Lune, J. (Jan) van de

Riele, H.J.J. (Herman) te

English

CWI's Institutional Repository

a the ma tisch cen trum AFDELING NUMERlEKE WISKUNDE (DEPARTMENT OF NUMERICAL MATHEMATICS) J. VAN DE LUNE & H.J.J. lE R!ELE ON THE ZEROS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP, I I I Preprint amst rdam NW 146/83 JANUARI stichting mathematisch centrum AFDELING NUMERIEKE WISKUNDE <DEPARTMENT OF NUMERICAL MATHEMATICS) J. VAN DE LUNE & H.J.J. TE RIELE ON THE ZEROS OF THE RIEMANN ZETA FUNCTION IN THE CRITICAL STRIP, I I I Preprint ~ MC NW 146/83 JANUARI kruislaan 413 1098 SJ amsterdam PrinreJ at the Marhemarical Centre, Kruislaan 413, Am.Herdam. The Nether/ands. The Mathematical Cemre, founded 11th Fehruwy 1946, is a 11011-pn~fit institlllion j<Jr the promotion of pure and applied mathematics and compurcr science. It is sponsored hy the Netherlands Covemmenr through the Netherlands Organi::ation .f1Jr the Advancement of Pure R eseorch ( Z. W. 0. ). 1980 Mathematics subject classification: Primary: 10H05 Secondary: 10-04, 65E05, 30-04 On the zeros of the Riemann zeta function in the critical strip, rrr*) by J. van de Lune & H.J.J. te Riele ABSTRACT We describe extensive computations which show that Riemann's zeta function ~(s) has exactly 300,000,001 zeros of the form o + it in the region 0 < t < 119,590,809.282. All these zeros are simple and lie on the line o = ½. (This extends a similar result for the first 200,000,001 zeros, established by Brent, van de Lune, te Riele and Winter in Math. Comp., v. 39, 1982, pp. 681-688.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser's rule" are given, together with some graphs of the func-tion Z(t) near the first observed failures of Rosser's rule. KEY WORDS & PHRASES: Gram bZoaks, Riemann hypothesis, Riemann zeta function, Rosser's rule *) This report will be submitted for publication elsewhere. I • INTRODUCTION This paper is a continuation of BRENT [I] and BRENT, VAN DE LUNE, TE RIELE & WINTER [2]. We have extended the computations described there to show that the first 300,000,001 zeros of Riemann's zeta function s(s) in the criticaZ strip are simple and lie on the line o =~-After separating the zeros of Z(t) in the range [g200,000,000'g300 ,000,000) with a slightly changed version of the FORTRAN/COMPASS program described in VAN DE LUNE, TE RIELE & WINTER [4], we ran the computation a little further, and found four Gram blocks (of length I) in [g300 000 000 ,g300 000 004 ). By applying ' , , ' Theorem 3.2 of [I) we completed the proof of our claim. A complete listing of our program may be found in VAN DE LUNE & TE . RIELE [5]. We also refer to [4] and [5] for a more detailed description of our slightly adapted strategy (compared with Brent's) for finding the required number of sign changes of Z(t) in Gram blocks of length~ 2. It may be noted here that the average number of Z-evaluations needed to separate two successive zeros of Z(t) varied slightly around 1.19. Comparing this with our previous computations, it appears that the oscillatory behaviour of Z(t) has not changed very much. Besides 126 exceptions to Rosser's rule of length 2, we also found three exceptions of length 3. The latter ones appeared as Gram blocks of length 3 containing only I zero, followed or preceeded by a Gram block of length I with 3 zeros. More details are given in the next section, including some graphs of Z(t) in the neighbourhood of various types of exceptions to Rosser's rule which have emerged. (Previously, two exceptions of length 3 were implicitly given by KARKOSCHKA & WERNER [3], viz. Bl 089 751 985 and BIO 008 051 629' ' , , ' ' , although these are far beyond the range covered by our systematic search.) In the next section we also give a number of tables which are (nearly) similar to those given in [I] and [2]. We intend to extend our computations in the near future. 