Lune, J. (Jan) van de

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

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stichting mathematisch centrum AFDELING NUMERIEKE WISKUNDE (DEPARTMENT OF NUMERICAL MATHEMATICS) J. VAN DE LUNE & H.J.J. TE RIELE RIGOROUS HIGH SPEED SEPARATION OF ZEROS OF RIEMANN'S ZETA FUNCTION, I I I ~ MC NN 29/83 FEBRUARI. kruislaan 413 1098 SJ amsterdam Printed ar the Mathematical Cenrre, Krnis/aan 413, AmsTerdam, The /\/ether/ands. The Mmhemalical Centre, founded I Ith Februa(r /946, is a 11011-prr~jit institllfion .few rhe promotion of pure and applied mathematics and computer science. It is sponsored l~r the Netherlands Government through the Netherlands Organi::.ation pJr the Advancement <?/ Pure Research ( Z. W. 0. ). 1980 Mathematics subject classification: 10H05, 10-04, 65GX:X Rigorous high speed separation of zeros of Riemann's zeta function, III by J. van de Lune & H.J.J. te Riele ABSTRACT This is an intermediate report for the purpose of announcing that the first 300,000,001 zeros of the Riemann zeta function s(s) in the critical strip are simple and lie on the line Re(s) =½.This extends our previous result that the first 250,000,000 zeros have this property, KEY WORDS & PHRASES: Riemann hypothesis~ Riemann zeta function~ Riemann-Siegei fornm.la~ Rosser's :nlie I • INTRODUCTION This note is a continuation of our report NN 26/82 [2]. The purpose of the present note is to announce that the first 300,000,001 zeros of the Riemann zeta function ~(s) in the critical strip are sirrrpZe and Zie on the Zine Re(s) = ½, This result was obtained by extending the computations (up to zero# 250,000,000) described in [2]. Extending the computations a little further we found 4 Gram blocks in [gJOO,OOO,OOO' gJ00,000,004), all of them satisfying Rosser's rule. By applying Theorem 3.2 of BRENT [1] we completed the proof of our claim. In ~ection 2 we present tables similar to those given in Section 2 of [2], this time for the range [g249 , 999 , 999 ,g300 ,000,000). The FORTRAN/ COMPASS program by which the zeros in this range were separated is the same as that explicitly presented in [2]. We hope to extend our computations in the near future. 2. THE TABLES 1 s In Table 2.1 we list the number of Gram blocks of type (L,k), Ls 8, 1 s ks L, in the interval [g 249 , 999 , 999 ,g300 ,ooo,ooo). as in NN 26/82 these counts are exact. Similarly On the lines with L = 2 and L = 3 in Table 2.1 we also list the number of exceptions to Rosser' s rule of length 2 (with "O O"-zero pattern) and th!)se of length 3 (with "O 1 O"-zero pattern), respectively. Moreover, on the line with L = 2 we also mention the three blocks with "2 2"-zero pattern found in relation to the exceptions of type 5 and 6 (cf. Table 2.3). One exception to Rosser's rule of length L = 3 was found. Note the occurrence of two blocks of type (7,1), not observed before, and one block of type (7,7), which is the second one observed (the first one being B195 , 610 , 937 ). The entries in parentheses are the approximate percentages with respect to the total number of blocks of length L, given in the final column. For all Gram blocks our strategy of finding the "missing two zeros" was the same as in [2]. Table 2.1 shows the justification of this strategy. 