Author Archives: Charles Li

The 100th post

Posted on 14/08/2012 by Charles Li

This is the 100th post of this blog. As a very senior (translated as old) blogger here, I take the liberty of posting the 100th post. The honor should have gone to other people, most notably KKK, who wrote almost … Continue reading →

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Uniform distribution of polynomials mod 1

Posted on 01/08/2012 by Charles Li

Last week Lam Wai Kit posted an excellent introductory article of ergodic theory and its applications to number theory. As stated in the note, one consequence of Birhoff recurrence theorem is that for any non-constant polynomial , the fractional part … Continue reading →

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Fourier Analysis and Number Theory III: The Poisson summation formula as a trace formula

Posted on 17/09/2011 by Charles Li

Yin and Yang are important concept in the Chinese philosophy. They usually describe two opposite things which are not only complement of each other but also have deep interrelationship. In mathematics, we called them duality. For example (1) primes vs … Continue reading →

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Fourier Analysis and Number Theory II: The Poisson Summation Formula and The Functional Equation of The Riemann-Zeta Function

Posted on 09/09/2011 by Charles Li

Riemann’s ten-page-long paper “Über die Anzahl der Primzahen unter einer gegebener Gröβe” has great influence on modern number theory. In the paper, he established two important properties of the Riemann-zeta function (the summation is absolutely convergent for ).

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Fourier Analysis and Number theory I: The self-dual function e^{-\pi x^2}

Posted on 03/09/2011 by Charles Li

During a lecture at UCLA, Serge Lang asked what is the most important function in mathematics. This question is quite personal and every person certainly has his own opinion. In fact, a professor spoke out loud that the constant function … Continue reading →

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