2. STATISTICS The counts given here are now exact (in contrast with the LR & W -counts given in [2]).This was realised by immediately computing ZB(gn) in case Jz (g) I was too small (rather than computing ZA(g -o) for a few small A n n 2 values of o, as we did in [2]). We never met a value oft for which our method B could not determine the sign of Z(t) rigorously, with the bounds given in Section 3 of [2]. In Table I we present a list of the 126 exceptions to Rosser's rule of length 2 and the 3 of length 3, in the range [g200 000 000 ,g300 000 OOO), ' , , , including their types and the local extreme values of S(t). Note the occurrence of a type 4 - exception (the first one observed) which yields the second Gram interval with four zeros (viz. G61,331,768 [I]). G237 516 724 ; the first one is ' , Table 2 gives a survey of the various types of exceptions to Rosser's rule observed until now, and their frequencies in [g200 000 000 ,g300 000 000). , , , , Table 3 gives the numbers of Gram blocks of length S 8 in [g200 000 00O' g300,000,000) for strings of 107 successive zeros. The last line giv;s the totals for the whole range. The average Gram block length for this range is 1.2039, against 1.2003 and 1.1900 for the ranges [gIOO,OOO,OOO'g200,000,000) and [g0 ,glOO,OOO,OOO), respectively. Note that the number of Gram blocks of length I is slowly decreasing, in favour of the numbers of Gram blocks of length~ 2 and~ 5. In Table 4 we present the numbers of Gram intervals G.=[g.,g. 1), 7 J J J+ n ~ j < n + 10 , which contain exactly m zeros of Z(t), 0 s m ~ 4. Note that the number of Gram intervals with precisely one zero is slightly decreasing in favour of the numbers of Gram intervals with no zeros and those with 2 zeros. In [I] and [2] we have listed the first occurrences of Gram blocks of various types. Here we have met only one new type of Gram block, viz. the type (7,1) - Gram block B258 , 779 , 892 • In Table 5 we list the numbers of Gram blocks of type (j,k), Is j ~ 8, I ~ k s j, in the range [g200,000,000'g300 ,000,000). We also give the num-bers of Gram blocks of length 2 with zero-pattern "0 0" and "2 2", and those of length 3 with zero-pattern "0 I 0". The "0 0" - blocks correspond to the 126 length 2 - exceptions to Rosser's rule of types I - 6 and the "2 2" -blocks correspond to the length 2 - exceptions of types 5 and 6. The "0 O" -blocks correspond to the 3 (newly introduced) length 3 - exceptions of types I and 2. The entries in parentheses denote the percentages with respect to the total number of blocks of length j, given in the final column. These percentages are nearly the same as those given in the corresponding table for the range [g156,800,000'g2oo,OOO,OOO) in [2], and we conclude that our strategy of dealing with Gram blocks of length j ~ 2 is successful for 2 ~ j ~ 5. 3 In order to give the reader an impression of the erratic behaviour of Z(t), we give in Figures 1.1 - 1.8 graphs of Z(t) in the neighbourhood of the first (observed) exceptions to Rosser's rule of length 2 and 3 and of various types. We have plotted Z(g) with x as a continuous independent variable. The excep-x tional Gram block is marked by two arrows pointing downwards. The adjacent Gram block where the "missing two" zeros are situated is marked by two arrows pointing upwards. A magnification of the latter block is shown in an accompanying graph. Some "critical" values of Z(t) are explicitly mentioned. REFERENCES [1] BRENT, R.P., On the zeros of the Riemann zeta function ~n the critical strip, Math. Comp., 33 (1979), pp. 1361-1372. [2] BRENT, R.P., J. VAN DE LUNE, H.J.J. TE RIELE & D.T. WINTER, On the zeros of the Riemann zeta function in the cY'iticai strip.II, Math. Comp., ~ (1982), pp.681-688. [3] KARKOSCHKA, E. & P. WERNER, Einige Ausnahmen zur Rosserschen RegeZ in der Theorie der Riemannschen Zetafunktion, Computing, 27 (1981), pp. 57-69. [4] LUNE, J. VAN