Given any positive integer n, let $A(n)$ denote the height of the $n\text{th}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes p and $p^{\prime }$ with $p⩾127$, we have $p^{\prime }-p<\sqrt{p}+1$. We also conjecture that every natural number occurs as the maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture holds, that is, that $\sqrt{p^{\prime }}-\sqrt{p}<1$ always holds. This is the first time, as far as the authors know that a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath–Brown on prime gaps, we show unconditionally that every natural number $m⩽x$ occurs as $A(n)$ value with at most $O_{\epsilon }(x^{3/5+\epsilon })$ exceptions. On the Lindelöf Hypothesis, we show there are at most $O_{\epsilon }(x^{1/2+\epsilon })$ exceptions and study them further by using deep work of Bombieri–Friedlander–Iwaniec on the distribution of primes in arithmetic progressions beyond the square‐root barrier.

中文翻译：

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具有规定高度和素数理论的循环多项式

给定任何正整数n，让$\mathrm{一种}（ñ）$ 表示高度 $ñ\text{日}$环多项式，即其绝对值的最大系数。众所周知$\mathrm{一种}（ñ）$是无界的。我们推测，每个自然数都可以作为$\mathrm{一种}（ñ）$并假设每对连续素数p和$p^{\prime }$ 与 $p⩾127$， 我们有 $p^{\prime }-p<\sqrt{p}+\mathrm{1个}$。我们还推测，每个自然数都是某个环多项式的最大系数，并且证明如果Andrica的猜想成立，这是真的。$\sqrt{p^{\prime }}-\sqrt{p}<\mathrm{1个}$总是成立。据作者所知，这是首次发现质子间隙与环多项式之间的联系。使用希斯布朗的质子缺口结果，我们无条件表明每个自然数$米⩽X$ 发生为 $\mathrm{一种}（ñ）$ 最多有价值 ${Ø}_{\epsilon }（X^{3/5+\epsilon }）$例外。关于Lindelöf假说，我们表明最多${Ø}_{\epsilon }（X^{\mathrm{1个}/2+\epsilon }）$ 例外情况，并通过使用Bombieri–Friedlander–Iwaniec在平方根障碍以外的算术级数中素数的分布的深入研究来对其进行进一步研究。