Erdős maximum-term constant

Description of constant

For any transcendental entire function $f(z)=\sum_{n\ge 0} a_n z^n$, define . \(M(r,f):=\max_{|z|=r}|f(z)|, \qquad \mu(r,f):=\max_{n\ge 0}|a_n|\,r^n.\) Following [Er1961], define \(\beta(f):=\liminf_{r\to\infty}\frac{\mu(r,f)}{M(r,f)}.\)

We define $C_{51} = B$ to be the supremum of $\beta(f)$ over all transcendental entire functions $f$.

Known upper bounds

Known lower bounds

Additional comments

• If polynomials were allowed, then $C_{51}=1$ (e.g. $f(z)=z+1$ gives $\mu(r,f)/M(r,f)\to 1$). The problem is interesting only in the transcendental class; this is the formulation used in [CH1964] and [HT2026].
• The quantity $\mu(r,f)/M(r,f)$ compares the size of the largest single term of the power series to the maximum modulus on $|z|=r$. In general it can be quite small (e.g. for $f(z)=e^z$, one has $\mu(r,f)/M(r,f)\to 0$ as $r\to\infty$), but Erdős conjectured that it cannot be too small, and in particular that $\beta(f)\le 1/2$ for all transcendental entire $f$. This was disproven by Kövári (unpublished), who showed that $\beta(f)$ can exceed $1/2$ for some transcendental entire function $f$.

• Clunie and Hayman [CH1964] constructed a specific transcendental entire function with $\beta(f)\ge 4/7$, and proved that this is the best possible using their construction. The problem of determining the exact value of $C_{51}$ and the Erdős problem asks how large it can be in the lim inf sense.

• The Clunie–Hayman lower bound $4/7$ is based on constructing a bilateral Laurent series $k$ satisfying a functional equation (“scaling identity”), then truncating it to an entire function $f$. The scaling identity allows one to compute $M(r,k)$ exactly on a geometric sequence of radii $r=K^m$, and hence relate $\beta(f)$ to $\max_{|z|=1}|k(z)|$ [CH1964].

• He and Tang [HT2026] generalize this to a two-parameter family $f_{K,\varepsilon}$ (scale $K>1$ and unimodular phase $|\varepsilon|=1$), prove an exact identity of the form \(\beta(f_{K,\varepsilon})=\frac{1}{\max_{|z|=1}|k_{K,\varepsilon}(z)|},\) and then certify a specific choice of $(K,\varepsilon)$ giving $\beta>0.58507$ via ball arithmetic.

• This problem is catalogued as Erdős Problem #513 (see [EP513]).

References

• [EP513] Bloom, T. F. Erdős Problem #513. https://www.erdosproblems.com/513 (accessed 2026-02-13).
• [CH1964] Clunie, J.; Hayman, W. K. The maximum term of a power series. J. Analyse Math. 12 (1964), 143–186. DOI: 10.1007/BF02807433.
• [Er1961] Erdős, P. Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254.
• [GS1963] Gray, A.; Shah, S. M. A note on entire functions and a conjecture of Erdős. Bull. Amer. Math. Soc. 69 (4) (1963), 573–577.
• [HL2019] Hayman, W. K.; Lingham, E. F. Research Problems in Function Theory: Fiftieth Anniversary Edition. Springer, 2019. (See Problem 2.14(c).)
• [HT2026] He, Yixin; Tang, Quanyu. Generalizing the Clunie–Hayman construction in an Erdős maximum-term problem. 2026. arXiv:2602.12217. https://arxiv.org/abs/2602.12217
• [HTcode] He–Tang certification code repository (linked from [HT2026]): https://github.com/QuanyuTang/ep513-arb-certification

Contribution notes

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