Vinogradov's mean value theorem is an important inequality in analytic number theory, named for I. M. Vinogradov. It relates to upper bounds for , the number of solutions to the system of simultaneous Diophantine equations in variables given by with . In other words, an estimate is provided for the number of equal sums of k-th powers of integers up to X. An alternative analytic expression for is where A strong estimate for , valid for different relative ranges of and . The classical form of the theorem applies when is very large in terms of .

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• Vinogradov's mean value theorem is an important inequality in analytic number theory, named for I. M. Vinogradov. It relates to upper bounds for , the number of solutions to the system of simultaneous Diophantine equations in variables given by with . In other words, an estimate is provided for the number of equal sums of k-th powers of integers up to X. An alternative analytic expression for is where A strong estimate for is an important part of the Hardy-Littlewood method for attacking Waring's problem and also for demonstrating a zero free region for the Riemann zeta-function in the critical strip. Various bounds have been produced for , valid for different relative ranges of and . The classical form of the theorem applies when is very large in terms of . On December 4, 2015, Jean Bourgain, Ciprian Demeter, and Larry Guth announced a proof of Vinogradov's Mean Value Theorem. (en)
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• Vinogradov's mean value theorem is an important inequality in analytic number theory, named for I. M. Vinogradov. It relates to upper bounds for , the number of solutions to the system of simultaneous Diophantine equations in variables given by with . In other words, an estimate is provided for the number of equal sums of k-th powers of integers up to X. An alternative analytic expression for is where A strong estimate for , valid for different relative ranges of and . The classical form of the theorem applies when is very large in terms of . (en)
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• Vinogradov's mean-value theorem (en)
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