TejasviTomar
October 2026 · Preprint

An explicit bound below 2/π for the maximum term of an entire function

Tejasvi Singh Tomar

In plain words

Consider a power series with infinitely many nonzero terms that converges in the whole complex plane, like the exponential function. On the circle of radius r around 0, its largest term is at most its largest absolute value there. In 1961 Erdős asked how large their ratio can stay as r grows. An argument of Clunie gives at most 2/π, about 0.6366. In 1964 Clunie and Hayman showed that the answer is smaller than 2/π by some fixed positive amount, which they did not compute. This paper proves that the ratio cannot stay above 0.63005.

Abstract

For a transcendental entire function f(z) = ∑ akzk, let μ(r) = maxk |ak|rk be its maximum term and M(r) its maximum modulus on |z| = r. Erdős asked for the largest possible value B of lim inf μ(r)/M(r) as r → ∞ (Erdős problem #513). An argument of Clunie gives B ≤ 2/π = 0.636619…, and in 1964 Clunie and Hayman proved B ≤ 2/π − c for an absolute constant c > 0, which they did not compute. The lower bound B > 0.5850788 is due to <a href="https://www.erdosproblems.com/forum/thread/513">Sothanaphan</a>, who improved the parameters of a construction of He and Tang. We prove B ≤ 0.63005. The proof bounds M(r) from below by duality: trigonometric polynomials are tested against the Fourier coefficients of f(reiθ) in a finite window. Two consecutive radii at which the central index jumps involve three consecutive vertices of the Newton polygon of log|ak|. If the two gaps between these vertices are equal, testing f at both radii leads to a problem in six real parameters. If they differ, a single radius is used, and averaging over roots of unity leads to a problem in seven parameters for each value p ≥ 2 of the larger gap divided by the greatest common divisor of the two gaps. Both problems are bounded by branch and bound, with dual vectors certified in ball arithmetic. Two independently written programs check the certificates, which have 176,240 leaves in the equal-gap case and 37,198 in the unequal-gap case.

Left, a teal rectangle divided into smaller and smaller rectangles that shrink towards a magenta dot near the centre, with a horizontal band of yellow rectangles whose top edge runs through the dot, deepening towards the sides, a narrow yellow column near the left edge, and a yellow strip along the right edge. Right, two small bar charts of five bars each, the upper one teal and the lower one yellow; the bars rise or fall from a solid baseline between dashed lines above and below. In each chart two neighbouring bars have thick outlines and reach the dashed lines: in the upper chart one reaches down to the lower line and the next up to the upper line, and in the lower chart both reach the upper line.
Illustration of the equal-gap certificate. Left: a slice through its partition of the six-parameter domain, at x = y = 0.417 and φ₋₁ = φ₂ = π, with σ (from 0 to 4) across and φ₋₂ up, taken modulo 2π so that φ₋₂ = π is in the middle. Both axes are stretched near the magenta dot (an inverse hyperbolic sine scale), so that the smallest rectangles of the slice, 512 times narrower than the largest, are visible. Each rectangle is the cross-section of one box: teal boxes carry a dual vector for the five-term window at the first of two consecutive radii where the largest term changes hands, yellow boxes one for the window at the second. The dot is the configuration where, numerically, the five-term bound is largest, about 0.630041, just below the certified 0.63005. Right: the five coefficients of that configuration, normalized so that the middle one is 1, at the first radius (top) and the second (bottom); the bars with thick outlines are the two tied terms, whose size is the maximum term, and the dashed lines mark ±1.