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Will Orrick (talk | contribs) →Explicit formulae for other arithmetical functions: I have added punctuation to this section and slightly rearranged wording in a couple of places. Also I have added three tags. Someone more knowledgeable should check that these fixes haven't altered any meanings. |
Will Orrick (talk | contribs) define Lambda as the von Mangoldt function and add wiki link |
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Weil's explicit formula can be understood like this. The target is to be able to write that :
: <math>\frac{d}{du} \left[ \sum_{n \le e^{|u|}} \Lambda(n) + \frac{1}{2} \ln(1-e^{-2|u|})\right] = \sum_{n=1}^\infty \Lambda(n) \left[ \delta(u+\ln n) + \delta(u-\ln n) \right] + \frac{1}{2}\frac{d\ln(1-e^{-2|u|})}{du} = e^u - \sum_\rho e^{\rho u} </math>,
where {{math|Λ}} is the [[von Mangoldt function]].
So that the Fourier transform of the non trivial zeros is equal to the primes power symmetrized plus a minor term. Of course, the sum involved are not convergent, but the trick is to use the unitary property of Fourier transform which is that it preserves scalar product :
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{{unreferenced section}}
The Riemann-Weyl formula{{clarify|reason=A formula by this name is not mentioned in the article.|date=September 2020}} can be generalized to arithmetical functions other than the
: <math> \sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\rho}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} </math>.
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