%I A117538
%S A117538 2,5,7,12,19,31,41,53,72,130,171,224,270
%N A117538 Locations of the increasing peak values of the integral of the absolute 
               value of the Riemann zeta function between successive zeros on the 
               critical line. This can also be defined in terms of the Z function; 
               if t and s are successive zeros of a renormalized Z function, z(x) 
               = Z(2 pi x/ln(2)), then take the integral between t and s of |z(x)|. 
               For each successively higher value of this integral, the corresponding 
               term of the integer sequence is r = (t+s)/2 rounded to the nearest 
               integer.
%C A117538 The fractional parts of the numbers r = (t+s)/2 above are very unevenly 
               distributed. For all of the values in the table, the integers are 
               in fact the unique integers contained in the interval of zeros [t, 
               s] of z(x). An interesting challenge to anyone wishing to do computations 
               related to the zeta function would be to find the first counterexample, 
               where in fact the peak value interval did not contain the corresponding 
               integer. Perhaps even more than the peak values of the zeta function 
               themselves, these integrals are extremely closely related to relatively 
               good equal divisions of the octave in music theory.
%D A117538 Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
%D A117538 Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised 
               (Heath-Brown) edition, Oxford University Press, 1986
%D A117538 Paris, R. B. and Kaminski, D., Asymptotics and Mellin-Barnes Integrals, 
               Cambridge University Press, 2001
%H A117538 <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 A117538 <a href="http://en.wikipedia.org/wiki/Z_function">Z function, Wikipedia</
               a>
%Y A117538 Cf. A117536, A117537, A117539, A054540.
%Y A117538 Sequence in context: A023564 A005895 A135525 this_sequence A001060 A042343 
               A042691
%Y A117538 Adjacent sequences: A117535 A117536 A117537 this_sequence A117539 A117540 
               A117541
%K A117538 hard,more,nonn
%O A117538 0,1
%A A117538 Gene Ward Smith (genewardsmith(AT)gmail.com), Mar 27 2006