Zeros of Sections of the Zeta Function. I

Zers f Sectins f the Zeta Functin. I By Rbert Spira 1. Intrductin. The sectins f the title are the Dirichlet plynmials: (1) U(s) = n"5 n = l We write s = a -f- it, and take M ^ 3. Turan [1], [2] shwed that the Riemann hypthesis wuld be true prvided the zers f "«(s) all had real parts ^1 +fc/ M1/2, fr sme psitive k. The authr verified that these real parts were 1 fr M í= 100 and t ^ 1000. Apóstl [3] generalized sme f Turán's results t L-series. The present authr's wrk als raises interest in these zers in relatin t the Rieniann hypthesis fr the functins (2) guis) = fv(s) + x(s)u(l - s) where x(s) is the functinal equatin multiplier fr the f-functin, i.e., f(s) = x(s)f(l - s). In this paper, therems n zer-free regins f fji/(s) are derived, the methds used fr calculating the zers are given, and the lcatins f the zers are described. The numerical values (t QD) f the zers may be fund in Spira [7]. The zers calculated are: (a) M: 3(1)12,0 < i ^ 100, (b) M = 10*, k: 2(1)5, -1 S a, 0 < i ^ 100, (c) M = 1010, a string f zers, 0 < i g 100, (d) Sequences f zers sm, where s.v+i is btained by applying Newtn's methd t tm+iis) with initial apprximatif sm The fllwing sequences were calculated: M = 4(1)50, Sa = lwest zer; Ü7 = 10(1)35, Si = next t lwest zer; M = 10(1)40, si =.35 4-14.50z; M = 11(1)50, sn =.54 + 37.65 ; six sequences, M = 11(1)25, su =.60 + 25.0O,.60 + 30.43t,.57 + 32.86,.57 + 40.86,.53 4-43.25,.54 4-48.10z'. Figures 1, 2, and 3 give the zers (a). Tw sequences (d) are als given in Figure 3. Figure 4 gives the zers (b) and (c). Papers by Langer [4] and Wilder [5] estimate the number f zers f "«(s) t be within M f T(lgilf)/2ir. Table I gives the number f zers fund and the values f 100(lg M)/2t fr cmparisn. 2. Zer-Free Regins. Therem 1. If ^ 1.85, f,w(s) ^ 0. Prf. (m(s) I ^ 1 - M -1 pm n è 1-2~ - x" _n-2 J J2 dx ^ 1 - [2"" + 2W/U - 1)] = 1-2-'(<r + 1)/U - 1). Received July 14, 1965. Revised April 18, 1966. 542

ZEROS OF SECTIONS OF THE ZETA FUNCTION. I 543 0 _Q-Q--O-&- O _O

544 ROBERT SPIRA O O O _a_ O O O " "cr L&xr t SI. {J-a*-TT- CM 00 II g b M OÓT 0 0 0 O O00 _Ü_CL _Q_2_ M 00T

ZEROS OF SECTIONS OF THE ZETA FUNCTION. I 545 O O O O -«-Û -xr 000 ~u~ - «w~ «~- - O O 0,9 O O

546 ROBERT SPIRA c H -i II n S SE=4 _- a M.. *

ZEROS OF SECTIONS OF THE ZETA FUNCTION. I 547 Table I M Number f zers < 100 100 (Lg M)/2tt Left limit f zers 3 4 56 7 8 9 10 11 12 17 22 26 29 31 33 35 157 38 40 17.485 22.064 25.615 28.517 30.970 33.095 34.970 36.647 38.164 39.549-1. -1.73050-2.42601-3.11889-3.81199-4.50517-5.19834-5.89151-6.58468-7.27784 73579 27644 57337 945S5 30392 67486 57183 12961 42797 102 103 104 105 10'» 73 110 146 183 366 73.294 109.940 146.587 183.234 366.468 The last five entries fr number f zers means the number fund fr a It is prbable that there are n thers fr i g 100. Since 2 " and (a 4- l)/(a 1) are decreasing functins, we need nly seek the rt f 1 = 2"'i -f- l)/(cr 1), which is easily seen t lie t the left f 1.85. Therem 2. If a ^ 1 - M, fm(s)? 0. Prf. Çvis) I è M" _ n=l > m~ [l 4-2- 4- f" ' x- dx] è M~' - [1 4-2" + (M - 1)1_7(1 - a)} ^ M~ - [1 4-2" + (M - I) ], since Af - 1 á 1 - «r. Thus f (s) ^ 0, prvided M~ > 1 4-2'" 4- im - l)-'. Dividing this last by i M Vf, and nting that the resulting right hand side is ^2.25 fr M ^ 3, ur therem will be true prvided [M/iM - l)]^"1 ^ 2.25. Putting m = M 1, we knw by the binmial therem that (14- l/m)m is an increasing functin, s that [M/iM 1)]M_1 is always 2:2.25 as it is s fr M = 3. One can btain clser bunds in particular cases. M l -a, Prpsitin. If M " > ^"=ï n " and ai < a, then M "l > /Z* i n Prf. Let a = ai + k,k> 0. Then AT AT = M~"Mk > M"^2%=}n- = Y^n=i Mkn^^ En-i'nV = = T.n=ln-'1, Q.E.D. Thus, the rt f AT* = ~ZYZn=i n~ gives a right hand bund fr a zer-free half plane f tm(s). Clumn 4 f Table I gives this rt, runded t ten places, fr 3 ^ M ^ 12. A crude upper bund fr this rt is M1/2, fr M ^ 18. T shw this, let a = M112, then (3) n > ÍJ xa dx + (M - l)a

