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Two remarks on a paper by Jan Moser. (English) Zbl 0896.11034

The authors prove the following theorem: There is an absolute constant \(C\) satisfying the following property: Let \(k\geq 1\) be an integer, \(q=k^{-1}\), \(q_1= q^2/4\), \(\lambda =q/4\), \(T\geq 10\) and \(H=C T^\lambda (1+q_1 (\log T)^{1-q_1})\). Then at least one of the following holds: (i) \((\zeta (s))^q\) is not regular in \(0\leq \sigma\leq 1\), \(T\leq t\leq T+H\). (ii) \((\zeta({1 \over 2} +it))^q\) has a zero of odd order in \(T\leq t\leq T+H\).
It is interesting to note that the above theorem implies that \(\zeta(s)\) always has a zero \(\rho= \beta+ i\gamma\); \((0\leq \beta\leq 1\), \(T\leq \gamma\leq T+H)\) of order not divisible by \(2k\), improving upon a conditional result (assuming Lindelöf’s Conjecture) due to J. Moser [Czech. Math. 44, 385-404 (1994; Zbl 0821.11041)].

MSC:

11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0821.11041
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