The Ramanujan Journal ( IF 0.7 ) Pub Date : 2020-07-25 , DOI: 10.1007/s11139-020-00288-5 Prapanpong Pongsriiam , Robert C. Vaughan
Let \(n, q, a\in \mathbb N\), d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over \(n\le x\) and \(n\equiv a\pmod q\). Previously we obtained an asymptotic formula for the second moment
$$\begin{aligned} V(x,Q) = \sum _{q\le Q}\sum _{a=1}^q \left| A(x,q,a)-M(x,q,a)\right| ^2, \end{aligned}$$where \(Q\le x\) and M(x, q, a) is a usual expected value for A(x, q, a). In this article, we compute the second moment similar to the previous one but M(x, q, a) is replaced by a better approximating function \(\rho _R(x,q,a)\) given by
$$\begin{aligned} \rho _R(x,q,a) = \sum _{\begin{array}{c} n\le x\\ n\equiv a(\bmod \, q) \end{array}}\sum _{r\le R}\frac{c_r(n)}{r}\left( \log \frac{n}{r^2}+2\gamma \right) , \end{aligned}$$where \(c_r(n)\) is Ramanujan’s sum. The advantage of using \(\rho _R(x,q,a)\) instead of M(x, q, a) is that the new second moment, for a suitable R, gives us practically the same main term and the error term of smaller or the same order of magnitude with much simpler and shorter calculation than that of V(x, Q).
中文翻译：
残差类别III的除数函数
令\（n，q，a \ in \ mathbb N \），d（n）为n的正数除数，而A（x， q， a）为d（n）与\（n \ le x \）和\（n \ equiv a \ pmod q \）。以前我们获得了第二个时刻的渐近公式
$$ \ begin {aligned} V（x，Q）= \ sum _ {q \ le Q} \ sum _ {a = 1} ^ q \ left | A（x，q，a）-M（x，q，a）\右| ^ 2，\ end {aligned} $$其中\（Q \ le x \）和M（x， q， a）是A（x， q， a）的通常期望值。在这篇文章中，我们计算类似于前一个，但第二个瞬间中号（X， q， 一）通过更好地逼近函数替换\（\ RHO _R（X，Q，A）\）由下式给出
$$ \ begin {aligned} \ rho _R（x，q，a）= \ sum _ {\ begin {array} {c} n \ le x \\ n \ equiv a（\ bmod \，q）\ end { array}} \ sum _ {r \ le R} \ frac {c_r（n）} {r} \ left（\ log \ frac {n} {r ^ 2} +2 \ gamma \ right），\ end {aligned } $$其中\（c_r（n）\）是Ramanujan的总和。使用\（\ rho _R（x，q，a）\）代替M（x， q， a）的优点是，对于合适的R，新的第二矩实际上给了我们相同的主项和误差项比V（x， Q）小或相同的数量级，且计算简单，短得多。
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