**Editors: J.P. Keating, Z. Rudnick and T.D. Wooley **

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half a century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. The articles in this volume explain some of the recent developments. Specifically, they focus on statistical properties of arithmetic functions in the two settings, on cancellation in arithmetic sums, and on moments of zeta functions.

*Cover image*: The image on the title page is a slightly modified phase plot of the Riemann zeta function in the domain –27 < Re *z* < 6, –4 < Im *z* < 29. The “rainbow points” where all colours meet are either zeros or poles, they can be distinguished by the orientation of colours in their neighbourhood. We see the pole at *z* = 1, nine trivial zeros at the negative even integers –2, –4, ..., –18, and the three zeros on the critical line Re *z* = 1/2 which were known already to Berhard Riemann. For further information on phase plots we refer to Semmler and Wegert [1], a comprehensive introduction is given in “Visual Complex Functions” [2], with additional material provided at www.visual.wegert.com.

*Image copyright*: Elias Wegert

*References*: [1] E. Wegert, G. Semmler, Phase Plots of Complex Functions: A Journey in Illustration. *Notices of the AMS* **58** (2011), 768–780. [2] E. Wegert, *Visual Complex Functions – An Introduction with Phase Portraits*. Springer, Basel 2012.

## Read

This issue is available to read online, including the introduction which is free to access.

**Don't have access?**

If you would like to recommend this journal to your Librarian please fill out this form or contact sales@royalsociety.org

## Buy

This issue is available to buy in print.

We offer discounts for bulk orders of the print version of this journal issue for educational use (£20 per copy for 10 or more). Please contact our sales team or more information.

## Also of interest

**Related issues**

- Partial differential equation models in the socio-economic sciences
- Complex patterns in wave functions – drums, graphs, and disorder
- Entropy and convexity for nonlinear partial differential equations
- Fractional calculus and its applications

**Read more content from all our journals on:**

*Philosophical Transactions A * publishes themed journal issues on topics of current scientific importance and general interest within the physical, mathematical and engineering sciences, edited by leading authorities and comprising original research, reviews and opinions from prominent researchers. To browse all our recent issues please see our previous issues page.

## New content

REGISTER for table of contents email alerts to see the latest content published in Royal Society journals

## Propose an issue

Find out more about guest editing a theme issue of *Philosophical Transactions A* from our Guest Editor information page.