The quantum life of π

Aninda Sinha, 
CHEP, Indian Institute of Science, Bangalore, India &
QHA, PHAS, University of Calgary, Canada.

#WolframTechConf

ABSTRACT

I will talk about an infinite new number of formulas for π, belonging to a family parametrized by the elasticity parameter. For large elasticities, we recover the Madhava series. The methods leading to this discovery lead to an infinite number of new mathematical identities, which can be investigated using Mathematica.

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Madhava Series

The Madhava (Leibniz) series takes the form:
π==4 (∑_(k=0)^∞)((((-1))^k)/(2 k+1))

madhavapi[nmax_]:=4 Sum[(((-1)^k)/(2k+1)),{k,0,nmax}]

N[madhavapi[20],20]

Very slow convergence

Stringy π

This formula is based on"Field theory expansions of string theory amplitudes," A.P. Saha and A. Sinha, published in Physical Review Letters '24   chosen by Scientific American as "one of the 7 coolest mathematical discoveries of 2024".

π==4+(∑_(n=1)^∞)((((1/(n+λ))-(4/(2 n+1))) (Pochhammer[((((2 n+1))^2)/(4 (n+λ)))-n, n-1]))/(n!))

PiPartial[510/9,20,20]

Much faster convergence

PiPartial[10^(10),20,20]

Goes to the Madhava series for large λ

Accuracy checks

Let us check the accuracy

(*---Error and “digits of accuracy”---*)ClearAll[PiAbsError,PiDigits]
PiAbsError[λ_?NumericQ,nmax_Integer?NonNegative,wp_:60]:=Module[{val=PiPartial[λ,nmax,wp]},If[val===$Failed,Indeterminate,Abs[val-N[Pi,wp]]]]

(*Digits of accuracy≈-Log10(absolute error)*)
PiDigits[λ_?NumericQ,nmax_Integer?NonNegative,wp_:60]:=Module[{e=PiAbsError[λ,nmax,wp]},If[NumericQ[e]&&e>0,-Log10[e],Indeterminate]]

(*---Plot:digits of accuracy vs.λ---*)
ClearAll[PiAccuracyPlot]
PiAccuracyPlot[nmax_Integer?Positive,{lamMin_?NumericQ,lamMax_?NumericQ},wp_:60]:=Module[{exclusions},exclusions=Table[-k,{k,1,nmax}];(*poles of 1/(n+λ)*)Plot[PiDigits[λ,nmax,wp],{λ,lamMin,lamMax},Exclusions->exclusions,ExclusionsStyle->Directive[Dashed,GrayLevel[0.5]],PlotRange->All,PlotPoints->75,MaxRecursion->3,AxesLabel->(Style[#,14]&/@{"λ","digits of accuracy  (-Log10 error)"}),PlotLabel->Style[Row[{"Accuracy of partial sum vs λ,  nmax = ",nmax}],14],ImageSize->Large]]

Plots

PiAccuracyPlot[30,{-0.9,125}]        (*30 terms,λ∈[-0.9,5]*)

frac[λ_,nmax_]:=((4+Sum[PiTerm[λ,n],{n,1,nmax}]-Pi)/Pi)

ls={};Do[AppendTo[ls,{nmax,λ,frac[λ,nmax]}],{nmax,10,100},{λ,1,20}];

ListPlot3D[ls,AxesLabel->{"nmax","λ","ϵ"}]

Euler-Beta function: New series

Euler Beta function: ((Γ(x)Γ(y))/(Γ(x+y))) has poles when x or y are negative integers. 

Standard series representations: 
               (Γ (x) Γ (y))/Γ(x + y)=-(Subscript[Σ, n=0])(1/(n!))((Pochhammer[-y+1,n])/(x+n))

Beta[x,y]-Sum[1/n! Pochhammer[1-y,n]/(x+n),{n,0,Infinity}]//FullSimplify

This is misleading! The truncated sum does not know about the poles in y.

diff[x_,y_]:=Beta[x,y]-Sum[1/n! Pochhammer[1-y,n]/(x+n),{n,0,50}]//N

diff[0.1,-0.9]

newBeta[x_,y_,nmax_,λ_]:=-Sum[(1/(n!))((1/(-x-n))+(1/(-y-n))+(1/(λ+n)))Pochhammer[1-λ+(((-x+λ)(-y+λ))/(n+λ)),n],{n,0,nmax}]

newdiff[x_,y_,nmax_,λ_]:=newBeta[x,y,nmax,λ]-Beta[x,y]

newdiff[0.1,-0.9,50,2]

Previously it was -34.11 close to the y pole.

Plot[newdiff[0.1,-0.9,50,λ],{λ,1,10},PlotRange->All]

Table[x=RandomReal[{-5.1,5.1}];y=RandomReal[{-5.1,5.1}];{x,y,newdiff[x,y,50,10]},{points,1,50}]

ListPlot3D[%]

Infinitely many new bizarre identities

Clear[x,y]

SeriesCoefficient[Beta[x,y],{x,0,0},{y,0,2}]//FullSimplify

New formula for ζ(3)

term[n_,λ_]:=-SeriesCoefficient[(1/(n!))((1/(-x-n))+(1/(-y-n))+(1/(λ+n)))Pochhammer[1-λ+(((-x+λ)(-y+λ))/(n+λ)),n],{x,0,0},{y,0,2}]

Sum[term[n,10],{n,1,10}]-Zeta[3]//N

Even weirder identities follow by demanding λ independence

dterm[n_,λ_]=D[term[n,λ],λ]//Simplify

Sum[dterm[n,121/10],{n,1,50}]//N

Wolfram language and theoretical physics

I have used mathematica since 2001.

For a theoretical physicist, mathematica provides the lab!

The new formulas we discovered relied crucially on both numerical and analytical checks performed using Mathematica

Unfortunately, I do not have time to tell you about the beautiful physics connections of the Ramanujan 1/pi series. Stephen Wolfram has a beautiful blog about Ramanujan and his time in Cambridge. I strongly encourage you to read it!