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OP-8774Status unassessedAnalysis

Riemann Ξ\Xi-function · Question 8.2

Stability, hyperbolicity, and zero localization of functions · Historical workshop question

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
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Problem statement

Are there any non-real zeros of

∫0∞Φα(t)cos⁡(zt)dtforα>0?\int_{0}^\infty \Phi^\alpha(t)\cos(zt)dt \qquad \text{for}\qquad \alpha>0 ?

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Background

Workshop introduction

A list of the problems from the workshop on stability, hyperbolicity, and zero localization of functions, held December 5-9, 2011, is presented below. The problems have been loosely categorized for convenience, although many of the problems could be listed under several different headings.

Section context

Let

Ξ(z):=∫−∞∞Φ(t)cos⁡(zt)dt,\Xi(z):=\int_{-\infty}^{\infty} \Phi(t)\cos(zt) dt,

where

Φ(t):=∑n=1∞(2n4π2e9t−3n2πe5t)exp⁡(−n2πe4t).\Phi(t):=\sum_{n=1}^\infty (2n^4\pi^2e^{9t}-3n^2\pi e^{5t})\exp(-n^2\pi e^{4t}).

Progress & source assessment

Historical workshop record. Current open/solved status has not been independently reassessed.

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References
  1. [1]Stability, hyperbolicity, and zero localization of functions link

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