DE, H.J.J. TE RIELE & D.T. WINTER, Rigorous high speed separation of zeros of Riemann's zeta function, Report NW 113/81, October 1981, Mathematical Centre, Amsterdam. [5] LUNE, J. VAN DE & H.J.J. TE RIELE, Rigorous high speed separation of ze:r>as of Riemann's zeta function.II, Report NN 26/82, June 1982, Mathematical Centre, Amsterdam. TABLE I The 129 exceptions to Rosser's rule in [g2oo,ooo,ooo•g300 , 000 , 000) 126_of_length_2.Notation: n(type)extreme S(t), 201007375(1) 201030606(2) 201184291 (2) 201685414(5) 202762876(2) 202860958(2) 203832578(2) 205880544(1) 206357111(1) 207159768(2) 207167344(2) 207669541(2) 208053426(1) 208110028(2) 209513827(2) 212623522(1) 213841715(1) 214012333(1) 214073567(1) 215170601(2) 215881040(2) 216274605(2) 216957121(2) 217323208(1) 218799264(1) 218803558(2) 219735146(1) 219830063(2) 219897904(1) 221205545(1) 223601929(1) 223907077(2) 223970397(1) 224874046(6) 225291157(1) 227481734(1) 228006443(2) 228357900(1) 228386399(1) 228907446(1) 228984551(2) 229140286(2) -2.002900 2.1113gs 2.001518 -2.016715 2.011439 2.018888 2.063611 -2.017679 -2.031216 2.033954 2.029320 2.020740 -2.073357 2.031212 2.023920 -2.010194 -2.024334 -2.010937 -2.009287 2.007728 2.021267 2.052279 2.032421 -2.013607 -2.040304 2.013448 -2.026815 2.015232 -2.081132 -2.0)4535 -2.101580 2.007094 -2.028754 2 .022804 -2 .152675 -2.018298 2.023042 -2.022758 -2.008899 -2.018338 2.032004 2.000109 where n is the index of the Gram block Bn = [gn,gn+2) with zero-pattern "O O" 23UH0024(1) 232838063(2) 234389089(2) 235588194(1) 236645695(1) 236962877 (2) 237516725(4) 240004911 (1) 240221307(2) 241549003(1) 241729717(1) 241743685(2) 243780201 (2) 243801 317(1) 244122072(1) 244691225(2) 244841577(1) 245813461 (1) 246299475(1) 2464501 77 ( 2) 249069349(1) 250076378(1) 252442158(2) 252904232(2) 255145220(1) 255285972(2) 2567D230(1) 257992082(1) 258447957(6) 259298046(2) 262141503(1) 263681744(2) 266617122(1) 266628045(2) 267305763(1) 267388405(2) 267441673(2) 267464886(1) 267554908(2) 269787480(1) 270881434(1) 270997584(2) -2. 026611 2.022488 2.106429 -2.001915 -2 .08%39 2.023259 2.108817 -2.000249 2.096293 -2.036151 -2.025503 2.()701':>5 2.025648 -2.020358 -2.035325 2.018927 -2.053021 -2.035731 -2.001039 2.116655 -2.020698 -2.036397 2.085094 2.112235 -2.002286 2.034861 -2.015377 -2.042307 2.0056.55 2 .0919.55 -2.006009 2.006016 -2.046423 2.048158 -2.028836 2.012716 2.085691 -2.006418 2.112706 -2.080890 -2.026487 2.021752 272096379(2) 272583009(1) 274190882(2) 274268747(1) 275297430(2) 275545477(2) 275898480(2) 275953000(1) 277117197(5) 277447311(2) 279059658(2) 279259145(2) 279513637(2) 279849070(2) 280291419(1) 281449426(2) 281507954(2) 281825600(1) 282547094(2) 283120964(2) 283323493(1) 2847645%(2) 286172640(2) 286688824(1) 287222173(2) 287235535(2) 287304862 ( 2) 287433571(1) 287823551(1) 287872423(2) 288766616(2) 290122964(2) 290450849(5) 291426142(2) 292810354(2) 293109862(2) 293398055(2) 294134427(2) 294216438(1) 295367142(2) 297834112(2) 299099970(2) 2.001554 -2.032037 2.008416 -2.018420 2.014738 2.032087 2.005068 -2.007296 -2.069283 2.058999 2.037126 2.033129 2.000375 2.048163 -2.021221 2.000609 2.001841 -2.033191 2.002833 2.028096 -2.032511 2.001422 2.042925 -2.046407 2.048065 2.024894 2.003208 -2.021945 -2.038399 2.016959 2 .024072 2.039001 -2.068090 2.075533 2.048278 2.013978 2.042772 2.043302 -2.005490 2.049246 2.022351 2.030191 3 _ of _length_ 3. Notation: n(type)extreme S(t), where n is the index of the Gram block B = [g g ) with zero-pattern "O I O" n n' n+3 207482540(2) 2.000431 241389213(1) -2.010430 266527881(2) -2,008550 4 (For the definition of the types in case of length 3 exceptions, see Table 2.) \'I' 0 2 3 4 TABLE 2 Various types of exaeptions to Rosser 1s rule and their frequenaies in [g200 000 000 ,g300 000 000 ) ' ' ' , lGram block of length 2 with "0 0" zero-pattern LENGTH= 2 type frequency \ii' '" ,, r ,v 0 