files / problems / riemann-hypothesis / compute / q2 / certificate / analytic / normalizer-correction.txt
PASS S1 h_y<0 on full N/y domain; sigma slope >=1/2 hprime=[-2.99999999999999999999999984727262576805695586209727891711092822929909204983758293, -2.99999999999999999999999970597640637309642830241132002921406754536457343151738155] PASS C1 worst-corner correction log-rate Xi(N0,y0)<0 Xi=[-1.3631121547576400423589522875967657300943872934719933440777628515908407394885546, -1.3631121547576400423589522875967657300943872934719933440777628515908407394879328] PASS C2 d_N[g+.5logN]<0 upper=[-0.000000723601567610466618224780870655602750923947519716522954202552971352876506557990926, -0.000000723601567610466618224780870655602750923947519716522954202552971352876506557990926] PASS C3 d_N[k(N)logN]<0 upper=[-224823583.649510003167366160723268141539313360384681650541665758844098821500331608, -224823583.649510003167366160723268141539313360384681650541665758844098821500331608] PASS C4 logN0>1/2 (closed-form monotonicity premise) logN0=[13.4458777364629791302032245676066198004659044542983165751017977332881292295766285, 13.4458777364629791302032245676066198004659044542983165751017977332881292295767667] PASS M-P11 M>=1 and D^+M<0 on leg M=[1.51393270042405750988245425007599770820343485596023167002360267859216592664715259, 1.76509711189419268716304920268550761388204888708747406811699008306657690849780972] derivative_upper=[-0.50451133027033545767081288699799837738201281661457641590143844067139205185153309, -0.298186444766117197461908520344419085775031305831846518549972431852028301030938116] PASS M-P7 M>=1 and D^+M<0 on leg M=[1.49315986739475837533355386984150345074798487270680901755618993358167541352826292, 1.72228723432395698193145350921734816449467254825901088402235616420700938632411514] derivative_upper=[-0.444633310108789136278890916088302330871665757295514957256544240859656708702981508, -0.272967828747259033173217250744431575923795904634044540945046234940732564758166665] PASS M-P5 M>=1 and D^+M<0 on leg M=[1.45060297811083708451531624102618343418240863073381085149910902458919999251678204, 1.6428045916942053278020603874293434117175678094108933562710515515998999893201159] derivative_upper=[-0.353466904496848037578046667607371289546089623087098349053550632560719810678340567, -0.229123345666402720566423197725816780621275950545966563252570910982396895787716654] PASS M-P23 M>=1 and D^+M<0 on leg M=[1.3563794186568371366070958011566003063209930121446021252098952818647043777456712, 1.52217455457556177183535322252604055221338406268809787190227460128848736586618088] derivative_upper=[-0.240977237420298681521323178098982423901742975710465151061975458778591270825818064, -0.156773986617550701870743383807263416187365218866608782456399175818146232548196619] ATOM u/(1-exp(-u)) <= 1+u for u>0 follows from 1+u<=exp(u); after u=k*y*log(m), correction term rate <=g+1/y+k*log(N)+.5*log(N). RESULT ALL PASS precision 256