A curve associated with the so-called Riemann zeta function. The zeta function Zeta[s] is defined as Sum[1/k s , {k, ∞ }] . The curve shown here is the so-called Riemann–Siegel Z function, which is essentially Zeta[1/2 +  t] .

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.

Cycles and zeta functions The number of sequences of n cells that can occur repeatedly, corresponding to cycles in the network, is given in terms of the adjacency matrix m by Tr[MatrixPower[m,n]] . These numbers can also be obtained as the coefficients of x n in the series expansion of x ∂ x Log[ ζ [m, x]] , with the so-called zeta function, which is always a rational function of x , given by ζ [m_, x_] := 1/Det[IdentityMatrix[Length[m]] - m x] and corresponds to the product over all cycles of 1/(1 - x n ) .

Ignoring parts that depend on particle masses the result (derived in successive orders from 1, 1, 7, 72, 891 diagrams) is 2 ( 1 + α /(2 π ) + (3 Zeta[3]/4 - 1/2 π 2 Log[2] + π 2 /12 + 197/144) ( α / π ) 2 + (83/72 π 2 Zeta[3] - 215 Zeta[5]/24 - 239 π 4 /2160 + 139 Zeta[3]/18 + 25/18 (24 PolyLog[ 4, 1/2] + Log[2] 4 - π 2 Log[2] 2 ) - 298/9 π 2 Log[2] + 17101 π 2 /810 + 28259/5184) ( α / π ) 3 - 1.4 ( α / π ) 4 + …), or roughly 2. + 0.32 α - 0.067 α 2 + 0.076 α 3 - 0.029 α 4 + … The comparative simplicity of the symbolic forms here (which might get still simpler in terms of suitable generalized polylogarithm functions) may be a hint that methods much more efficient than explicit Feynman diagram evaluation could be used.

A still better approximation is obtained by subtracting Sum[LogIntegral[n r i ], {i, - ∞ , ∞ }] where the r i are the complex zeros of the Riemann zeta function Zeta[s] , discussed on page 918 .

(The probability for s randomly chosen integers to be relatively prime is 1/Zeta[s] .)

Many numbers associated with Zeta and Gamma can readily be generated, though apparently for example  and EulerGamma cannot.

In the mid-1800s it became clear that despite their different origins most of these functions could be viewed as special cases of Hypergeometric2F1[a, b, c, z] , and that the functions covered the solutions to all linear differential equations of a certain type. ( Zeta and PolyLog are parametric derivatives of Hypergeometric2F1 ; elliptic modular functions are inverses.)

The best-known algorithms for evaluating Zeta[1/2 +  x] (see page 918 ) to fixed precision take roughly √ x operations—or 2 n/2 operations if x is an n -digit integer.