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Heuristic limitations of the circle method

One of the most basic methods in additive number theory is the Hardy-Littlewood circle method. This method is based on expressing a quantity of interest to additive number theory, such as the number of representations of an integer as the sum of three primes, as a Fourier-analytic integral over the unit circle involving exponential [...]
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We have seen in previous notes that the operation of forming a Dirichlet series or twisted Dirichlet series is an incredibly useful tool for questions in multiplicative number theory. Such series can be viewed as a multiplicative Fo...

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Williams’ P+1 Factorization Algorithm

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