# Arithmetic structure and dependent randomness

## Abstract/Contents

- Abstract
- In recent decades, many great ideas from probability theory have significantly improved our understanding of many classical number theory problems and have brought new aspects to the area. The interaction between probability theory and number theory has grown fascinating and promising. It is exciting to see the interactions between the randomness and arithmetic structures.\\ In analytic number theory, many fundamental open problems like the Riemann hypothesis involve the study of multiplicative functions, such as the M\"obius function. Thus the study of multiplicative functions is one of the central topics. As a first attempt, which turned out to be successful, mathematicians use probabilistic models to model and predict the behavior of multiplicative functions. Nowadays, people call such models random multiplicative functions. The success and fruitful outcomes of studying such probabilistic objects have exceeded mathematicians' expectations. From purely probabilistic aspects, we believe many problems arising here have their own attraction. In particular, the probabilistic model here usually has some tricky dependence relation. The randomness is not naively independent as we need to consider multiplicative structures, for example. As for arithmetic applications, there are at least two good reasons for studying such probabilistic objects. Firstly, it may be a good model for predicting the truth of the deterministic objects. This is also perhaps the mathematicians' original motivation and initial desire for such study. Secondly, it turns out that such models can also give guidance and insights into solving problems in the arithmetic setting. Successful examples include Granville-Soundararajan on large character sums and Harper's work on typical character sums. We should emphasize that applications here can even include seemingly unrelated arithmetic problems. \\ This thesis focuses on several fundamental aspects of random multiplicative functions, including limiting distribution, typical size, threshold phenomenon, high moments, low moments, and large values. We also study some related problems in arithmetic combinatorics, which are connected at the level of techniques used in the proof.

## Description

Type of resource | text |
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Form | electronic resource; remote; computer; online resource |

Extent | 1 online resource. |

Place | California |

Place | [Stanford, California] |

Publisher | [Stanford University] |

Copyright date | 2024; ©2024 |

Publication date | 2024; 2024 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Author | Xu, Wenqiang |
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Degree supervisor | Fox, Jacob, 1984- |

Degree supervisor | Soundararajan, Kannan, 1973- |

Thesis advisor | Fox, Jacob, 1984- |

Thesis advisor | Soundararajan, Kannan, 1973- |

Thesis advisor | Conrey, J. B |

Degree committee member | Conrey, J. B |

Associated with | Stanford University, School of Humanities and Sciences |

Associated with | Stanford University, Department of Mathematics |

## Subjects

Genre | Theses |
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Genre | Text |

## Bibliographic information

Statement of responsibility | Wenqiang Xu. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis Ph.D. Stanford University 2024. |

Location | https://purl.stanford.edu/wr403pk2531 |

## Access conditions

- Copyright
- © 2024 by Wenqiang Xu
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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