The sets of numbers that can be obtained by applying elementary functions like Exp , Log and Sin seem in various ways to be disjoint from algebraic numbers.

(Typically one needs to generalize formulas that are initially set up with integer numbers of terms; examples include taking Power[x, y] to be Exp[Log[x] y] and x!

The picture in the main text shows RiemannSiegelZ[t] , defined as Zeta[1/2 +  t] Exp[  RiemannSiegelTheta[t]] , where RiemannSiegelTheta[t_] = Arg[Gamma[1/4 +  t/2]] - t Log[ π ]/2 The first term in an approximation to RiemannSiegelZ[t] is 2 Cos[RiemannSiegelTheta[t]] ; to get results to a given precision requires summing a number of terms that increases like √ t , making routine computation possible up to t ~ 10 10 .

If one computes the product of Exp[  (j 1 + j 2 - j 3 )] for all triangles, then it turns out for example that this quantity is extremized exactly when the whole surface is flat. … And from this it turns out that limits of products of 6j symbols correspond essentially to Exp[  s] , where s is the discrete form of the Einstein–Hilbert action—extremized by flat 3D space.

Note that isotropy can also be characterized using analogs of multipole moments, obtained in 2D by summing r i Exp[  n θ i ] , and in higher dimensions by summing appropriate SphericalHarmonicY or GegenbauerC functions.

In general, the probability distribution for the displacement of a particle that executes a random walk is With[{ σ = 1}, (d/(2 π σ t)) d/2 Exp[-d r 2 /(2 σ t)]] The same results are obtained, with a different value of σ , for other random microscopic rules, so long as the variance of the distribution of step lengths is bounded (as in the Central Limit Theorem).

When a = 2 it is Exp[x] and when a = -2 it is 2 Cos[(1/3) ( π - √ 3 x)] .

For large n , DivisorSigma[1, n] is known to grow at most like Log[Log[n]] n Exp[EulerGamma] , and on average like π 2 /6 n (see page 1093 ).

(The high-order terms often seem to be associated with asymptotic series for things like Exp[-1/ α ] .)

His bound was roughly Exp[(c s) 10 5 —but later work in essence reduced this, and by the 1990s practical algorithms were being developed.