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An Introduction to Probabilistic Number Theory
  • Emmanuel Kowalski
  • Published online:
    04 May 2021
    Print publication:
    06 May 2021
  • Despite its seemingly deterministic nature, the study of whole numbers, especially prime numbers, has many interactions with probability theory, the theory of random processes and events. This surprising connection was first discovered around 1920, but in recent years the links have become much deeper and better understood. Aimed at beginning graduate students, this textbook is the first to explain some of the most modern parts of the story. Such topics include the Chebychev bias, universality of the Riemann zeta function, exponential sums and the bewitching shapes known as Kloosterman paths. Emphasis is given throughout to probabilistic ideas in the arguments, not just the final statements, and the focus is on key examples over technicalities. The book develops probabilistic number theory from scratch, with short appendices summarizing the most important background results from number theory, analysis and probability, making it a readable and incisive introduction to this beautiful area of mathematics.

  • Kloosterman paths and the shape of exponential sums

  • Part of
  • Emmanuel Kowalski, William F. Sawin
  • Journal:
    Compositio Mathematica / Volume 152 / Issue 7 / July 2016
    Published online by Cambridge University Press:
    15 April 2016, pp. 1489-1516
    Print publication:
    July 2016
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  • We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums $\text{Kl}_{p}(a)$, as $a$ varies over $\mathbf{F}_{p}^{\times }$ and as $p$ tends to infinity. Using independence of Kloosterman sheaves, we prove convergence in the sense of finite distributions to a specific random Fourier series. We also consider Birch sums, for which we can establish convergence in law in the space of continuous functions. We then derive some applications.

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