The distribution of k-free numbers and the derivative of the Riemann zeta-function
2016 ◽
Vol 162
(2)
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pp. 293-317
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Keyword(s):
AbstractUnder the Riemann Hypothesis, we connect the distribution of k-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of ζ(s). Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-{x}/{\zeta(k)}$ and μk(n) is the characteristic function of k-free numbers. Finally, we make a conjecture about the maximum order of Mk(x) by heuristic analysis on the tail of the limiting distribution.
2017 ◽
2017 ◽
Keyword(s):
2001 ◽
Vol 71
(1)
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pp. 113-121
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Keyword(s):
QUESTIONS AROUND THE NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION. COMPUTATIONS AND CLASSIFICATIONS
2011 ◽
Vol 16
(1)
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pp. 72-81
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