Processing math: 100%

Thursday, May 29, 2025

Entry 117

For fundamental discriminants d=4m with class number h(d)=16, there are exactly 60 m that are even. The largest is m=3502=2×17×103 hence has 23=8 divisors. But this set has no m with 32 divisors so it seems one can't express their modular lambda function λ(m) with 16 quadratic units. (Unlike for h(d)=8 where we can express a few λ(m) with 8 quadratic units.) However, using another function, we can have four quartic units. Given the nome q=eπiτ, τ=n, and the Ramanujan G and g functions 21/4Gn=q124k>0(1+q2k1)=η2(τ)η(τ2)η(2τ)21/4gn=q124k>0(1q2k1)=η(τ2)η(τ)

discussed in Entry 116. Then u=(g3502)4=(η(τ2)21/4η(τ))4=(a+a21)2(b+b21)2(c+c21)(d+d21)1.43×1013
where τ=3502 and (a,b,c,d) are a=12(23+434)b=12(192+717)c=(429+3042)d=12(627+4422)
A version of this was first found by Daniel Shanks in 1980 (Quartic Approximations for Pi) but this one is slightly different with smaller integers since I simplified the first two expressions as squares. These radicals imply a very close approximation to pi,π13502ln((2u)6+24)
which differs by just 10161.

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