Abstract
Suppose that g(n) is a real-valued additive function and τ(n) is the number of divisors of n. In this paper, we prove that there exists a constant C such that \(\sup \limits_a \sum\limits_{n<N}{g(n) \in [a,a+1)} \tau(N-n) \leqslant C \frac{N \log N}{\sqrt{W(N)}},\) where \(W(N) = 4 + \mathop {min}\limits_\lambda \left( {\lambda ^2 + \sum\limits_{p < N} {\frac{1}{p}} min(1,(g(p) - \lambda log p)^2 )} \right).\). In particular, it follows from this result that \(\mathop {\sup }\limits_a |\{ m,n:mn < N,g(N - mn) = a\} | \ll N\log N\left( {\sum\limits_{p < N,g\left( p \right) \ne 0} {(1/p)} } \right)^{ - 1/2} .\) The implicit constant is absolute.
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Timofeev, N.M., Khripunova, M.B. The Concentration Function of Additive Functions with Nonmultiplicative Weight. Mathematical Notes 75, 819–835 (2004). https://doi.org/10.1023/B:MATN.0000030991.37899.cc
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DOI: https://doi.org/10.1023/B:MATN.0000030991.37899.cc