548 ROBERT SPIRA and.m-2 \ xa dx = (AT - 2)a+1/ia 4-1) = (a-l)((af - 2)/(Af - l))(af - 2)" J > UM - 2)112-1)(6/7)(Ai - 2)a = (6/7)(Af - 2)a+m - (6/7)(Af - 2)a. We can drp the secnd term as it is verpwered by the secnd term f (3), s we need nly shw f(af - 2)a+m > AT. After dividing by (A/ - 2)a and setting m = M - 2, this transfrms t fm1/2 > (1 + 2/m)(m+2>. Nw (m + 2)1'2 < m/2 fr m 2: 6, s the right hand side f the last inequality is bunded by e, which the left hand side will surely exceed if m ^ 16, r Af ^ 18. As shwn in Turan [2], the 1.85 bund f Therem 1 can be reduced t 1 fr M g 5. Fr M = 3, ne can d slightly better. If f3(s) = 0, then (4) Squaring 2" cs it lg 2) + 3"" cs (i lg 3) = -1, 2~* sin (i lg 2) 4-3"" sin (i lg 3) = 0. and adding, we btain (5) cs (ilgf) = {6' - [(*)' 4- (#) )}/2 = gia) and (6) g'ia) = (6>g6 -f- (lgf)(9- - 4T')]\/2. Fr a Ú 0, 9" ^ -T", s g'ia) > 0, and fr a > 0, lg 6 > 4"'lgf, s g'ia) > 0 fr all a and gia) is strictly increasing. Frm (5), gia)\ ^ 1, and a detailed calculatin gives 1 and.78788 49110 as the limits n a. A curius cnsequence f the abve analysis is the fllwing : Prpsitin. If j \, f3(s) and f3(l s) cannt bth be zer. Prf. Let f3(s) = 0. We can take t ^ 0. Frm the secnd equatin f (4), it fllws that sin ( i lg 2 ) and sin ( i lg 3 ) are bth zer r bth nnzer. If they bth vanished, we wuld have i lg 2 = kw, t lg 3 = jv, k and j nnnegative integers. If i 5 0, (and hence k, j ^ 0), we can divide these last tw equatins, btaining lg 2/lg 3 = k/j, r 3* = 21, which is impssible fr k, j > 0. If i = 0, we culd deduce frm the first equatin f (4) that 2" + 3~" = 1, which is impssible. Hence sin (i lg 2) and sin (i lg 3) are nnzer and we can write (t)' = -(sin(ilg3))/(sin(ilg2)). Thus, a is determined as a functin f i, s that tw distinct values f a are impssible fr a given t. This shws that if gzis) is zer ff a = 5, it cannt happen fr the reasn his) = f3(l s) = 0. In Spira [6], it was shwn that fr i sufficiently large, (7i(s) and g2is) satisfy the Riemann hypthesis. In Spira [10], the calculatins are described indicating zers ff the critical line f gmis) fr M =ï 3. 3. Methd f Calculatin. The zers fr M ^ 1010 were fund by lcating a zer within a square, searching the square by abslute value tests fr small functinal values with the fur pssibilities f signs fr the real and imaginary parts, clsing in with linear interplatin, and then a final tightening with a high precisin Newtn's

ZEROS OF SECTIONS OF THE ZETA FUNCTION. I 549 methd. Fr the zers with M S 12, it was pssible t search the entire strip within which the zers were knwn t be lcated. Fr larger M? 1010, nly the regin t the right f a = 1 was searched. Frm the general appearance f the zers, and the fact that the numbers f zers fund are s clse t 100(lg M)/2w, it seems likely that all the zers fr 0 á i á 100 have been fund fr the M cnsidered. An integratin was perfrmed ver the bundaries f regins searched, verifying the number f zers btained. The zers fr M 1010 were calculated in a sequence using Newtn's methd, with initial apprximants : s = 1 4-27Tz71g 1010, si = 1 4-4«/lg 1010, Sm+1 == Sm -p v. Sm Sm 1 /. Fr large M, the direct use f series ( 1 ) wuld invlve an impractical amunt f time. Thus, it was necessary t use the asympttic expansin derived frm the Euler-McLaurin frmula :,.?T -., N~' N1-. AT*. M1'' iitis) = 2-, n + - 4- - 4- -,- n=l 2 1 s 2 1 s I2v 2 m n /2r 2 \ t r> /2v 2\ A- errr. One als needs, fr Newtn's methd, a similar asympttic expansin fr tm' is). The prgrams fr these were btained as mdificatins f prgrams fr f(s) and f (s), available as a separate reprt [8]. Nte that the term Af1-S/(1 s) is very large near a = 0, and als changes argument very rapidly as i varies. The direct and asympttic series were checked against each ther fr M = 100, as were mst f the ther prgrams. The set f zers finally btained was differenced and als resubstituted t verify that they were gd apprximatins t the true zers. The cmputatins were carried ut at the University Cmputing Center, University f Tennessee (NSF-G13581). 4. The Zers. Fr Af g 12, the zers appear t have a pattern which has tw parts. One part is a line f Af 1 zers stretching upward in the left half plane with a negative slpe which increases negatively with M. The ther part cnsists f zers which at first frm a line near a = \, the zers f fm(s) lying quite clse t the zers f f(s), and then this line disappearing in a general scattering r blssming. Fr M = 3, this scattering appears almst immediately, while fr increasing M, the start f scattering mves prgressively upward, being arund 70 r 80 units up at M = 12. Since fis) = U-i(s) + M-/2 + M'-'/is - l) 4- Z ( m p /2v 1 \ II (s +j) laf1 2" + R v=i i v)! \,-- /

