Bounding \(S(t)\) and \(S_1(t)\) on the Riemann hypothesis. (English) Zbl 1325.11084
We quote the authors’ perfect abstract: “Let \(\pi S(t)\) denote the argument of the Riemann zeta-function, \(\zeta(s)\), at the point \(s={1\over 2}+it\). Assuming the Riemann hypothesis, we present two proofs of the bound
\[
|S(t)|\leq \left({1\over 4}+o(1)\right){{\log t}\over {\log \log t}}
\]
for large \(t\). This improves a result of D. A. Goldston and S. M. Gonek [Bull. Lond. Math. Soc. 39, No. 3, 482–486 (2007; Zbl 1127.11058)] by a factor of \(2\). The first method consists of bounding the auxiliary function \(S_1(t)=\int_{0}^{t}S(u)\,du\) using extremal functions constructed by E. Carneiro et al. [Trans. Am. Math. Soc. 365, No. 7, 3493–3534 (2013; Zbl 1276.41018)]. We then relate the size of \(S(t)\) to the size of the functions \(S_1(t\pm h)-S_1(t)\) when \(h\asymp 1/\log \log t\). The alternative approach bounds \(S(t)\) directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function \(f(x)=\arctan\left({1\over x}\right)-{x\over {1+x^2}}\). This draws upon recent work by E. Carneiro and F. Littmann [Constr. Approx. 38, No. 1, 19–57 (2013; Zbl 1280.41018)].”
Reviewer: Giovanni Coppola (Avellino)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 11M36 | Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) |
| 41A30 | Approximation by other special function classes |
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