[seqfan] Sequences Related to the Riemann Zeta Zeros

[Paul D Hanna]

        I would like to describe a set of sequences that could be
generated from the non-trivial zeros of the Riemann Zeta function.

        While it is obvious that the following sum does not converge:
           sum(n>=1, 1/n^(1/2 + i*y) )
where (1/2 + i*y) is a Riemann Zeta Zero, perhaps certain subsets of the
integers would allow this to converge to zero.

        Suppose we define such an integer sequence {a(n)} such that

(*)        sum(n>=1, 1/a(n)^(1/2 + i*y) ) = 0    

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 (*) to be less in magnitude than the (n-1)-th partial sum.
        
        Further, it would be interesting to derive a table of related row
sequences satisfying (*) 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.  

        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.

        Here are a few sequences that satisfy (*) for six different Zeta
zeros, and would form the first row of the above tables.

        Thanks,
                Paul
------------------------------------------------------

PARI code:
? \p1000
? y=...(500+ digits of imaginary value of Zeta zero)

? S=0;i=sqrt(-1);w=1;a=0;
?
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,","))
------------------------------------------------------

Sequences that satisfy (*) for the k-th Riemann Zeta Zero, Z_k = 1/2
+i*y.

Z_1:
y=14.134725141734693790457251983562470270784257115699243175685567460149

1,2,3,4,5,6,11,13,16,20,25,30,36,44,54,65,78,93,110,
130,153,178,205,234,266,300,337,376,418,462,509,559,
611,666,723,783,845,910,978,1048,1122,1198,1277,1359,
1444,1532,1623,1717,1814,1914,2017,2123,2232,2344,2458,
2576,2696,2819,2945,3074,3205,3339,3476,3616,3759,3904,
4052,4203,4357,4514,4673,4835,5000,5168,5339,5512,5688,
5867,6049,6234,6421,6611,6805,7001,7199,7401,7605,7813,
8023,8236,8451,8670,8891,9116,9343,9572,9805,10041,10279,10520,...
 
Z_2:
y=21.022039638771554992628479593896902777334340524902781754629520403587

1,2,3,4,5,6,7,8,12,18,49,55,62,94,105,118,134,153,173,
194,217,243,272,304,339,377,418,462,509,559,612,668,727,
789,854,922,993,1067,1144,1224,1307,1393,1482,1574,1669,
1767,1868,1972,2080,2190,2304,2421,2541,2664,2791,2920,
3053,3188,3327,3469,3614,3762,3914,4069,4227,4389,4554,
4722,4894,5069,5247,5429,5614,5802,5994,6189,6387,6589,
6794,7002,7214,7429,7647,7869,8094,8322,8554,8789,9027,
9269,9514,9762,10014,10269,10527,10789,11054,11322,11594,11869,...
 
Z_3:
y=25.010857580145688763213790992562821818659549672557996672496542006745

1,3,4,7,38,56,64,72,80,89,99,110,123,138,154,171,189,
208,228,249,271,295,322,352,384,418,454,493,534,577,622,
669,719,771,825,881,939,1000,1063,1129,1197,1267,1340,
1415,1493,1574,1657,1743,1831,1921,2014,2109,2207,2308,
2411,2517,2625,2736,2849,2965,3083,3204,3328,3454,3583,
3714,3848,3985,4124,4266,4410,4557,4707,4859,5014,5171,
5331,5494,5659,5827,5997,6170,6346,6524,6705,6888,7074,
7263,7454,7648,7844,8043,8245,8449,8656,8865,9077,9292,9509,9729,...
 
Z_4:
y=30.424876125859513210311897530584091320181560023715440180962146036993

1,2,4,6,7,15,20,27,37,50,55,61,67,73,80,108,118,129,141,
154,168,184,202,221,241,262,284,307,331,356,383,413,446,
481,518,557,598,641,687,736,788,843,901,962,1025,1091,
1159,1230,1303,1379,1457,1538,1621,1707,1795,1886,1979,
2075,2173,2274,2377,2483,2591,2702,2815,2931,3049,3170,
3293,3419,3547,3678,3811,3947,4085,4226,4369,4515,4663,
4814,4967,5123,5281,5442,5605,5771,5939,6110,6283,6459,
6637,6818,7001,7187,7375,7566,7759,7955,8153,8354,...
 
Z_5:
y=32.935061587739189690662368964074903488812715603517039009280003440784

1,2,4,5,20,58,64,84,91,99,108,118,129,142,156,170,185,
201,219,238,257,277,299,323,348,374,402,432,463,495,529,
566,606,649,695,744,796,851,909,969,1031,1095,1162,1232,
1305,1381,1459,1540,1623,1709,1797,1888,1981,2077,2175,
2276,2379,2485,2594,2705,2819,2935,3054,3175,3299,3426,
3555,3687,3821,3958,4097,4239,4384,4531,4681,4834,4989,
5147,5308,5471,5637,5806,5977,6151,6328,6507,6690,6875,
7063,7254,7448,7644,7844,8046,8251,8459,8670,8883,9100,9319,...
 
Z_100:
y=236.52422966581620580247550795566297868952949521218912370091896098781

1,3,5,9,12,23,28,46,86,92,101,108,125,161,177,205,257,
282,318,331,344,358,363,368,373,388,426,456,475,535,542,
564,587,595,619,644,670,716,745,775,806,838,849,884,920,
957,995,1008,1049,1091,1135,1181,1228,1243,1293,1345,1362,
1380,1398,1416,1473,1492,1512,1532,1552,1615,1680,1702,
1725,1748,1771,1794,1817,1841,1866,1891,1916,1941,1966,
1992,2019,2046,2073,2100,2127,2155,2184,2213,2242,2271,
2300,2330,2361,2392,2423,2454,2485,2517,2550,2583,...
------------------------------------------------------

        The Riemann Zeta zeros above were obtained from the website:
"The first 100 (non trivial) zeros of the Riemann Zeta function" by
Andrew M. Odlyzko:
        http://pi.lacim.uqam.ca/piDATA/zeta100.html
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