A zero-density theorem for the Riemann zeta-function. (English) Zbl 0548.10024

Let \(N(\sigma\),T) denote as usual the number of zeros of \(\zeta\) (s) in the box \(\sigma \leq Re s\leq 1\), \(| Im s|\leq T\). The paper considers estimates \(N(\sigma,T)<<T^{A(\sigma)(1-\sigma)+\epsilon},\) and shows that one may take \(A(\sigma)\leq 3/(7\sigma -4)\) for 3/4\(\leq\sigma \leq 10/13\) and \(A(\sigma)\leq 9/(8\sigma -2)\) for 10/13\(\leq\sigma \leq 1\). These improve on the bound \(A(\sigma)\leq 3/(3\sigma -1)\) due to M. N. Huxley [Invent. Math. 15, 164-170 (1972; Zbl 0241.10026)], throughout the range 3/4\(\leq\sigma \leq 1\). It is claimed that these are the first bounds to do so. However M. Jutila [Acta Arith. 32, 55-62 (1977; Zbl 0307.10045)] showed that one may take \(A(\sigma)\leq 3k/((3k-2)\sigma +2-k)\) if \[ (*)\quad (5k-4)/(6k- 4)\leq\sigma \leq (3k^ 2-2k+1)/(4k^ 2-3k+1) \] and if \(k\geq 2\) is an integer. The author comments that for any fixed k Huxley’s bound is better, if \(\sigma\) is sufficiently close to 3/4. None the less by choosing \(k=[(11\sigma -8)/(4\sigma -3)]\) (in which case \(\sigma\) lies in the range (*)) one obtains \[ A(\sigma)\leq 3(11\sigma -8)/(25\sigma^ 2- 21\sigma +2)\quad for\quad 3/4\leq\sigma \leq 10/13; \] this is superior to the bound 3/(7\(\sigma\) -4) given in the present paper. The case \(k=5\) of Jutila’s result yields \(A(\sigma)\leq 15/(13\sigma -3) (<9/(8\sigma -2))\) for 10/13\(\leq\sigma \leq 21/26\); and the reviewer’s estimate \(A(\sigma)\leq 9/(7\sigma -1)\) for 11/14\(\leq\sigma \leq 1\) [J. Lond. Math. Soc., II. Ser. 19, 221-232 (1979; Zbl 0393.10043)] is superior to \(9/(8\sigma -2)\) for the remaining range 21/26\(\leq\sigma \leq 1\). Thus the author’s results are inferior to those already known, for the whole range 3/4\(\leq\sigma \leq 1\).
Reviewer: D.R.Heath-Brown


11M06 \(\zeta (s)\) and \(L(s, \chi)\)