Search: id:A117539
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%I A117539
%S A117539 12,19,31,41,46,53,58,65,72,77,87,94,99,103,111
%N A117539 Integrals of the absolute value of the Z function between successive 
               zeros greater than or equal to the integral corresponding to 12. 
               If we define the normalized Z function by z(x) = Z(2 pi x/ln(2)), 
               then the 33rd and 34th zeros are approximately 11.82 and 12.25. Integrating 
               |z(x)| between these values gives a quantity I and the above sequence 
               is defined as the midpoints of all successive zeros of z(x) such 
               that the integral of |z(x)| is greater than or equal to I.
%C A117539 The reason for the choice of 12 as a starting point is from musical practice; 
               12 is the standard equal division of the octave of Western music. 
               The subsequent values where this integral is greater than it is for 
               12 are also equal divisions. While all the values tabulated are such 
               that the integer of the integer sequence is actually contained in 
               the interval between two successive zeros, it must eventually happen 
               that a counterexample would be found. Another interesting question 
               is the density of this sequence; it is not clear if it is increasing 
               in density or not.
%D A117539 Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
%D A117539 Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised 
               (Heath-Brown) edition, Oxford University Press, 1986
%H A117539 <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), Odlyzko, Andrew </a>
%H A117539 <a href="http://en.wikipedia.org/wiki/Z_function">Z function, Wikipedia</
               a>
%Y A117539 Cf. A117536, A117537, A117538, A054540.
%Y A117539 Sequence in context: A045067 A043882 A093330 this_sequence A136770 A155574 
               A119382
%Y A117539 Adjacent sequences: A117536 A117537 A117538 this_sequence A117540 A117541 
               A117542
%K A117539 hard,more,nonn
%O A117539 0,1
%A A117539 Gene Ward Smith (genewardsmith(AT)gmail.com), Mar 27 2006

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