Note (b) for Mathematical Functions…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] . … In 1972 Sergei Voronin showed that Zeta[z + (3/4 +  t)] has a certain universality in that there always in principle exists some t (presumably in practice usually astronomically large) for which it can reproduce to any specified precision over say the region Abs[z] < 1/4 any analytic function without zeros.

Note (a) for Mathematical Functions

Functions that can be used to formulate logic. In each case the minimal combinations of primitive functions necessary to reproduce each of the 16 logical functions of two arguments is given. … Nand and Nor are the only primitive functions that work on their own.

Note (e) for Mathematical Functions

Note (g) for Mathematical Functions

functions together with factorials and multinomial coefficients then it appears that there is not. But if one also allows higher mathematical functions then it turns out that such a formula can in fact be found: as indicated in the table above each coefficient is given by a particular value of a so-called Gegenbauer or ultraspherical function. … But for most nested patterns there seems to be no obvious way to relate them to ordinary mathematical functions.

But unlike in example (e), this Turing machine is not the only one that computes the function it computes. … And it turns out that there are about 33,000 distinct functions that one or more of these machines computes. Most of the time the fastest machine at computing a given function again exhibits linear or at most quadratic growth.

But at least in this case it seems fairly clear that none of the simple functions shown below can for example ever lead to results that go beyond ones that could readily be generated by the evolution of ordinary discrete systems. And the same is presumably true if one works with essentially any of what are normally considered standard mathematical functions. But what happens if one assumes that one can set up a system that not only finds values of such functions but also finds solutions to arbitrary equations involving them?

Standard mathematical functions There are an infinite number of possible functions with integer or continuous arguments. … The so-called elementary functions (logarithms, exponentials, trigonometric and hyperbolic functions, and their inverses) were mostly introduced before about 1700. … Rather few new special functions have been introduced over the past century.

Universal logical functions The fact that combinations of Nand or Nor are adequate to reproduce any logical function was noted by Charles Peirce around 1880, and became widely known after the work of Henry Sheffer in 1913. … Nand and Nor are the only 2-input functions universal in this sense. ( {Equal} can for example reproduce only functions {9, 10, 12, 15} , {Implies} only functions {10, 11, 12, 13, 14, 15} , and {Equal, Implies} only functions {8, 9, 10, 11, 12, 13, 14, 15} .) … For large n roughly 1/4 of all n -input functions are universal.