Riemann-von Mangoldt Formula

Rho values

rhovalues = {14.134725141734693790457251983562470270784257115699243175685567460149 I, 21.022039638771554992628479593896902777334340524902781754629520403587 I, 25.010857580145688763213790992562821818659549672557996672496542006745 I, 30.424876125859513210311897530584091320181560023715440180962146036993 I, 32.935061587739189690662368964074903488812715603517039009280003440784 I, 37.586178158825671257217763480705332821405597350830793218333001113622 I, 40.918719012147495187398126914633254395726165962777279536161303667253 I, 43.327073280914999519496122165406805782645668371836871446878893685521 I, 48.005150881167159727942472749427516041686844001144425117775312519814 I, 49.773832477672302181916784678563724057723178299676662100781955750433 I, 52.970321477714460644147296608880990063825017888821224779900748140317 I, 56.446247697063394804367759476706127552782264471716631845450969843958 I, 59.347044002602353079653648674992219031098772806466669698122451754746 I, 60.831778524609809844259901824524003802910090451219178257101348824808 I, 65.112544048081606660875054253183705029348149295166722405966501086675 I, 67.079810529494173714478828896522216770107144951745558874196669551694 I, 69.546401711173979252926857526554738443012474209602510157324539999663 I, 72.067157674481907582522107969826168390480906621456697086683306151488 I, 75.704690699083933168326916762030345922811903530697400301647775301574 I, 77.144840068874805372682664856304637015796032449234461041765231453151 I, 79.337375020249367922763592877116228190613246743120030878438720497101 I, 82.910380854086030183164837494770609497508880593782149146571306283235 I, 84.735492980517050105735311206827741417106627934240818702735529689045 I, 87.425274613125229406531667850919213252171886401269028186455557938439 I, 88.809111207634465423682348079509378395444893409818675042199871618814 I, 92.491899270558484296259725241810684878721794027730646175096750489181 I, 94.651344040519886966597925815208153937728027015654852019592474274513 I, 95.870634228245309758741029219246781695256461224987998420529281651651 I, 98.831194218193692233324420138622327820658039063428196102819321727565 I};

(*b=100;
For[i=1,i≤Length[rhovalues],i++,
Print[
{
N[Exp[Log[b](1/2 + rhovalues[[i]])]/(1/2 + rhovalues[[i]])+Exp[Log[b](1/2 - rhovalues[[i]])]/(1/2 - rhovalues[[i]])],
N[2Abs[Exp[Log[b](1/2 + rhovalues[[i]])]/(1/2 + rhovalues[[i]])]]
}]
]*)

Report both sides of the equality

The next module assumes b is not a prime power

ReportBothSides[b_] := Module[{Lb, sum, n, p, Lp, c},
   Print["Riemann-von Mangoldt for β = ", b];
   Lb = Log[b];
   
   (* Right side *)
c = Sum[b^(1/2 + rhovalues[[i]])/(1/2 + rhovalues[[i]]) + b^(1/2 - rhovalues[[i]])/(1/2 - rhovalues[[i]]), {i, 1, Length[rhovalues]}];
   Print["Approximate right side: ", b - c - N[Log[2 Pi]] - 1/2 N[Log[1 - 1/b^2]]];
   
   (* Left side *)
   sum = 0;
   n = 1; p = Prime[n];
   While[p < b,
    Lp = Log[p];
    sum += Lp*Floor[Lb/Lp];
    n++;
    p = NextPrime[p]
    ];
   If[b <= 100, Print["Left side symbolically: ", sum]];
   Print["Left side numerically: ", N[sum]];
   ];

Calls

The bound can't be a prime power

ReportBothSides[10];

ReportBothSides[100];

ReportBothSides[1000];

ReportBothSides[10000];

ReportBothSides[50000];

ReportBothSides[100000];

Report π(x)/(x/ln(x))

xmax=10000000;
Plot[PrimePi[x]/(x/Log[x]),{x,3,xmax}]