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- Title
Zero-free regions near a line.
- Authors
Bickel, Kelly; Pascoe, J. E.; Sargent, Meredith
- Abstract
We analyze metrics for how close an entire function of genus one is to having only real roots. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any potential non-real zeros are forced to be farther and farther away from the critical line.
- Subjects
RIEMANN hypothesis; INTEGRAL functions; POWER series; ZETA functions
- Publication
Mathematische Zeitschrift, 2023, Vol 305, Issue 3, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-023-03364-w