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<DIV> I would like to describe a set
of sequences that could be generated from the non-trivial zeros of
the Riemann Zeta function.</DIV>
<DIV> </DIV>
<DIV> While it is obvious that the
following sum does not converge:</DIV>
<DIV> sum(n>=1,
1/n^(1/2 + i*y) )</DIV>
<DIV>where (1/2 + i*y) is a Riemann Zeta Zero, perhaps certain subsets of the
integers would allow this to converge to zero.</DIV>
<DIV> </DIV>
<DIV> Suppose we define such an
integer sequence {a(n)} such that</DIV>
<DIV> </DIV>
<DIV><STRONG>(*) </STRONG>
<STRONG>sum(n>=1, 1/a(n)^(1/2 + i*y) ) =
0 </STRONG> </DIV>
<DIV> </DIV>
<DIV>by requiring that the modulus of the partial sums be always decreasing in
magnitude, so that the sum approaches zero as a limit. The n-th
term a(n) is to be the least positive integer that causes
the n-th partial sum of <STRONG>(*)</STRONG> to be less in magnitude than
the (n-1)-th partial sum.</DIV>
<DIV> </DIV>
<DIV> Further, it would be
interesting to derive a table of related row
sequences satisfying <STRONG>(*)</STRONG> for the
same Zeta zero, such that they collectively would form a permutation
of the natural numbers. There would then be a separate table defined
for each Riemann Zeta zero. </DIV>
<DIV> </DIV>
<DIV> Perhaps someone would like to
derive some of these tables? I can generate the first row sequences, but
the subsequent rows are beyond my calculating capability at present.</DIV>
<DIV> </DIV>
<DIV> Here are a few sequences that
satisfy <STRONG>(*)</STRONG> for six different Zeta zeros, and would
form the first row of the above tables.</DIV>
<DIV> </DIV>
<DIV> Thanks,</DIV>
<DIV>
Paul</DIV>
<DIV>------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV><STRONG>PARI code:</STRONG></DIV>
<DIV><STRONG>?</STRONG> \p1000</DIV>
<DIV><STRONG>?</STRONG> y=...(500+ digits of imaginary value of Zeta zero)</DIV>
<DIV> </DIV>
<DIV><STRONG>?</STRONG> S=0;i=sqrt(-1);w=1;a=0;</DIV>
<DIV><STRONG>?</STRONG>
for(n=1,100,b=a+1;while(abs(S+exp(i*y*log(b))/sqrt(b))>w,b++);S=S+exp(i*y*log(b))/sqrt(b);w=abs(S);a=b;print1(b,","))<BR>------------------------------------------------------</DIV>
<DIV> </DIV>
<DIV><STRONG>Sequences that satisfy (*) for the k-th Riemann Zeta Zero, Z_k
= 1/2 +i*y.</STRONG></DIV>
<DIV> </DIV>
<DIV><STRONG>Z_1:</STRONG><BR>y=14.134725141734693790457251983562470270784257115699243175685567460149</DIV>
<DIV> </DIV>
<DIV>1,2,3,4,5,6,11,13,16,20,25,30,36,44,54,65,78,93,110,<BR>130,153,178,205,234,266,300,337,376,418,462,509,559,<BR>611,666,723,783,845,910,978,1048,1122,1198,1277,1359,<BR>1444,1532,1623,1717,1814,1914,2017,2123,2232,2344,2458,<BR>2576,2696,2819,2945,3074,3205,3339,3476,3616,3759,3904,<BR>4052,4203,4357,4514,4673,4835,5000,5168,5339,5512,5688,<BR>5867,6049,6234,6421,6611,6805,7001,7199,7401,7605,7813,<BR>8023,8236,8451,8670,8891,9116,9343,9572,9805,10041,10279,10520,...</DIV>
<DIV> </DIV>
<DIV><STRONG>Z_2:</STRONG><BR>y=21.022039638771554992628479593896902777334340524902781754629520403587</DIV>
<DIV> </DIV>
