Content deleted Content added
Link suggestions feature: 3 links added. |
|||
Line 55:
====Proof of Euler's product formula====
The [[fundamental theorem of arithmetic]] states that if {{math|''n'' > 1}} there is a unique expression <math>n = p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}, </math> where {{math|''p''<sub>1</sub> < ''p''<sub>2</sub> < ... < ''p''<sub>''r''</sub>}} are [[prime number]]s and each {{math|''k''<sub>''i''</sub> ≥ 1}}. (The case {{math|1=''n'' = 1}} corresponds to the [[empty product]].) Repeatedly using the multiplicative property of {{mvar|φ}} and the formula for {{math|''φ''(''p''<sup>''k''</sup>)}} gives
:<math>\begin{array} {rcl}
Line 104:
&=& 4 .
\end{array}
</math>Unlike the [[Euler product]] and the divisor sum formula, this one does not require knowing the factors of {{mvar|n}}. However, it does involve the calculation of the greatest common divisor of {{mvar|n}} and every positive integer less than {{mvar|n}}, which suffices to provide the factorization anyway.
===Divisor sum===
Line 414:
===Riemann hypothesis===
The [[Riemann hypothesis]] is true [[if and only if]] the inequality
:<math>\frac{n}{\varphi (n)}<e^\gamma \log\log n+\frac{e^\gamma (4+\gamma-\log 4\pi)}{\sqrt{\log n}}</math>
is true for all {{math|''n'' ≥ ''p''<sub>120569</sub>#}} where {{mvar|γ}} is [[Euler's constant]] and {{math|''p''<sub>120569</sub>#}} is the [[Primorial|product of the first]] {{math|120569}} primes.<ref>{{Cite book |last1=Broughan |first1=Kevin |title=Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents |publisher=Cambridge University Press |year=2017 |edition=First |isbn=978-1-107-19704-6}} Corollary 5.35</ref>
| |||