Zeros of Jensen polynomials and asymptotics for the Riemann xi function. (English) Zbl 1479.11146
Summary: The classical criterion of Jensen for the Riemann hypothesis is that all of the associated Jensen polynomials have only real zeros. We find a new version of this criterion, using linear combinations of Hermite polynomials, and show that this condition holds in many cases. Detailed asymptotic expansions are given for the required Taylor coefficients of the xi function at 1/2 as well as related quantities. These results build on those in the recent paper of Griffin, Ono, Rolen and Zagier [M. Griffin et al., Proc. Natl. Acad. Sci. USA 116, No. 23, 11103–11110 (2019; Zbl 1431.11105)].
MSC:
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Keywords:
Riemann xi functionCitations:
Zbl 1431.11105References:
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