On the frequency of Titchmarsh's phenomenon for \(\zeta{}(s)\). VIII
Let \(z=e^{i\theta}\) where \(\theta\) is a constant in \([0,2\pi)\) and let \(G(\sigma,t)=|(\zeta(\sigma+it))^ z|\) where \(\sigma\) is a constant satisfying \({1\over 2}\leq\sigma\leq 1\). Also \(t\geq 1\). Then the authors prove the following three theorems. Theorem 1. We have \[ \max_{T\leq t\leq T+H} G(1,t)\geq e^ \gamma \lambda(\theta)(\log\log H-\log\log\log H)+O(1) \] where \(\gamma\) is the Euler's constant and \(\lambda(\theta)=\prod_ p \lambda_ p(\theta)\) where \(\lambda_ p(\theta)=(1-1/p)((1-s^ 2/p^ 2)^{1/2}(-{c/p})^{- c}\exp(s\sin^{-1} (s/p))\), where \(c+is=\exp(i\theta)\). The conditions on \(H\) and \(T\) are \(T\geq H\geq C\log\log\log T\), \(T\geq T_ 0\) where \(C\) and \(T_ 0\) are certain positive constants. Theorem 2. Let \(\alpha\) be a constant in \((1/2,1)\). Then \[ \max_{T\leq t\leq T+H}G(\alpha,t)>\exp(C_ 1(log H)^{1-\alpha}/\log\log H) \] where \(C_ 1\) is a certain positive constant depending on \(\alpha\) and \(T^{1/3}\leq H\leq T\). If \(\theta=0\) then the conditions on \(H\) and \(T\) are \(T\geq H\geq C'\log\log T\); where \(C'\) is a certain positive constant. Theorem 3. (On Riemann hypothesis) In Theorem 2 the condition on \(H\) can be relaxed to \(T\geq H\geq C'\log\log T\). Also under this condition on \(H\), there holds \(\max_{T\leq t\leq T+H} G(1/2,t)>\exp(C_ 2(\log H/\log\log H)^{1/2})\) where \(C_ 2\) is a positive numerical constant. Remark 1. Where \(\theta=0\), Theorem 3 is true even if we do not assume Riemann hypothesis. Remark 2. The results of this paper were inspired by the result \[ \max_{0\leq t\leq T} G(\alpha,t)>\exp(C_ 3(\log H)^{1- \alpha}/(\log\log H)^ \alpha) \] where \(C_ 3>0\) depends on \(\alpha\) and \(1/2<\alpha<1\) due to H. L. Montgomery. It may be noted that K. Ramachandra and A. Sankaranayanan have obtained recently \(C_ 3=C_ 4(\alpha-1/2)^{1/2}(1-\alpha)^{-1}\) by the method of Montgomery. Here \(C_ 4>0\) is a numerical constant. Also the quantity \((\alpha- 1/2)^{1/2}\) can be replaced by 1 if we assume Riemann hypothesis. The proofs of Theorems 1, 2 and 3 depend on a theorem of K. Ramachandra on what he calls Titchmarsh series. These results have been quoted in the paper and were first published by him in 1981. Two important results are announced at the end of the paper namely: Theorem 4. (\textit{R. Balasubramananian}) We can take \(C_ 2=3/4\) in Theorem 3 [cf. Hardy-Ramanujan J. 9, 1-10 (1986; Zbl 0662.10030)]. Theorem 5 (\textit{K. Ramachandra}). Let \(f(H)\) denote \(\min_{| I|=H} \max_{t\in I} G(1,t)\), the minimum being taken over all intervals \(I\) of length \(H\). Then for all \(H\geq H_ 0(\theta)\) we have \[ | f(H)e^{-\gamma}(\lambda(\theta))^{-1}-\log\log H|\leq\log\log\log H+O(1) \] [cf. Hardy-Ramanujan J. 13, 28-33 (1990; Zbl 0704.11026)]. Remark. The proof of Theorem 5 depends on some investigations on what the author calls ``weak Titchmarsh series''.
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