) nα n“1 ÿ where ℜpsqą 0, and ηp1q“ lnp2q by continuity. Using the same reasoning as above, the Dirichlet eta function is symmetrical with respect to the real axis of the complex plane. Furthermore, the factor 1 ´ 21´s “ 1 ´ 21´α pcospβ ln 2q´ i sinpβ ln 2qq also has mirror symmetry with respect to the real axis. As the product of holomorphic functions which are` symmetrical˘ with respect to the real axis yields a holomorphic function 4 Yuri Heymann which is also symmetrical with respect to the real axis, we get ζps¯q “ ζpsq when ℜpsq Ps0, 1r. Let us consider the Riemann zeta functional (4), and introduce the function ξ : C Ñ C such that: sπ ξpsq“ 2Γ psqp2πq´s cos . (9) 2 ´ ¯ If ξpsq is symmetrical with respect to the real axis, then by analytic continu- ation, the Riemann zeta function is also symmetrical with respect to the real axis, when ℜpsqă 0. From formula 6.1.23 p. 256 in [1], we have Γ psq“ Γ ps¯q. ´s cospβ ln 2π´i sinpβ ln 2πqq ´s Furthermorre, p2πq “ p2πqα , hence p2πq is symmetrical with respect to the real axis. From the formula 4.3.56 p. 74 ins [1], we have sπ απ βπ απ βπ cos 2 “ cos 2 cosh 2 ´ i sin 2 sinh 2 . As coshp´xq“ coshpxq sπ and` sinh˘p´xq“´` sinh˘ pxq´, cos¯ 2 is` also˘ symmetrical´ ¯ with respect to the real axis. Hence, ξpsq has mirror symmetry with respect to the real axis. Therefore, ` ˘ ζps¯q“ ζpsq when ℜpsqă 0. As the Riemann zeta function is a meromorphic function, by continuity ζps¯q“ ζpsq on the lines ℜpsq“ 0 and ℜpsq“ 1. Proposition 3 Given s a complex number ands ¯ its complex conjugate, we define a holomorphic function ν : C Ñ C such that: ζpsq“ νpsqζp1 ´ s¯q . (10) A property of νpsq is that it is equal to one if ℜpsq“ 1{2. Proof Let us set s “ α ` iβ where α and β are reals. When α “ 1{2, we get 1 1 ζpsq“ ζ 2 ` iβ and ζp1 ´ s¯q“ ζ 2 ` iβ . Hence, ζpsq“ ζp1 ´ s¯q provided that ℜpsq“ 1 . For a given α ‰ 1 , we have to introduce a complex factor νpsq ` 2 ˘ 2 ` ˘ because we can no longer satisfy the two equations ℜpζpsqq “ ℜpζp1 ´ s¯qq and ℑpζpsqq “ ℑpζp1 ´ s¯qq with one degree of freedom given by β. Proposition 4 Given s a complex number ands ¯ its complex conjugate, we have: 1 Γ ps¯q πs¯ u2 ´ v2 ´2u v “ 2 cos ` i , (11) νpsq p2πqs¯ 2 u2 ` v2 u2 ` v2 ´ ¯ „ ˆ ˙ where u “ ℜpζpsqq and v “ ℑpζpsqq. Proof This formula is derived from the Riemann zeta functional. By the def- 1 ζp1´s¯q inition in proposition 3, we have νpsq “ ζpsq . From (4), the Riemann zeta Γ psq πs functional is expressed as ζp1 ´ sq “ p2πqs 2cos 2 ζpsq. Hence, ζp1 ´ s¯q “ ` ˘ An investigation of the non-trivial zeros of the Riemann zeta function 5 Γ ps¯q πs¯ p2πqs¯ 2cos 2 ζps¯q. From proposition 2, we have ζps¯q“ ζpsq, where ζpsq is the complex conjugate of ζpsq. ` ˘ 2 2 u ´v ´2u v We use proposition 1 and get ζps¯q “ ζpsq u2`v2 ` i u2`v2 where u “ ℜpζpsqq and v “ ℑpζpsqq. Eq. (11) follows. ” ´ ¯ı Proposition 5 Given s a complex number, the function νpsq as defined above is equal to zero only at the points s “ 0, ´2, ´4, ´6, ´8, ..., ´n where n is an even integer.