0 3 1 52 0 0 2 68 0 0 4 0 3 0 0 0 I 4 I 0 0 2 2 5 3 5 2 0 0 6 2 Gram block of length 3 l with "0 1 0" zero-pattern r 0 0 TABLE 3 Number of Gram bloaks of given length 7 J'(k,n) := J(k,n+I0) - J(k,n) LENGTH= 3 type frequency 2 : _] n J' (1,n) J' (2,n) J' (3,n) J' (4,n) J 1 (5,n) J' (6,n)J' (7,n)J' (8,n) 200,000,000 6,973,019 1,056,242 236,180 44,997 4,838 272 21 0 210,000,000 6,971,273 1,055,810 236,438 45,249 4,976 297 17 220,000,000 6,966,636 1,056,779 236,573 45,864 4,921 312 22 0 230,000,000 6,965,176 1,056,494 236,663 46,250 4,979 295 26 0 240,000,000 6,961,469 1,057, 1 20 237,279 46,150 5,156 327 16 0 250,000,000 6,957,609 1,057,612 237,551 46,633 s, 145 341 29 260,000,000 6,955,568 1,056,920 238,418 46,694 5,300 306 30 2 270,000,000 6,951,895 1,057,635 238,483 47, 21 1 5,257 353 20 0 280,000,000 6,950,241 1,056,940 238,974 47,373 5,420 366 24 0 290,000,000 6,948,974 1,058,297 238,885 47,295 5,254 354 29 ·o ----------------------- --------------Totals 69,601,860 10,569,849 2,375,444 463,716 51,246 3,223 234 5 1 2 3 4 5 6 7 8 TABLE 4 Number of Gram intervals containing exactly m zeros n m = 0 m = 1 m = 2 m = 3 m = 4 200,000,000 1,360,273 7,297,177 1,324,827 17,723 0 210,000,000 1,360,333 7,296,878 1,325,245 17,544 0 220,000,000 1,362,066 7,293,463 1,326,876 17,595 0 230,000,000 1,362,561 7,292,733 1,326,852 17,853 240,000,000 1,363,647 7,290,305 1,328,449 17,599 0 250,000,000 I ,365 ,010 7,287,678 1,329,614 17,698 0 260,000,000 1,365,698 7,286,632 1 , 329,642 18,028 0 270,000,000 1,366,741 7,284,300 1,331,176 17,783 0 280,000,000 1,367,037 7,283,866 1,331,158 17,939 0 290,000,000 1,367,806 7,282,080 1,332,422 17,692 0 ---------------------------------------. --------------------Totals 13,641,172 72,895,112 13,286,261 177,454 Number of Gram blocks of type (j,k), I s j s 8, 1 s ks j, in the range [g200,ooo,ooo•g300,ooo,ooo) k -+ I 69,601,860 5,285,566 (50) I, 125,098 (4 7) 212,286 (46) 20,326 (40) 488 ( 15) 2*) 0 2 3 4 5 6 7 8 Totals 5,284,152 (50) 126 "0 0" - blocks 5 "2 2" - blocks 124,535 l ,125,808 (S) (47) 19,862 19,649 (4) (4) 4,252 (8) 866 (27) 73 0 2, l 77 (4) 263 (8) 42 2 3 "0 I 0" - blocks 21 l, 919 (46) 4,346 20,145 (8) (39) 286 859 461 (9) (27) (14) 9 0 40 67 1*) 0 3 0 0 69,601,860 10,569,849 2,375,444 463,716 5 I ,246 3,223 234 5 *)viz. Bn' for n 258,779,892, 282,307,390 and 299,608,968, 6 10.0 Z(gx) t..,. X -ss.o 0,88 BIJ,999,525 type=I + length=2 + +see + below Z(g 13,999,527)=-.004539 -0, 16 Fi!1i• 1, l B13, 999,525 x= 13,999, 528/1 25,0 B30,930,927 -1- type=2 -l-lengl;:h=2 below I -40,0 0,4 x=30,930,927 i -~--~r-----+- -----1-------f x=30,930,926 --4--~~~ -0.12 Fig. 1,2 B30,930,927 --..J 18,0 + + length=2 +see t type=3 beiow B6l,33l,766 -8,0 0,056 I r•61,33J,768 ? 0~7:;/,; 7 ~✓ I I I /" \ I / x=61,331,769 -0.048 Fig. 1.3 B61,331,766 6.0 -20.0 0,040 B237,516,725 type=4 + length=2 + +see + below ,:;_ I I ~ 4,-~ ;( x•237,5J6,725 x /237,516,724 ~I I /~\ -J -0,064 Fig. 1,4 B237,516,725 (P 7.0 -45.0 0, I Bl9I,297,535 type=5 Hength=2 + bel-0/.,J + 1 x=J9l,297,537 x=l91,297,538 x=l91,297,539.j, ..J 7 k:: I I I ~I /c::r-,,\ +--~ Z(g 191,297,537,16)=+.001976 -0.55 Fig. 1.5 BJ 91,297,535 25.0 Bl82,105,257 type=6 + length=2 + + see bel-OI;.) + -40.0 0.24 x=l82, 105,255 -L~-~t---0.8 Z(g 182,105,257)=-.008240 xz I 82, I 05 ,256 x=l82,105,257t r I Fig, 1.6 BJ82,I05,257 \D 20.0 length=3 tsee t bel(Jl,) -45.0 o. 16 itx=24l ,389,216 x=241,389,217¾ Z(g24J,389,216)=+.005933 -0.36 Fig. I, 7 8241,389,213 20.0 -32,0 0,050 + tsee t length=3 \type=2 below B 207,482,54 Z(g207,482,540)= +. 000322 x=207,482,539 I- x=207,482,540,y l-'.---------1---+---A-----t--t-----+-----t--'t--t----t----0.054 Fig. 1.8 B207,482,540 0 

On the zeros of the Riemann zeta function in the critical strip III

