In complex analysis, we have the following well-known theorem: Theorem If , then the Fourier inversion holds, namely, given     We have     The proof is not easy for a beginner who just touched this theorem. As for myself, I spent a tons of time on managing the proof. So I would like […]

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Our goal is to prove the following theorem by Carlo Sanna. The proof is adapted and changed slightly from his paper. Theorem. For any prime number and any , we have     Lemma 1 If is prime, , and , then     and     Proof. . Each of the sum is by […]

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The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first  natural numbers:     This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as     Then many problems raise. Here, we have a problem […]

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Let be a continuous integrable real function on , then     Solution 1 By Cauchy-Schwarz inequality,     Solution 2 By Jensen’s inequality, since is a convex function, we have     So     Solution 3 By Hölder’s Inequality, we get     Choose , we get the case of solution 1. We […]

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