2 Table2.l Number of Gram blocks of type (L,k), ls Ls: 8, ls: ks L, in the interval [g249,999,999' 8300,000,000) k+ L + 2 3 34,764,288 2 2,643,484 2,643,854 + 63 blocks (50) (SO) + 3 blocks 3 565,054 62,319 564,937+1 (47) (5) (47) 4 107,657 9,992 9,979 (46) (4) (4) 5 10,511 2,159 I , 1 23 (40) (8) (4) 6 284 442 137 ( I 7) (26) (8) 7 2*) 44 21 8 0 0 1**) *)viz. B n' for n = 258,779,892 **) viz. B n' for n = 264,680,348 4 5 6 7 8 Total 34,764,288 with 0 0 zero-pattern 5,287,404 with 2 2 zero-pattern block with O l O zero-pattern 1,192,311 107,578 235,206 (46) 2,195 10,388 26,376 (8) (39) 153 448 256 1 , 7 20 (9) (26) (I 5) 4 26 34 1*) 132 0 0 2**) 0 0 3 282,307,390 258,666,950 299,608,968 respectively. 262,831,140 respectively. In Table 2.2 we present the 64 exceptions to Rosser's rule in the range [g 249 , 999 , 999 ,g300 ,000,000)' including the local extreme values of S(t). (The definitions of the various types are implicitly given in Table 2. 3) . Table 2.2 The exceptions to Rosser's rule of length 2 and 3 in the interval [g249,999,999'g300,000,000) 3 63 exceptions of length 2 ' with zero pattern "O O". 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) 2.091955 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 Notation: n (type) local extreme S(t), where n is the index of the Gram block Bn = [gn,gn+2) containing no zeros 272096379(2) 2.001554 284764536(2) 2.001422 272583009(1) -2.032037 286172640(2) 2.042925 274190882(2) 2.008416 286688824(1) -2.046407 274268747(1) -2.018420 287222173(2) 2.048065 275297430(2) 2.014738 287235535(2) 2.024894 275545477(2) 2.032087 287304862(2) 2.003208 275898480(2) 2.005068 287433571(1) -2.021945 275953000(1) -2.007296 287823551(1) -2.038399 277117197(5) -2.069283 287872423(2) 2.016959 277447311(2) 2.058999 288766616(2) 2.024072 279059658(2) 2.037126 290122964(2) 2.039001 279259145(2) 2.033129 290450849(5) -2.068090 279513637(2) 2.000375 291426142(2) 2.075531 279849070(2) 2.048163 292810354(2) 2.048278 280291419(1) -2.021221 293109862(2) 2.013978 281449426(2) 2.000609 293398055(2) 2.042772 281507954(2) 2.001841 294134427(2) 2.043302 281825600(1) -2.033191 294216438(1) -2.005490 282547094(2) 2.002833 295367142(2) 2.049246 283120964(2) 2.028096 297834112(2) 2.022351 283323493(1) -2.032511 299099970(2) 2 .030191 1 exception of length 3 with zero pattern "O l O". B266 , 527 , 881 , foUowed by a Gram block of length 1 containing 3 zeros; local extreme S(t) = -2.008550. Table 2.3 contains the frequencies of occurrence of the various types of exceptions to Rosser's rule in the range [g249 , 999 , 999 ,g300 , 000 , 000). Note that the number of exceptions of type 2 is twice the number of excep-tions of type l. There are neither exceptions of type 3 nor of type 4 in the range under consideration. Table 2.3 Vill'ious types of ex~eptions to Rosser's rule and their frequencies in [g249,999,999'g300,000,000) Exceptions of length 2 3 0 4 2 2 Gram block of length 2 without any zeros 0 0 0 0 0 0 0 0 0 0 0 0 Exceptions of length 3 } 3 Gram block of length 3 containing zero 3 4 0 2 2 frequency 1 2 3 4 5 6 frequency 0 l 20 40 0 0 2 1 4 Finally, from Tables 2.1 and 2.2 we have counted the following numbers of Gram intervals in [g 249 , 999 , 999 ,g300 ,000,000) containing exactly ro (O::;m:s:;4) zeros. 5 m 0 2 3 4 # 6,832,292 36,424,554 6,654,012 89 143*) 0 ' % 13. 7 72.8 13.3 0.2 0.0 4. REFERENCES [1] BRENT, R.P., On the zeros of the Riemann zeta function in the critical strip, Math. Comp., 33 (1979) pp. 1361-1372. [2] LUNE, J. VAN DE & 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. *)we take the opportunity to correct a misprint in the corresponding table in [2]: the printed number 88,312 should read 88,314. 

Rigorous high speed separation of zeros of Riemann's zeta function III

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