MATHEMATICS OF COMPUTATION VOLUME 41, NUMBER 164 OCTOBER 19~3. PAGES 759- 767 On the Zeros of the Riemann Zeta Function in the Critical Strip. HI By J. van de Lune and H.J. J. te Riele Abstract. We describe extensive computations which show that Riemann's zeta function .\(s) has exactly 300,000,00 I zeros of the form a + ii in the region 0 < 1 < 119,590,809.282. All these zeros are simple and lie on the line a "-' j. (This extends a similar result for the first 200,000.001 zeros, established by Brent. van de Lune. te Riele and Winter in Math. Comp., v. 39, 1982, pp. 681-688.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser's rule" are given, together with some graphs of the function Z( l) near the first observed failures of Rosser's rule. 1. Introduction. This paper is a continuation of Brent [ l] and Brent, van de Lune, te Riele and Winter [2]. We have extended the computations described there to show that the first 300,000,001 zeros of Riemann's zeta function rcs) in the critical strip are simple and lie on the line a = 1- After separating the zeros of Z( t) in the range [ g200.ooo.(Xl0• g300.()(XJ.()()()) with a slightly changed version of the FORTRAN/COM-PASS program described in van de Lune, te Riele and Winter [4], we ran the computation a little further and found four Gram blocks (of length 1) in [ g30o.cxxJ,()()()• g300 .()()().cx>4 ). By applying Theorem 3.2 of [1], we completed the proof of our claim. A complete listing of our program may be found in van de Lune and te Riele [5]. We also refer to [4] and [5] for a more detailed description of our slightly adapted strategy (compared with Brent's) for finding the required number of sign changes of Z( t) in Gram blocks of length;:;. 2. It may be noted here that the average number of Z-evaluations needed to separate two successive zeros of Z( t) varied slightly around 1.19. Comparing this with our previous computations, it appears that the oscillatory behavior of Z( t) has not changed very much. Besides 126 exceptions to Rosser's rule of length 2, we also found three exceptions of length 3. The latter ones appeared as Gram blocks of length 3 containing only 1 zero, followed or preceded by a Gram block of length 1 with 3 zeros. More details are given in the next section, including some graphs of Z(t) in the neighborhood of various types of exceptions to Rosser's rule which have emerged. (Previously, two exceptions of length 3 were implicitly given by Karkoschka and Werner [3], viz., B 1,089.751 .985 andB 10,008 ,051 .629 , although these are far beyond the range covered by our systematic search.) In the next section we also give a number of tables which are (nearly) similar to those given in [ 1] and [2]. We intend to extend our computations in the near future. Received February 8, 1983. 1980 Mathematics Subject Classification. Primary !OHOS; Secondary 10-04, 65EOS, 30-04. Key words and phrases. Gram blocks, Riemann hyoothesis, Riemann zeta function, Rosser's rule. 759 COl983 American Mathematical Society 0025-5718/83 $1.00 + $.25 per page 760 J. VAN DE LUNE AND H.J. J. TERIELE 2. Statistics. The counts given here are now exact (in contrast with the LR & W-counts given in [2]). This was realized by immediately computing Z8 (gn) in case I ZA(gn) I was too small (rather than computing ZA(gn - 8) for a few small values of 6, as we did in [2]). We never met a value of t for which our method B could not determine the sign of Z(t) rigorously, with the bounds given in Section 3 of [2]. TABLE 1 The 129 exceptions to Rosser' s rule in [ g200.ooo.ooo, g300.ooo.ooo) 126 of length 2. Notation: n (type) extreme S(t), 201007375(1) -2.002900 201030606(2) 2.111895 201184291(2) 2.001518 201685414(5) -2.016715 202762876(2) 2.011439 202860958(2) 2.018888 203832578(2) 2.063611 205880544(1) -2.017679 206357111(1) -2.031216 207159768(2) 2.033954 207167344(2) 2.029320 207669541(2) 2.020740 208053426(1) -2.073357 208110028(2) 2.031212 209513827(2) 2.023920 212623522(1) -2.010194 213841715(1) -2.024334 214012333(1) -2.010937 