Research Problems in Function Theory — Problem 2.44

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 $\log0=-\infty$. For a nonconstant entire $f$ of order $\rho$, define \[\nu_f(r)=\#\{z:|z|=r,\ |f(z)|=1\}.\] This counts distinct points. Use $\log0=-\infty$ and $\log(+\infty)=+\infty$ if an exceptional radius has infinitely many such points.

Statement recorded in the supplied source.

Must \[\limsup_{r\to\infty}\frac{\log\nu_f(r)}{\log r}=\rho(f)?\] [S2] Update 2.44 disproves the proposed identity with $f(z)=e^z$: there are exactly two unit-modulus points on each circle of positive radius, while $\rho(f)=1$. [S2]

Short English statement

Does the number of unit-modulus points on large circles grow with the same exponent as the entire function's order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2302044 (AMR-022-2044). 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 2.44 and Update 2.44. https://arxiv.org/abs/1809.07200.