In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

中文翻译：

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连续素数偏差的分布和锯齿随机变量的总和

在最近的工作中，我们考虑了连续素数模式的频率（modq) 并在数字上发现了对某些模式和其他模式的偏见。我们提出了一个猜想来解释这些偏差，这是允许简单描述但无法区分许多看似频率非常不同的模式的主要因素。我们的猜想中有一个次要因素解释了这种额外的变化，但它只是由一个复杂的表达式给出，其分布不容易理解。在这里，我们研究了这个项，它被证明与经典 Dedekind 和的傅里叶变换以及 φ 之和的渐近公式中的误差项（n）。