214073567(1) -2.009287 215170601(2) 2.007728 215881040(2) 2.021267 216274605(2) 2.052279 216957121(2) 2.032421 217323208(1) -2.013607 218799264(1) -2.040304 218803558(2) 2.013448 219735146(1) -2.026815 219830063(2) 2.015232 219897904(1) -2.081132 221205545(1) -2.014535 223601929(1) -2.101580 223907077(2) 2.007094 223970397(1) -2.028754 224874046(6) 2.022804 225291157(1) -2.152675 227481734(1) -2.018298 228006443(2) 2.023042 228357900(1) -2.022758 228386399(1) -2.008899 228907446(1) -2.018338 228984553(2) 2.032004 229140286(2) 2.000109 where n is the index of the Gram block Bn = [gn, gn+ 2) with zero-pattern "O O" 231810024(1) -2.026611 232838063(2) 2.022488 234389089(2) 2.106429 235588194(1) -2.001915 236645695(1) -2.089639 236962877(2) 2.023259 237516725(4) 2.108817 240004911(1) -2.000249 240221307(2) 2.096293 241549003(1) -2.036151 241729717(1) -2.025503 241743685(2) 2.070155 243780201(2) 2.025648 243801317(1) -2.020358 244122072(1) -2.035325 244691225(2) 2.018927 244841577(1) -2.053021 245813461(1) -2.035731 246299475(1) -2.001039 246450177(2) 2.116655 249069349(1) -2.020698 250076378(1) -2.036397 252442158(2) 2.085094 252904232(2) 2.112235 255145220(1) -2.002286 255285972(2) 2.034861 256713230(1) -2.015377 257992082(1) -2.042307 258447957(6) 2.005655 259298046(2) 2.0919~5 262141503(1) -2.006009 263681744(2) 2.006016 266617122(1) -2.046423 266628045(2) 2.048158 267305763(1) -2.028836 267388405(2) 2.012716 267441673(2) 2.085691 267464886(1) -2.006418 267554908(2) 2.112706 269787480(1) -2.080890 270881434(1) -2.026487 270997584(2) 2.021752 272096379(2) 2.001554 272583009(1) -2.032037 274190882(2) 2.008416 274268747(1) -2.018420 275297430(2) 2.014738 275545477(2) 2.032087 275898480(2) 2.005068 275953000(1) -2.007296 277117197(5) -2.069283 277447311(2) 2.058999 279059658(2) 2.037126 279259145(2) 2.033129 279513637(2) 2.000375 279849070(2) 2.048163 280291419(1) -2.021221 281449426(2) 2.000609 281507954(2) 2.001841 281825600(1) -2.033191 282547094(2) 2.002833 283120964(2) 2.028096 283323493(1) -2.032511 284764536(2) 2.001422 286172640(2) 2.042925 286688824(1) -2.046407 287222173(2) 2.048065 287235535(2) 2.024894 287304862(2) 2.003208 287433571(1) -2.021945 287823551(1) -2.038399 287872423(2) 2.016959 288766616(2) 2.024072 290122964(2) 2.039001 290450849(5) -2.068090 291426142(2) 2.075533 292810354(2) 2.048278 293109862(2) 2.013978 293398055(2) 2.042772 294134427(2) 2.043302 294216438(1) -2.005490 295367142(2) 2.049246 297834112(2) 2.022351 299099970(2) 2.030191 3 of length 3. Notation: n (type) extreme S(t), where n is the index of the Gram block Bn = [ gn, gn+ 3 ) with zero-pattern "O 1 O" 207482540(2) 2.000431 241389213(1) -2.010430 266527881(2) -2.008550 {For the definition of the types in case of length 3-exceptions, see Table 2.) 0 2 n 200,000,000 210,000,000 220,000,000 230,000,000 240,000,000 250,000,000 260,000,000 270,000,000 280,000,000 290,000,000 \II 3 4 2 ZEROS OF THE RIEMANN ZETA FUNCTION TABLE 2 Various types of exceptions to Rosser 's rule and their frequencies in [g200,000,000 , g3oo.ooo.ooo) Gram block of length 2 with "O O" zero-pattern '/ \ii 0 0 0 0 0 0 0 0 0 0 0 0 'ii 3 4 2 Gram block of length 3 with "O I O" zero-pattern 1V LENGTH = 2 type frequency \V I 52 2 68 0 3 0 4 I 2 5 3 6 2 LENGTH = 3 type frequency 1,I:I I:I 3 2 TABLE 3 Number of Gram blocks of given length J'(k, n) := J(k, n + 107 ) - J(k, n) 761 J 1 (I , n) J' (2 ,n) J' (3,n) J' (4,n) J' (5,n) J 1 (6,n)J'(7,n)J'(8,n) 6,973,019 I ,056,242 236, I 80 44,997 4,838 27 2 21 0 6,971,273 I ,055,810 236,438 45,249 4,976 297 I 7 2 6,966,636 I ,056, 779 236 ,573 45,864 4,921 312 22 0 6,965, 176 I , 056, 494 236,663 46,250 4, 979 295 26 0 6,961,469 1,057, 120 237 ,279 46,150 5, 156 327 16 0 6,957,609 I , OS 7, 6 12 237,551 46,633 5, 145 341 29 6,955,568 1,056,920 238.,418 46, 694 5,300 306 30 2 6,951,895 1,057,635 238,483 47 ,211 5,257 353 20 0 6,950,241 