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Crystal symmetry governs the physical properties of materials and provides the framework for interpreting their microscopic excitations and macroscopic responses. In altermagnets, the crystal symmetries that relate oppos...
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We study the exponential sum involving multiplicative function f under milder conditions on the range of f , which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of ad...
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Let P + y (n) denote the largest prime factor p of n with p y. We prove that there exists a positive proportion of integers n such that P + y (n) < P + y (n + 1) for y = x α when α is small. Especially, the proportion is...