Talk:Explicit formulae for L-functions: Difference between revisions

Revision as of 22:30, 28 February 2019

Revision as of 05:33, 1 March 2019

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=== Error? ===

"...Riemann found an explicit formula for the number of primes π(x) less than a given number x."

Should this not ''''''Bold text'''''''''Bold text'''''''''Bold text'''''''''Bold text'''''''''Bold text'''''''''Bold text'''''''''Bold text'''''''''Bold text''''''''Bold text''''''Italic text''''''Italic text''''''Italic text''''''Italic text''''''Italic text''''''Italic text''''''Italic text'''''''''''''''''''''''''''''''''''''''''''not be the number of primes '''less than or equal to''' a given number x?

I tried to verify mathematically that the formula gives this, but I am not adept enough to work with it much...I did find that the f(x) formula can work either way (if pi(x) includes x, then f(x) includes x; and if pi(x) excludes x, then f(x) excludes x). This was quite simple using induction.

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Some of the formulas under this topic don't look correct to me.

'''Question (1)''': In the formula <math>\frac{d}{du} \left[ \sum\limits_{n \le e^{|u|}} \Lambda(n) + \frac{1}{2} \ln(1-e^{-2|u|})\right] <\/math><math>= \sum\limits_{n=1}^\infty \Lambda(n) \left[ \delta(u-\ln n) + \delta(u-\ln n) \right] + \frac{d\ln(1-e^{-2|u|})}{du} = e^u - \sum{\rho} e^{\rho u} <\/math>,

'''(1a)''' Should <math>\left[ \delta(u-\ln n) + \delta(u-\ln n) \right]<\/math> be <math>\left(\delta(u-\ln n)+\delta(u+\ln n)\right)<\/math>?

'''(1b)''' Should <math>\frac{d\ln(1-e^{-2|u|})}{du}<\/math> be <math>\frac{1}{2}\frac{d\ln(1-e^{-2|u|})}{du}<\/math>?

'''(1c)''' Should <math>\sum{\rho} e^{\rho u} <\/math> be <math>\sum\limits_{\rho}{\rho}\,e^{\rho u}$</math> or $<math>\sum\limits_{\rho}\,e^{\rho u}<\/math>?

'''Question (2)''': In the last paragraph should <math>g(u) = \sum_{n=1}^\infty \Lambda(n) \left[ \delta(u-\ln n) + \delta(u-\ln n) \right] <\/math> be <math>g(u)=\sum_{n=1}^\infty\Lambda(n)\left(\delta(u-\ln n)+\delta(u+\ln n)\right)<\/math>?

[[User:StvC|StvC]] ([[User talk:StvC|talk]]) 22:30, 28 February 2019 (UTC)