On Studying the Phase Behavior of the Riemann Zeta Function Along the Critical Line
OOn Studying the Phase Behavior of the RiemannZeta Function Along the Critical Line
Henrik StenlundJune 1st, 2018
Abstract
The critical line of the Riemann zeta function is studied from a newviewpoint. It is found that the ratio between the zeta function at any zeroand the corresponding one at a conjugate point has a certain phase andits absolute value is unity. This fact is valid along the whole critical lineand only there. The common functional equation is used with the aid ofthe function ratio between any zero and its negative side pair, a complexconjugate. As a result, an equation is obtained for solving the phase alongthe critical line.
Riemann zeta function, zeros of zeta function, recursion relation of zetafunction, functional equation of zeta function
MSC: 11M26
Developing methods for studying of the nature of the nontrivial zeros on thecritical line of the Riemann zeta function are very important. One is attemptingin this paper to find a way of eliminating the zeta function out of the functionalequations allowing possibly new expressions to be developed in terms of moreelementary functions. We are attempting to find an expression for the phase ofthe zeta function along the critical line. Visilab Report The author is with Visilab Signal Technologies Oy, Finland. The author is grateful forfull support from the organization a r X i v : . [ m a t h . G M ] J un .2 Preliminaries As is well known, the Riemann zeta function has a number of zeros, along thenegative real axis at even integer points and along the critical line x = . Theselatter appear at points whose exact positions are not known beforehand butrequire a lot of numerical work, the points being of a transcendental nature.Every zero in the upper half of the complex plane has a pair mirroring on thenegative half plane. This work is now focusing in these nontrivial zeros. Thefollowing treatment is relying on the validity of the Riemann hypothesis [4]. Itrequires all nontrivial zeros to reside on the critical line x = and none outsideit. We are living the times of seeing the hypothesis finally proven, if not alreadydone.Now what will happen if a zeta function approaching a zero is divided byanother zeta approaching a corresponding pairing zero on the negative halfplane? As both of them go to zero one would expect the ratio to becomesingular. While investigating numerically the behavior of the Riemann zetafunction approaching some of its zeros on the critical line, it was noted that thephase behavior of the function has a particular feature. When the argument s approaches any zero s of the zeta function ζ ( s ). The ratio seems to belim s → s ζ ( s ) ζ (¯ s ) = e iθ (1)with s = + it . One can evaluate this ratio while approaching the zero at s from any direction and it appears that the ratio will not be singular. It can beproven as done in the following section. This leads one to think that there couldexist another way to study the zeros of the zeta function. Since the zeta function is real on the real axis and meromorphic elsewhere, onehas according to Schwarz’s reflection principle ζ (¯ s ) = ¯ ζ ( s ) (2)The function has a general form ζ ( s ) = e iφ ρ (3)over the complex plane (with φ, ρ ∈ R ). Especially one does have along thecritical line ζ ( s ) = e iφ ρ (4)and then one has the following ¯ ζ ( s ) = ρ e − iφ (5)But according to equation (2) this is equal to ζ ( ¯ s ) = ρ e − iφ (6)2herefore, ζ ( s ) ζ ( ¯ s ) = e iφ (7)proving the assertion and verifying the original numerical observation. Thisholds along the entire critical line. The function ratio (1) can be processed asfollows at or very near a zero s ζ ( s ) ζ ( ¯ s ) = ζ ( s ) ζ (1 − s ) (8)Approaching the zero can be done along the critical line from either direction.Then the ratio’s amplitude remains at unity and the phase angle alone is varying.It can be done from a close distance from other directions on the complex planewhile the amplitude is not unity and the phase angle is varying. The correctvalue is finally obtained at the zero and the amplitude reaches unity.The functional equation of the