550 ROBERT SPIRA we can expect a reasnable prximity f the zers f f (s) and "m-i(s) whenever the remaining terms are small. Fr fixed s, the B2v terms becme small when M exceeds í/2tt (Lehmer [9]). Near a = \, M1_"/(s - 1) ~ A/1/2/i which will certainly nt be small when Af <~ i2. Turning the argument abut, we can see that t(s) is rughly apprximated by fj/(s) near the critical line fr i < 27TÜ7, but i als large enugh s that Af1/2/i is small. In Figure 3, ne can bserve the zers f "aí(s), fr successive Ü7, circling first in ne directin and then in the ther arund the zer s0 =.5 A- 37.586z f t(s) (dented by a small square). This circling can be explained by setting sm t be such a zer, then f.tf+i(s.w) = (Af 4-1), and sm can be mved slightly t s.«s that the vectrs MsM' have vercme the disturbance (Af 4-1) 'M. The disturbance fr each M will be apprximately M~s", and as this rtates depending n M, the zers Sm will be rtating ne way r anther depending n the quadrant f M"H. The successive lwest zers, als n Figure 3, appear t have imaginary part slightly less than 2ir/lg Af. Thus, the vectrs n _s have cnsecutive arguments spread between 0 and 2ir e, fr this lwest zer. This appears t be true fr every AÍ. The real parts appear t be strictly increasing with upper limit 1. As nted in Turan [2], every pint n the line a = 1 is an accumulatin pint f the zers f CmÍs), but t see the apprach ne must prceed t very large Af, as in Figure 4. Fr such large Af, the zers lie n a line which sags t the left between i lcatins f zers f f (s), and at such i lcatins there is a frcing t the right as well as a shrtening f the intervals between zers. Thus, we have described the empirical behavir f the zers f "«(s). In Turan [2] there is given a prf by Jessen that fif(s) 5 0 fr M ^ 5, by shwing that Re f.w(s) > 0 fr f^l, The present authr was able t extend this result t Af = 6 and M = 8. Hwever, ne cannt shw Re his) > 0 fr a ^ 1, but it appears that a different methd can settle this case. These matters require extensive calculatin, and further study, and will be taken up in part II f this paper. University f Tennessee Knxville, Tennessee 37916 1. P. Turan, "Nachtrag zu meiner Abhandlung 'On sme apprximative Dirichlet plynmials in the thery f zeta-functin f Riemann'," Acta Math. Acad. Sei. Hungar, v. 10, 1959, pp. 277-298. MR 22 #6774; MR 22, 2546. 2. P. Turan, "On sme apprximative Dirichlet plynmials in the thery f the zetafunctin f Riemann," Danske Vid. Selsk. Mat.-Fys Medd., v. 24, n. 17, 1948, pp. 3-36. MR 10, 286. 3. T. M. Apóstl, "Sets f values taken by Dirichlet's L-series," Prc. Symps. Pure Math., Vl. 8, pp. 133-137, Amer. Math. Sc, Prvidence, R.I., 1965. MR31 #1229. 4. R. E. Langer, "On the zers f expnential sums and integrals," Bull. Amer. Math. Sc, v. 37, 1931, pp. 213-239. 5. C. E. Wilder, "Expansin prblems f rdinary linear differential equatins with auxiliary cnditins at mre than tw pints," Trans. Amer. Math. Sc, v. 18,1917, pp. 415-442. 6. R. Spira, "Apprximate functinal apprximatins and the Riemann hypthesis," Prc. Amer. Math. Sc, v. 17, 1966, pp. 314-317. 7. R. Spira, "Table f zers f sectins f the zeta functin," UMT files. 8. R. Spira, Check Values, Zers and Frtran Prgrams fr the Riemann Zeta Functin and its First Three Derivatives, Reprt N. 1, University Cmputatin Center, University f Tennessee, Knxville, Tennessee. 9. D. H. Lehmer, "Extended cmputatin f the Riemann zeta-functin," Mathematika, v. 3, 1956, pp. 102-108. MR 19, 121, 1431. 10. R. Spira, "Zers f apprximate functinal apprximatins" Math. Cmp. (T appear.)