Dirichlet’s Theorem for arithmetic progressions with difference 4

From antiquity, Euclid’s Theorem asserts that the number of primes is infinite. Since all primes except are odd, a re-phrasement of Euclid’s theorem is that the set of odd integers contains an infinite number of primes. A natural generalization, proved by Dirichlet, is that every arithmetic progression of the form contains an infinite number of primes, provided that and are relatively prime. (Dirichlet’s Theorem) If , then the arithmetic progression contains infinitely many primes, where denotes the greatest common divisor of and . The proof is one of the first application of Fourier analysis on finite abelian groups. Here in this post, we prove Dirichlet’s theorem when , i.e., both the arithmetic […]

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Exponential sum and the equidistribution theorem

Consider the sequence , where gives the fractional part of a real number. The equidistribution theorem states that is irrational iff for any sub-interval of the unit interval, for sufficiently large , roughly of the numbers fall inside . More precisely, (Equidistributed sequence) A sequence with is equidistributed mod 1 if for every ,     (Equidistribution theorem) The sequence is equidistributed mod 1 iff is irrational. Let us try to understand why the sequence behaves differently depending on whether is rational or not. If for some integers and , then roughly of the first multiples of are integers, i.e., . In fact, it is not difficult to see that for , […]

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Summability, Tauberian theorems, and Fourier series

Consider a periodic function . Its Fourier series always exists formally, but questions of whether its Fourier series converges to itself can be rather subtle. In general, may be different from its Fourier series: a Fourier coefficient is the evaluation of an integral, so can be changed pointwise without affecting the value of any Fourier coefficient. It may be natural to believe that convergence holds when is continuous. However, an example constructed by du Bois-Reymond showed that continuity is not sufficient. The most general result in this area is Carleson’s theorem, showing that -functions have convergent Fourier series almost everywhere. Here we will prove that the Fourier series of converges to in […]

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The Basel problem via Fourier series

The Basel problem asks to compute the summation of the reciprocal of the natural numbers squared, and it is known that     While there are numerous known proofs (our presentation resembles Proof 9 most closely), we give a self-contained one that motivates the usage of Fourier analysis for periodic functions. Recall that a function is -periodic if for every , . From Fourier analysis, every such function can be expressed as a series of cosines and sines. More precisely, If is continuous and -periodic, then     where , and is the -th Fourier coefficient equal to . We say that is the Fourier series of . When we define the […]

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