<DIV>1,2,3,4,5,6,7,8,12,18,49,55,62,94,105,118,134,153,173,<BR>194,217,243,272,304,339,377,418,462,509,559,612,668,727,<BR>789,854,922,993,1067,1144,1224,1307,1393,1482,1574,1669,<BR>1767,1868,1972,2080,2190,2304,2421,2541,2664,2791,2920,<BR>3053,3188,3327,3469,3614,3762,3914,4069,4227,4389,4554,<BR>4722,4894,5069,5247,5429,5614,5802,5994,6189,6387,6589,<BR>6794,7002,7214,7429,7647,7869,8094,8322,8554,8789,9027,<BR>9269,9514,9762,10014,10269,10527,10789,11054,11322,11594,11869,...</DIV>
<DIV> </DIV>
<DIV><STRONG>Z_3:</STRONG><BR>y=25.010857580145688763213790992562821818659549672557996672496542006745</DIV>
<DIV> </DIV>
<DIV>1,3,4,7,38,56,64,72,80,89,99,110,123,138,154,171,189,<BR>208,228,249,271,295,322,352,384,418,454,493,534,577,622,<BR>669,719,771,825,881,939,1000,1063,1129,1197,1267,1340,<BR>1415,1493,1574,1657,1743,1831,1921,2014,2109,2207,2308,<BR>2411,2517,2625,2736,2849,2965,3083,3204,3328,3454,3583,<BR>3714,3848,3985,4124,4266,4410,4557,4707,4859,5014,5171,<BR>5331,5494,5659,5827,5997,6170,6346,6524,6705,6888,7074,<BR>7263,7454,7648,7844,8043,8245,8449,8656,8865,9077,9292,9509,9729,...</DIV>
<DIV> </DIV>
<DIV><STRONG>Z_4:</STRONG><BR>y=30.424876125859513210311897530584091320181560023715440180962146036993</DIV>
<DIV> </DIV>
<DIV>1,2,4,6,7,15,20,27,37,50,55,61,67,73,80,108,118,129,141,<BR>154,168,184,202,221,241,262,284,307,331,356,383,413,446,<BR>481,518,557,598,641,687,736,788,843,901,962,1025,1091,<BR>1159,1230,1303,1379,1457,1538,1621,1707,1795,1886,1979,<BR>2075,2173,2274,2377,2483,2591,2702,2815,2931,3049,3170,<BR>3293,3419,3547,3678,3811,3947,4085,4226,4369,4515,4663,<BR>4814,4967,5123,5281,5442,5605,5771,5939,6110,6283,6459,<BR>6637,6818,7001,7187,7375,7566,7759,7955,8153,8354,...</DIV>
<DIV> </DIV>
<DIV><STRONG>Z_5:</STRONG><BR>y=32.935061587739189690662368964074903488812715603517039009280003440784</DIV>
<DIV> </DIV>
<DIV>1,2,4,5,20,58,64,84,91,99,108,118,129,142,156,170,185,<BR>201,219,238,257,277,299,323,348,374,402,432,463,495,529,<BR>566,606,649,695,744,796,851,909,969,1031,1095,1162,1232,<BR>1305,1381,1459,1540,1623,1709,1797,1888,1981,2077,2175,<BR>2276,2379,2485,2594,2705,2819,2935,3054,3175,3299,3426,<BR>3555,3687,3821,3958,4097,4239,4384,4531,4681,4834,4989,<BR>5147,5308,5471,5637,5806,5977,6151,6328,6507,6690,6875,<BR>7063,7254,7448,7644,7844,8046,8251,8459,8670,8883,9100,9319,...</DIV>
<DIV> </DIV>
<DIV><STRONG>Z_100:</STRONG><BR>y=236.52422966581620580247550795566297868952949521218912370091896098781</DIV>
<DIV> </DIV>
<DIV>1,3,5,9,12,23,28,46,86,92,101,108,125,161,177,205,257,<BR>282,318,331,344,358,363,368,373,388,426,456,475,535,542,<BR>564,587,595,619,644,670,716,745,775,806,838,849,884,920,<BR>957,995,1008,1049,1091,1135,1181,1228,1243,1293,1345,1362,<BR>1380,1398,1416,1473,1492,1512,1532,1552,1615,1680,1702,<BR>1725,1748,1771,1794,1817,1841,1866,1891,1916,1941,1966,<BR>1992,2019,2046,2073,2100,2127,2155,2184,2213,2242,2271,<BR>2300,2330,2361,2392,2423,2454,2485,2517,2550,2583,...</DIV>
<DIV>------------------------------------------------------</DIV>
<DIV>
<DIV> </DIV>
<DIV> The Riemann Zeta zeros
above were obtained from the website:</DIV>
<DIV>"The first 100 (non trivial) zeros of the Riemann Zeta function" by Andrew
M. Odlyzko:</DIV>
<DIV> <A
href="http://pi.lacim.uqam.ca/piDATA/zeta100.html">http://pi.lacim.uqam.ca/piDATA/zeta100.html</A></DIV></DIV></BODY></HTML>