Differential equations [for sine sums] The function Sin[x] + Sin[ √ 2 x] can be obtained as the solution of the differential equation y''[x] + 2 y[x] - Sin[x]  0 with the initial conditions y[0]  0 , y'[0]  2 .

Some regularity is evident, and in cases (a) and (b) it can be understood from trigonometric sum formulas of Ramanujan discussed above (see also pages 586 and 1081 ).

But it is straightforward to define versions of entropy that take account of probabilities—and indeed the closest analog to the usual entropy in physics or information theory is obtained by taking the probabilities p[i] for the k n blocks of length n (assuming k colors), then constructing -Limit[Sum[p[i] Log[k, p[i]], {i, k n }]/n, n  ∞ ] I have tended to call this quantity measure entropy, though in other contexts, it is often just called entropy or information, and is sometimes called information dimension. The quantity Limit[Sum[UnitStep[p[i]], {i, k n }]/n, n  ∞ ] is the entropy discussed in the notes above, and is variously called set entropy, topological entropy, capacity and fractal dimension. An example of a generalization is the quantity given for blocks of size n by h[q_, n_]:= Log[k, Sum[p[i] q , {i, k n }]]/(n(q - 1) where q = 0 yields set entropy, the limit q  1 measure entropy, and q = 2 so-called correlation entropy.

Non-power bases One can consider representing numbers by Sum[a[n] f[n], {n, 0, ∞ }] where the f[n] need not be k n .

Unsolved problems [in number theory] Problems in number theory that are simple to state (say in the notation of Peano arithmetic) but that so far remain unsolved include: • Is there any odd number equal to the sum of its divisors? … (Perfect cuboid; 1719) • Is there any even number which is not the sum of two primes?

The diffusion and wave equations are linear, so that results are linear sums of those with single Gaussians.

Mathematical properties [of branching model] If an element c of the list b is real, so that there is a stem that goes straight up, then the limiting height of the center of the pattern is obtained by summing a geometric series, and is given by 1/(1 - c) .

Zeta function For real s the Riemann zeta function Zeta[s] is given by Sum[1/n s , {n, ∞ }] or Product[1/(1 - Prime[n] s ), {n, ∞ }] . The zeta function as analytically continued for complex s was studied by Bernhard Riemann in 1859, who showed that PrimePi[n] could be approximated (see page 909 ) up to order √ n by LogIntegral[n] - Sum[LogIntegral[n^r[i]], {i, - ∞ , ∞ }] , where the r[i] are the complex zeros of Zeta[s] . … 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 .

With completely random input, the output will on average be longer by a factor Sum[2 -(n + 1) r[n], {n, 1, ∞ }] where r[n] is the length of the representation for n .

Musical chords [from sine sums] In a so-called equal temperament scale the 12 standard musical notes that make up an octave have a progression of frequencies 2 n/12 .