1,056,940 238,974 47,373 5,420 366 24 0 6,948,974 I ,058,297 238,885 47,295 5,254 354 29 0 ------------------------------------------------------------------------------------Totals 69,601 ,860 10,569,849 2,375,444 463, 716 51'246 3,223 234 5 762 J. VAN DE LUNE AND H.J. J. TE RIELE TABLE 4 Number of Gram intervals containing exactly m zeros n m = 0 m :::::: 1 m = 2 m = 3 m = 4 200,000,000 1,360,273 7,297,177 1,324,827 17 '723 0 210,000,000 1,360,333 7,296,878 1 ,325,245 17,544 0 220,000,000 1 , 362, 066 7,293,463 1,326,876 17,595 0 230,000,000 1,362,561 7,292,733 1,326,852 17,853 240,000,000 1,363,647 7,290,305 1,328,449 17,599 0 250,000,000 1,365,010 7,287,678 1,329,614 17,698 0 260,000,000 1, 365 ,698 7,286,632 1 ,329,642 18,028 0 270,000,000 1,366,741 7,284,300 I, 331, 1 76 17,783 0 280,000,000 I ,367 ,037 7,283,866 1, 331, 1 58 17,939 0 290,000,000 1,367,806 7,282,080 I ,332,422 17,692 0 -----------------------------------------------------------------------Totals 13,641,172 72,895,112 13,286,261 177 ,454 TABLE 5 Number of Gram blocks of type ( j, k), I ,,;;;,.} ,,,;: 8, I,,,;: k ~j, in the range [ g2(x1.cxx1,1xxp K3<X1,<xx1,<xx1) k + t j I 4 5 6 7 8 Totals 69,601 ,860 69,601 ,860 2 5,285,566 5,284,152 126 11 0 0 11 - blocks 10,569,849 (SO) (SO) 5 11 2 2 11 - blocks 3 I, 12S,098 124,535 1, 125, 808 3 "O 1 O" - blocks 2,375,444 ( 4 7) (5) (47) 4 212,286 19,862 19,649 211, 919 463,716 (46) (4) (4) ( 46) 20,326 4,252 2, 177 4,346 20, 14S 51 ,246 (40) (8) (4) (8) (39) 6 488 866 263 286 8S9 461 3,223 ( 15) (27) (8) (9) (27) ( 14) 2*) 13 42 9 40 67 I*) 234 8 0 0 2 0 0 0 0 5 *) . Vl.Z. Bn, for n = 258, 779, 892, 282,307,390 and 299,608,968. In Table I we present a list of the 126 exceptions to Rosser's rule of length 2 and the 3 of length 3, in the range [ g200,000,000 , g300,000,000 ), including their types and the local extreme values of S(t). Note the occurrence of a type 4-exception (the first one observed) which yields the second Gram interval with four zeros (viz. G237•516.724 ; the first one is G61 ,331 ,768 [l]). Table 2 gives a survey of the various types of exceptions to Rosser's rule, observed until now, and their frequencies in [ g200,000,000 , g300,000,000 ). BJ3,999,5l5 type= I length=~ .,, =110,0 I-' /\ ~~~/f \1 - Z(gx) \ t --:-~~ - t I I ~ , \ - t \ I \ I . I -I =I I l-55. 0 j -~ 0.88 \ I I I \ I \ ,1 ' I \_-' ~/~~I 3 ·-~9:, 527 )=-. 004539 - ~, - -r- - r--~---1==-_j x=13,999,527 --_j_O. I 6 t De Loi) t- - ---+--- + - ---j x=J3,999,5281' FIGURE I. I. B 13,999.525 25.0 t type=2 ; BJ0,930,927 I ~------~L~~~-eng{fl=~--t -r l tsee t \ beloUJ j-40.0 --o.4 ' '\ \ '\\, \ -+--t x=JD,930,926 -0. I 2 ~ x=30,930,92: I·~~-='-~=,~ FIGURE 1.2. BJ0.930.927 N m ~ 0 en 0 "I1 m ~ I ~ "I1 c:: z 8 0 z -.::i a> ~ i 18,0 -/\ I \ I . =1l I \ I , -I ! 1 t (\ I~ I I + --- +-- -- + -~~~ -f- _2 tsee t -1 I I -!/ I ~r . -8.0 -10. 056 -[ ~i\ I· - i \ length- belOi,, type=3 Bbl ,331,766 i ~ _J \ -1. ,-x=61,331,76H ~=r \+ .. t -+--;.+--+ --+---1 \ / -1 \ - ! '--..._,..,,/ _ jJ-0.048 --+- - I I x=61,331,769 FIGURE 1.3. B61,331,766 6.0 -20,0 ~ 0.040 j 8 237 ,516, 725 type=4 + length=2 + >+---k--t---t--+see t bel01J x=237,516,725 ,j,. ---1 -- -+- -+--+--r---t--',1--"i=! 237,516,724 -0,064 FIGURE 1.4. B237,516,725 ...., a> II>-'.-< ~ tl tT1 r c z m ~ tl ~ ~ ~ @ ~ ~ m -45.0 0. I Bl91,297,535 type=S Hength=2 ' ~--t--r.