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A somewhat better approximation is LogIntegral[n] , equal to Integrate[1/Log[t], {t, 2, n}] . This was found empirically by Carl Friedrich Gauss in 1792, based on looking at a table of primes. ( PrimePi[10 9 ] is 50,847,534 while LogIntegral[10 9 ] is about 50,849,235.) … According to the Riemann Hypothesis, the difference between PrimePi[n] and LogIntegral[n] is of order √ n Log[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 Riemann Hypothesis then states that all r[i] satisfy Re[r[i]]  1/2 , which implies a certain randomness in the distribution of prime numbers, and a bound of order √ n Log[n] on PrimePi[n] - LogIntegral[n] . The Riemann Hypothesis is also equivalent to the statement that a bound of order √ n Log[n] 2 exists on Abs[Log[Apply[LCM, Range[n]]] - n] .
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. … One can also ask what numbers can be generated by integrals (or by solving differential equations). … Integrals of rational functions over regions defined by polynomial inequalities have recently been discussed under the name "periods".
2 : BesselI[0, 2] - 1 ; n 2 n : Log[2] ; n 2 : π 2 /6 ; (3n - 1)(3n - 2): π √ 3 /9 ; 3 - 16n + 16n 2 : π /8 ; n n!: ExpIntegralEi[1] - EulerGamma .
[History of] exact solutions Some notable cases where closed-form analytical results have been found in terms of standard mathematical functions include: quadratic equations (~2000 BC) ( Sqrt ); cubic, quartic equations (1530s) ( x 1/n ); 2-body problem (1687) ( Cos ); catenary (1690) ( Cosh ); brachistochrone (1696) ( Sin ); spinning top (1849; 1888; 1888) ( JacobiSN ; WeierstrassP ; hyperelliptic functions); quintic equations (1858) ( EllipticTheta ); half-plane diffraction (1896) ( FresnelC ); Mie scattering (1908) ( BesselJ , BesselY , LegendreP ); Einstein equations (Schwarzschild (1916), Reissner–Nordström (1916), Kerr (1963) solutions) (rational and trigonometric functions); quantum hydrogen atom and harmonic oscillator (1927) ( LaguerreL , HermiteH ); 2D Ising model (1944) ( Sinh , EllipticK ); various Feynman diagrams (1960s-1980s) ( PolyLog ); KdV equation (1967) ( Sech etc.); Toda lattice (1967) ( Sech ); six-vertex spin model (1967) ( Sinh integrals); Calogero–Moser model (1971) ( Hypergeometric1F1 ); Yang–Mills instantons (1975) (rational functions); hard-hexagon spin model (1979) ( EllipticTheta ); additive cellular automata (1984) ( MultiplicativeOrder ); Seiberg–Witten supersymmetric theory (1994) ( Hypergeometric2F1 ). When problems are originally stated as differential equations, results in terms of integrals ("quadrature") are sometimes considered exact solutions—as occasionally are convergent series.
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