Nifty formulas. By coincidence, earlier this morning I just happened to be scanning the last section (7-5) of Topics in Number Theory*, vol. II. by William LeVeque, where a related issue is addressed with a rather hairy proof: Let B(x) := the number of positive integers <= x that are each representable as the sum of two squares. Then B(x) is asymptotic to b*x/sqrt(log(x)) as x -> oo, where b is the constant b := sqrt(1/2) * Prod 1/sqrt(1-1/p^2) over all primes p == 3 (mod 4). --Dan _________________________________________________________________________________________________________________________________________________________ *A terrific and deep 2-volume set, where vol. II proves Kummer's theorem, the Tbue-Siegel-Roth, theorem, the (Hilbert-)Gelfond-Schneider theorem, and the Prime Number theorem. Gene wrote: << Warren wrote: << N=a^2+b^2>0 where a,b are integers (of either sign) #representations = 4*sum(over all divisors d of N) ( [1 if (d mod 4)=1 else 0] - [1 if (d mod 4)=3 else 0] ) I am not sure who first invented this nifty formula (Gauss?). It makes it clear, though, that #representations(N) < #divisors(N) < exp(log(2+o(1))*lnN/lnlnN).
Let r[s](n) be the number of representations of n as the sum of s squares. This is counted as the number of distinct s-tuples (x[1], ..., x[s]) such that x[1]^2 + ... + x[s]^2 = n. Then sum(r[s](k), n = 0 ... n) ~ (πn)^(s/2) / (s/2)! . Proof: The sum is the number of integer lattice points in an s-dimensional ball of radius sqrt(n).
________________________________________________________________________________________ It goes without saying that .