-::4--~ 1 t see beiow t I >tx=J9J,297,537 x;;~S38 x=J91,297,539.J, ----"--- --+---+---+--;- \--+- +---+ \~/ \ Z(g I 9 I , 2 9 7 , 5 3 7 , J 6) =+ • 00 I 9 7 6 -0.55 FIGURE 1.5. Bl9l,297,535 I 25,0 ~ ''" "" "" ~-I type=6 --1 + length=2 ~ ---+ -K-+- --+--++ t see below t -40,0 0.24 x=l82, 105,255 "- + - t--0,8 Z(gl82,JD5,257)=-.008240 j- - ,,, "'~',"-''"17 FIGURE 1.6. B1s2,ws,2s1 N m "" 0 tJl 0 'T1 i m "" m ~ z N ~ 'T1 c:: z g 0 z ...., "' <:JI 20.0 -45.0 o. 16 tsee t below x=241,389,2!6 x=241,389,217~ ~.t, I :J-- --=r===r=----- ..J I --I-- ----\ Z(g241,389,216)=+.005933 ----------0,36 FIGURE 1.7. B241,389,213 20.0 c+-below -32,0 ~0.050 x=207,482,539 -/. -0.054 B207 482 541 . ' --t ----+--- ~ {\~" =2 Z(g207,482,540)= +.000322 x=207 ,482,540¥ +--\--+- ·- +----FIGURE 1.8. B207,482.540 -.:i Cl> Cl> ..... < > z 0 tn r c z tn > z 0 ;i: ..... ..... @ ::0 m r tn ZEROS OF THE RIEMANN ZETA FUNCTION 767 Table 3 gives the numbers of Gram blocks of length.;;;; 8 in [ g200.000,000 , g300.000,000 ) for strings of 107 successive zeros. The last line gives the totals for the whole range. The average Gram block length for this range is 1.2039, against 1.2003 and 1.1900 for the ranges [ g100,000.000 , g200,000•000 ) and [ g0 , g100,ooo,ooo ), respectively. Note that the number of Gram blocks of length 1 is slowly decreasing in favor of the number of Gram blocks of lengths ;;;;., 2 and .;;;; 5. In Table 4 we present the number of Gram intervals G1 = [g1, g1+ 1), n o;;;j < n + 107, which contain exactly m zeros of Z(t), 0.;;;; m.;;;; 4. Note that the number of Gram intervals with precisely one zero is slightly decreasing in favor of the number of Gram intervals with no zeros and those with 2 zeros. In [ l] and [2] we have listed the first occurrences of Gram blocks of various types. Here we have met only one new type of Gram block, viz. the type (7, 1)-Gram block B258,779,892 • In Table 5 we list the number of Gram blocks of type {j, k ), 1 .;;;,. j ..;; 8, I ..;; k ..;; j, in the range [g200,000,000 , g300,000,000 ). We also give the number of Gram blocks of length 2 with zero-pattern "O O" and "2 2", and those of length 3 with zero-pattern "O 1 O". The "O O"-blocks correspond to the 126 length 2-exceptions to Rosser's rule of types 1-6, and the "2 2"-blocks correspond to the length 2-exceptions of types 5 and 6. The "O 1 O"-blocks correspond to the 3 (newly introduced) length 3-excep-tions of types 1 and 2. The entries in parentheses denote the percentages with respect to the total number of blocks of length j, given in the final column. These percentages are nearly the same as those given in the corresponding table for the range [ g156.800.000 , g200,cl00,ooo) in [2], and we conclude that our strategy of dealing with Gram blocks of length) ;;;;., 2 is successful for 2 .;;;,. j ..;; 5. In order to give the reader an impression of the erratic behavior of Z( t ), we give in Figures 1.1-1.8 graphs of Z(t) in the neighborhood of the first (observed) exceptions to Rosser's rule of lengths 2 and 3 and of various types. We have plotted Z(gx) with x as a continuous independent variable. The exceptional Gram block is marked by two arrows pointing downwards. The adjacent Gram block where the "missing two" zeros are situated is marked by two arrows pointing upwards. A magnification of the latter block is shown in an accompanying graph. Some "critical ... values of Z( t) are explicitly mentioned. Mathematical Centre Kruislaan 413 1098 SJ Amsterdam, The Netherlands I. R. P. BRENT, "On the zeros of the Riemann zeta function in the critical strip," Math. Comp., v. 33, 1979, pp. 1361-1372. 2. R. P. BRENT, J. VAN DE LUNE, H.J. J. TE RIELE & D. T. WINTER, "On the zeros of the Riemann zeta function in the critical strip. II," Math. Comp., v. 39, 1982, pp. 681-688. 3. E. KARKOSCHKA & P. WERNER, "Einige Ausnahmen zur Rosserschen Regel in der Theorie der Riemannschen Zetafunktion," Computing, v. 27, 198,l, pp. 57-69. 4. J. VAN DE LUNE, H. J. J. TE RIELE & D. T. WINTER, Rigorous High Speed Separation of Zeros of Riemann's Zeta Function, Report NW 113/81, October 1981, Mathematical Centre, Amsterdam. 5. J. VAN DE LUNE & H.J. J. TE RIELE, Rigorous High Speed Separation of Zeros of Riemann's Zeta Function, II, Report NN 26/82, June 1982, Mathematical Centre, Amsterdam. 

On the zeros of the Riemann zeta function in the critical strip III

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