Riemann's Paper on Understanding the Riemann Hypothesis
https://riemannhypothesis.info/categories/riemanns-paper/
Recent content in Riemann's Paper on Understanding the Riemann HypothesisHugo -- gohugo.iomarkus@recommend.games (Markus Shepherd)markus@recommend.games (Markus Shepherd)Sun, 23 Nov 2014 10:47:17 +0000Applying the Explicit Formula
https://riemannhypothesis.info/2014/11/applying-the-explicit-formula/
Sun, 23 Nov 2014 10:47:17 +0000markus@recommend.games (Markus Shepherd)https://riemannhypothesis.info/2014/11/applying-the-explicit-formula/It’s quite some time since we arrived at Riemann’s main result, the explicit formula
\[ J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2, \]
where \(J(x)\) is the prime power counting function introduced even earlier. It’s high time we applied this!
First, let’s take a look at \(J(x)\) when calculating it exactly:
You see how this jumps by one unit at prime values (\(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\)), by half a unit at squares of primes (\(4\), \(9\)), by a third at cubes (\(8\)), and by a quarter at fourth powers (\(16\)), but is constant otherwise.Integral Madness
https://riemannhypothesis.info/2014/04/integral-madness/
Sun, 13 Apr 2014 15:25:54 +0000markus@recommend.games (Markus Shepherd)https://riemannhypothesis.info/2014/04/integral-madness/We’ve seen the calculus version
\[ J(x)=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\log\zeta(s)x^s\frac{\mathrm{d}s}{s}, \]
of the Euler product, and we know how to express \(\xi(s)\) as a product over its roots
\[ \xi(s)=\xi(0)\prod_\varrho\left(1-\frac{s}{\varrho}\right), \]
where
\[ \xi(s) = \frac{1}{2} \pi^{-s/2} s(s-1) \Pi(s/2-1) \zeta(s) \newline = \pi^{-s/2} (s-1) \Pi(s/2) \zeta(s). \]
High time we put everything together – the reward will be the long expected explicit formula for counting primes!First, let’s bring the two formulae for \(\xi(s)\) together and rearrange them such that we obtain a formula for \(\zeta(s)\):Counting Primes Functionally
https://riemannhypothesis.info/2014/01/counting-primes-functionally/
Sun, 12 Jan 2014 11:31:23 +0000markus@recommend.games (Markus Shepherd)https://riemannhypothesis.info/2014/01/counting-primes-functionally/After all this playing with the \(\zeta\)-function it is time to return to the overall objective of this whole exercise: counting prime numbers. The idea behind analytic number theory is that primes are unpredictable on the small scale, but actually surprising regular on the large scale. This is why we’ll look at certain functions that behave pretty erratically when we look at every single value, but become smooth and “easy” to calculate once we “zoom out” and consider the global properties, the so-called asymptotic.More symmetry and Another Product
https://riemannhypothesis.info/2013/12/more-symmetry-and-another-product/
Sun, 01 Dec 2013 21:29:17 +0000markus@recommend.games (Markus Shepherd)https://riemannhypothesis.info/2013/12/more-symmetry-and-another-product/We’ve seen that \(\zeta(s)\) satisfies the functional equation
\[ \zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s). \]
(Well, it still needs to be proved, but let’s just assume it’s correct for now.) The goal of this post is an even more symmetrical form that will yield the function \(\xi(s)\) which we can develop into an incredibly useful product expression.
On our wish list for \(\xi(s)\) we find three items:
It’s an entire function, i.e., a function that’s holomorphic everywhere in \(\mathbb{C}\) without any poles.