Riemann zeta function with the argument s ∈ C is well known [1], [2], [3], [4]. One applies it in the following form. ζ ( s ) ζ (1 − s ) = (2 π ) s cos ( πs )Γ( s ) (9)From this it follows that at any nontrivial zero along the critical line ζ ( s ) ζ ( ¯ s ) = (2 π ) s cos ( πs )Γ( s ) = e iφ +2 πiN (10)This is true for any point s on the complex plane and the right side equalityis to be used in the following. One has eliminated the zeta function from thisequation. One has implemented the M od (2 πi ) , N ∈ N to be carried furtheron the way to the final expressions. The Weierstrass formula for the Gammafunction Γ( s ) is valid for s ∈ C s ) = se γs (cid:89) k =1 (1 + sk ) e − sk (11)and one can substitute it getting e iφ +2 πiN = (2 π ) s s e γs (cid:81) k =1 (1 + s k ) e − s k cos ( πs ) (12)Taking the logarithm of the equation above one will obtain2 iφ + 2 πiN = s ( γ + ln (2 π )) + ln ( s − ln ( cos ( πs (cid:88) k =1 ( − s k + ln (1 + s k ))(13)This equation with a complex variable s is valid for all nontrivial zeros.3reaking down the s as s = + it where t ∈ R , and substituting it willlead to2 iφ + 2 πiN = ( 12 + it )( γ + ln (2 π )) + ln ( + it − ln ( cos ( π ( + it )2 ))+ (cid:88) k =1 [ − + itk + ln (1 + + itk )] (14)The complex logarithms and the cosine term can further be broken down to realand imaginary parts as well and then one will get2 iφ +2 πiN = ( 12 + it )( γ + ln (2 π ))+ ln ( √ t − ln ( (cid:115) cosh ( πt ) + sinh ( πt )2 )+ i · artan (2 t ) + iπM − i · artan ( − tanh ( πt iπL + (cid:88) k =1 [ − k − itk + ln ( (cid:114) (1 + 12 k ) + t k ) + i · artan ( tk + )] + iπK (15)Obviously, the real and imaginary parts of each term succeeded to get a linearform allowing a trivial separation. For studying the zeros, one would be interested in the real part, equation (15)0 = 12 ( γ + ln (2 π )) + ln ( √ t − ln (cid:115) cosh ( πt ) + sinh ( πt )2 ++ (cid:88) k =1 [ − k + 12 ln (1 + 1 k + 14 k + t k )] (16)This can be developed further simplifying to γ + ln ( π ln ( cosh ( πt )1 + 4 t ) − (cid:88) k =1 ln [ e − k (1 + 1 k + 14 k + t k )] (17)In spite of its apparent complexity, this equation is an identity which is validfor all t . 4 .2 Solving the Phases The imaginary part of the equation (15) can be used for getting a general ex-pression for the limit phase φ at a point along the critical line φ + πM t γ + ln (2 π )) + 12 artan (2 t ) − artan ( − tanh ( πt (cid:88) k =1 [ − tk + artan ( tk + )] (18)The result has become M od ( π ) , M ∈ N . The resulting function is odd withrespect to t . A crude calculation of the ratio from equation (7) shows a value of -80.95 degrees ±
90 degrees for the first nontrivial zero and for the second one 77.36 degrees.One can plot the curve as a function of t along the axis x = 0 . φ as such. The M od ( π ) of the angle is more natural andFigure 1: Phase in degrees along x = 0 . A few interesting results have been obtained. The first is that according toequation (7) the ratio between the zeta function at any zero and its conjugate isnot singular, but always with unity absolute value and with a particular phase.5igure 2:
M od ( π ) of the phase in degrees along x = 0 . x = ) but fails immediately outsideof it. The failure is not dramatic but will cause an error which increases whilethe point of focus is moving farther from the critical line. The second is thatthe equation (10) has only elementary functions left for studying the zeros ofthe zeta function. The third interesting finding is ensuing from the previoustwo, equation (18) presenting a simplified expression for calculation of phase ata zero. These equations form the main results of this work. References [1]
Riemann, Bernhard : ¨Uber die Anzahl der Primzahlen unter einen gegebe-nen Grosse , Monatsb. der Berliner Akad., 1858/60, 671-80 (=GesammelteMathematische Werke. 2nd edn,Teubner, Leipzig. 1892. No VII, 145-55.(1858)[2] Siegel, C.L. : ¨Uber Riemanns Nachlass zur analytischen Zahlentheorie ,Quellen Studien zur Geschichte der Math. Astron. und Phys. Abt. B., Stu-dien 2. 45 - 80 (1932)[3] Edwards, H.M. : Riemann’s Zeta-Function , Dover Publications, 1st edi-tion, Mineola, New York(2001), ISBN 0-486-41740-9[4]