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Abstract Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non‐zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non‐universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).
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Zero-free regions near a line
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 poten- tial non-real zeros are forced to be farther and farther away from the critical line.
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- Award ID(s):
- 2000088
- PAR ID:
- 10509853
- Publisher / Repository:
- Mathematische Zeitschrift
- Date Published:
- Journal Name:
- Mathematische Zeitschrift
- Volume:
- 305
- Issue:
- 3
- ISSN:
- 0025-5874
- Subject(s) / Keyword(s):
- Zero-free regions entire functions of finite order Laguerre- Polya class
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00209-023-03364-w
