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On two conjectures in the theory of numbers. (English) Zbl 0063.02974

The conjectures referred to are that \(\displaystyle M(x) = \sum_{n\le x} \mu(n) = O(x^{\frac12})\) and \(\displaystyle L(x) = \sum_{n\le x} \lambda(n)\ge 0\), \(x\ge 2\). A consequence of either of these conjectures is the Riemann hypothesis and the fact that all the zeros of the zeta-function are simple. In this paper it is further shown that the imaginary parts of all the zeros above the real axis must be linearly dependent. It is proved that if the imaginary parts of the zeros are connected by no, or only a finite number of, linear relations with integral coefficients, then \(\varliminf x^{-\frac12}M(x)= -\infty\), \(\varlimsup ^{-\frac12}M(x)= +\infty\), \(\varliminf x^{-\frac12}L(x)= -\infty\), \(\varlimsup ^{-\frac12}M(x)= +\infty\), as \(x\to\infty\).

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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