Functions of Number Theory - 27.12 Asymptotic Formulas: Primes
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 27.12.E1 | \lim_{n\to\infty}\frac{p_{n}}{n\ln@@{n}} = 1 |
|
limit((p[n])/(n*ln(n)), n = infinity) = 1
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Limit[Divide[Subscript[p, n],n*Log[n]], n -> Infinity, GenerateConditions->None] == 1
|
Failure | Failure | Skip - No test values generated | Failed [10 / 10]
Result: -1.0
Test Values: {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: -1.0
Test Values: {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 27.12.E2 | p_{n} > n\ln@@{n} |
|
p[n] > n*ln(n)
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Subscript[p, n] > n*Log[n]
|
Failure | Failure | Failed [6 / 10] Result: 3.295836867 < -1.500000000
Test Values: {p[n] = -3/2, n = 3}
Result: 3.295836867 < 1.500000000
Test Values: {p[n] = 3/2, n = 3}
... skip entries to safe data |
Failed [25 / 30]
Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0]
Test Values: {Rule[n, 1], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 1.3862943611198906]
Test Values: {Rule[n, 2], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |