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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/log(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.

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`%I #8 Jan 30 2016 16:07:52
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`%S 12,19,31,41,46,53,58,65,72,77,87,94,99,103,111
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`%N 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/log(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.
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`%C 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.
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`%D Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
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`%D Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
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`%H Andrew 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>
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`%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Z_function">Z function</a>
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`%Y Cf. A117536, A117537, A117538, A054540.
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`%K hard,more,nonn
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`%O 0,1
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`%A _Gene Ward Smith_, Mar 27 2006
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