Research Problems in Function Theory — Problem 2.43

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an entire function $f$, write \[ M(r,f)=\max_{|z|=r}|f(z)|,\qquad m_0(r,f)=\min_{|z|=r}|f(z)|. \] For nonconstant $f$, its order is $\rho(f)=\limsup_{r\to\infty}\log\log M(r,f)/\log r$; the lower order is defined using $\liminf$ instead. Every constant entire function, including the zero function, has order and lower order zero. In logarithmic modulus ratios, use the extended-real convention $\log0=-\infty$. A function permutes a set $S$ if $f|_S:S\to S$ is bijective. For an entire function of order one, its exponential type is $\tau(f)=\limsup_{r\to\infty}\log M(r,f)/r$.

Statement recorded in the supplied source.

If $f$ is transcendental entire and permutes $\mathbb Z$, must $\rho(f)\ge1$, with $\tau(f)\ge\pi$ whenever $\rho(f)=1$? Ask the same question when $f$ permutes $\mathbb N=\{1,2,\ldots\}$. [S2] Update 2.43 supplies the counterexample $f(z)=z+4\cos(\pi(z-2)/3)$, which permutes both sets and has order one and type $\pi/3$. It also asks for examples of reasonably small type that substantially rearrange the positive integers, without specifying a quantitative scrambling criterion. [S2]

Short English statement

Do entire functions that bijectively rearrange either all integers or the positive integers require the proposed order and exponential type?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302043 (AMR-022-2043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), Chapter 2, Problem 2.43 and Update 2.43. https://arxiv.org/abs/1809.07200.