Search: id:A117537
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%I A117537
%S A117537 2,3,5,7,12,19,31,46,53,72,270,311,954,1178,1308,1395,1578,3395,4190
%N A117537 These are the locations of the midpoints of consecutive zeros of Riemann 
               zeta function on the critical line with increasingly large normalized 
               spacing; equivalently of consecutive real zeros of the Z function. 
               If t and s are consecutive zeros of the Z function, we define their 
               normalized spacing as (s-t)/(2 pi ln((s+t)/(4 pi))). The sequence 
               above is found by taking r = ln(2)(s+t)/(4 pi) and rounding to the 
               nearest integer. These values r have a marked tendency to be close 
               to integer values and all of the terms of the above sequence are 
               actually contained in the intervals [s, t]*ln(2)/(2 pi).
%C A117537 So far as the first 100000 zeros take us, the integers of the above sequence 
               actually fall inside of the normalized intervals of zeros of Z; that 
               is, they fall between two zeros of Z(2 pi t/ln(2)). It would be a 
               worthwhile project to push this computation far enough to find a 
               counterexample. The integers above, while slightly less clearly linked 
               to music than A117536 and A117538 are nevertheless very clearly closely 
               related to equal divisions of the octave. Large gaps between consecutive 
               zeros, in other words, seem to correspond to good scale divisions, 
               though less exactly than peak values or high integrals do.
%D A117537 Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
%D A117537 A. Ivic (1985). The Riemann Zeta Function, John Wiley & Sons. ISBN 0-471-80634-X.
%D A117537 Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised 
               (Heath-Brown) edition, Oxford University Press, 1986
%H A117537 A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/zeta.html">
               On the distribution of spacings between zeros of the zeta function</
               a>, Math. Comp., 48 (1987), 273-308.
%H A117537 A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html">
               The first 100,000 zeros of the Riemann zeta function, accurate to 
               within 3*10^(-9)</a>
%H A117537 Wikipedia, <a href="http://en.wikipedia.org/wiki/Z_function">Z Function</
               a>
%Y A117537 Cf. A117536, A117538, A117539.
%Y A117537 Sequence in context: A143642 A060986 A054540 this_sequence A137713 A018065 
               A048818
%Y A117537 Adjacent sequences: A117534 A117535 A117536 this_sequence A117538 A117539 
               A117540
%K A117537 hard,nonn
%O A117537 0,1
%A A117537 Gene Ward Smith (genewardsmith(AT)gmail.com), Mar 27 2006

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