#C8.txt, Feb. 12, 2018, Exp. Math. (Rutgers). Checking Riemann's famous memoir
Help:=proc():print(` xi(s), Check5B(s), ZZ(), Check5C() `): end:

#xi(s): the left side of the FE
xi:=proc(s):
Pi^(-s/2)*GAMMA(s/2)*Zeta(s):
end:

ZZ:=proc():

<<14.134725142>,<     21.022039639>,<     25.010857580>,<     30.424876126>,<     32.935061588>,<     37.586178159>,<     40.918719012>,<     43.327073281>,<     48.005150881>,<     49.773832478>,<     52.970321478>,<     56.446247697>,<     59.347044003>,<     60.831778525>,<     65.112544048>,<     67.079810529>,<     69.546401711>,<     72.067157674>,<     75.704690699>,<     77.144840069>,<     79.337375020>,<     82.910380854>,<     84.735492981>,<     87.425274613>,<     88.809111208>,<     92.491899271>,<     94.651344041>,<     95.870634228>,<     98.831194218>,<    101.317851006>,<    103.725538040>,<    105.446623052>,<    107.168611184>,<    111.029535543>,<    111.874659177>,<    114.320220915>,<    116.226680321>,<    118.790782866>,<    121.370125002>,<    122.946829294>,<    124.256818554>,<    127.516683880>,<    129.578704200>,<    131.087688531>,<    133.497737203>,<    134.756509753>,<    138.116042055>,<    139.736208952>,<    141.123707404>,<    143.111845808>,<    146.000982487>,<    147.422765343>,<    150.053520421>,<    150.925257612>,<    153.024693811>,<    156.112909294>,<    157.597591818>,<    158.849988171>,<    161.188964138>,<    163.030709687>,<    165.537069188>,<    167.184439978>,<    169.094515416>,<    169.911976479>,<    173.411536520>,<    174.754191523>,<    176.441434298>,<    178.377407776>,<    179.916484020>,<    182.207078484>,<    184.874467848>,<    185.598783678>,<    187.228922584>,<    189.416158656>,<    192.026656361>,<    193.079726604>,<    195.265396680>,<    196.876481841>,<    198.015309676>,<    201.264751944>,<    202.493594514>,<    204.189671803>,<    205.394697202>,<    207.906258888>,<    209.576509717>,<    211.690862595>,<    213.347919360>,<    214.547044783>,<    216.169538508>,<    219.067596349>,<    220.714918839>,<    221.430705555>,<    224.007000255>,<    224.983324670>,<    227.421444280>,<    229.337413306>,<    231.250188700>,<    231.987235253>,<    233.693404179>,<    236.524229666>,<    237.769820481>,<    239.555477573>,<    241.049157796>,<    242.823271934>,<    244.070898497>,<    247.136990075>,<    248.101990060>,<    249.573689645>,<    251.014947795>,<    253.069986748>,<    255.306256455>,<    256.380713694>,<    258.610439492>,<    259.874406990>,<    260.805084505>,<    263.573893905>,<    265.557851839>,<    266.614973782>,<    267.921915083>,<    269.970449024>,<    271.494055642>,<    273.459609188>,<    275.587492649>,<    276.452049503>,<    278.250743530>,<    279.229250928>,<    282.465114765>,<    283.211185733>,<    284.835963981>,<    286.667445363>,<    287.911920501>,<    289.579854929>,<    291.846291329>,<    293.558434139>,<    294.965369619>,<    295.573254879>,<    297.979277062>,<    299.840326054>,<    301.649325462>,<    302.696749590>,<    304.864371341>,<    305.728912602>,<    307.219496128>,<    310.109463147>,<    311.165141530>,<    312.427801181>,<    313.985285731>,<    315.475616089>,<    317.734805942>,<    318.853104256>,<    321.160134309>,<    322.144558672>,<    323.466969558>,<    324.862866052>,<    327.443901262>,<    329.033071680>,<    329.953239728>,<    331.474467583>,<    333.645378525>,<    334.211354833>,<    336.841850428>,<    338.339992851>,<    339.858216725>,<    341.042261111>,<    342.054877510>,<    344.661702940>,<    346.347870566>,<    347.272677584>,<    349.316260871>,<    350.408419349>,<    351.878649025>,<    353.488900489>,<    356.017574977>,<    357.151302252>,<    357.952685102>,<    359.743754953>,<    361.289361696>>;

end:

#xi(s): Checking B  on p. 21
Check5B:=proc(s)
evalf(xi(s)-xi(1-s)):
end:

#Checks fact C on p. 21
Check5C:=proc(n) local T:
T:=ZZ()[n][1]:
 if n>177 then
  RETURN(FAIL):
 fi:

evalf((n-T/(2*Pi)*log(T/(2*Pi*exp(1))))/log(T)):

end: