Lesson plan
Chebyshev function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by : \vartheta(x) = \sum_{p \le x} \log p where \log denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to . The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding : \psi(x) = \sum_{k \in \mathbb{N}}\sum_{p^k \le x}\log p = \sum_{n \leq x} \Lambda(n) = \sum_{p \le x}\left\lfloor\log_p x\right\rfloor\log p, where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (see the exact formula below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem. Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function: : f_{Tchb}(x,w) = \max_i w_i f_i(x). By minimizing this function for different values of w , one obtains every point on a Pareto frontier, even in the nonconvex parts.…
Article summary
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by : \vartheta(x) = \sum_{p \le x} \log p where \log denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to . The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding : \psi(x) = \sum_{k \in \mathbb{N}}\sum_{p^k \le x}\log p = \sum_{n \leq x} \Lambda(n) = \sum_{p \le x}\left\lfloor\log_p x\right\rfloor\log p, where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (see the exact formula below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem. Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing function is used when one has several functions to be minimized and one wants to "scalarize" them to a single function: : f_{Tchb}(x,w) = \max_i w_i f_i(x). By minimizing this function for different values of w , one obtains every point on a Pareto frontier, even in the nonconvex parts.…
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Discussion questions
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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