Sums of fractional parts of functions of a special form (Q1385936)

In this article the author claims that by the method in his earlier work [\textit{A. A. Karatsuba}, Russ. Acad. Sci., Dokl., Math. 48, 452-454 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 333, 138-139 (1993; Zbl 0826.11032] the following results concerning the \(f(x)\) defined below are valid. Let \(m\) and \(p\) denotes a positive integer and a prime respectively; \(a,b\) and \(x\) be integers with \((a,m)=1=(x,m)\); \(xx^*\equiv 1\pmod m\) with \(1\leq x^*\leq m\); \(0<\varepsilon\leq 0.001\); \(m^\varepsilon\leq X\leq m\); \(L=\log m\) and \(f(x)=\left\{{ax^*+bx\over m}\right\}\) where \(\{z\}\) means the fractional part of \(z\). Then there are \(c_j=c_j(\varepsilon)>0\) such that if \(E_j\) denotes \(1=O(L^{-c_j})\), \(j=1, 2, 3, 4,\) we have \[ \sum_{1\leq x\leq X}f(x)={\varphi(m)X\over 2m}E_1\quad\text{and}\quad\sum_{p\leq X}f(p)={\pi(X)\over 2}E_2\tag{1} \] (2) for any given real \(\alpha,\beta\) with \(0\leq\alpha<\beta\leq 1\), \[ N(\alpha,\beta,X)=(\beta-\alpha){\varphi(m)X\over m}E_3\quad\text{and}\quad N_1(\alpha,\beta,X)=(\beta-\alpha)\pi(X)E_4 \] where \(N(\alpha,\beta,X)\) (or \((N_1(\alpha,\beta,X))\) is the number of positive integers \(x\leq X\) (or \(p\leq X)\) such that \(\alpha\leq f(x)<\beta\) (or \(\alpha\leq f(p)<\beta)\). No detailed proof of the above results is given in this article.