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The bound of sums of multiplicative functions with shifted arguments. (English. Russian original) Zbl 0830.11035

Math. Notes 54, No. 5, 1138-1146 (1993); translation from Mat. Zametki 54, No. 5, 84-98 (1993).
The author’s starting point is the unproven conjecture that \(S(\lambda, \lambda, x)= o(x)\) for the Liouville function \(\lambda\), where \[ S(f, g, x)= \sum_{|a|< n\leq x} f(n) g(n+ a), \] where \(a\neq 0\) is an integer. Under suitable assumptions, the author aims for estimates of the form \[ |S(f, g, x)|\leq \theta x, \text{ with some } \theta, \quad 0< \theta< 1, \] where \(f\), \(g\) are multiplicative functions satisfying \(f= g=1\).
Theorem 1 states that \[ |S(f, g, x)|\leq \biggl( 1- \Bigl( {\textstyle {1\over 2}- \sqrt {{k \over r}}} \Bigr)^2 {\textstyle {4\over r^2}} \biggr) x+R, \] with some complicated estimate for \(R\), if \(g^r =1\) and if \(f\) takes at least \(k< {1\over 4}r\) distinct values at primes \(p\), \(\sqrt {x}\leq p\leq x\).
From this theorem it follows, for example, that (for \(x\geq x_0 (\varepsilon))\) \[ |S(f, g, x)|\leq \biggl( 1- \Bigl( {\textstyle {1\over 2}- \sqrt {{k\over r}}} \Bigr)^2 {\textstyle {4\over r^2}}+ \varepsilon \biggr) x, \] if in addition all series \(\sum_p {1\over p} (1- \text{Re} (g^s (p) p^{it} \chi^*_d (p)))\) diverge for any real \(t\), any \(s= 1,2, \dots, r-1\), and any primitive character \(\chi^*_d\). A second theorem gives an estimate for \(S(f, g, x)\) for multiplicative functions \(f(n)= \exp (2\pi i\alpha(n))\), \(g(n)= \exp (2\pi i\beta (n))\), where \(0\leq \alpha (n)< 1\), \(0\leq \beta(n)< 1\), and where (for some \(\sigma\), \(0< \sigma< {1\over 12} \sigma_1)\) \[ |\{p;\;p\leq y,\;|\alpha (p)- \alpha_0 |< \sigma\} |\geq \sigma_1 \pi (y) \] in \(\sqrt {x}\leq y\leq x\), where \(|x|\) is the distance to the nearest integer.
The proof uses sieve estimates and Bombieri-type estimates for “bilinear” expressions of the form \[ \sum_{\substack{ nm\leq t, nm\equiv \ell\bmod d,\;\text{gcd} ({{mn-\ell} \over d}, \prod_{y< p\leq y_1} p) =1}} a_n b_m, \] and for \(\sum_n f(n)\) with similar conditions of summation.

MSC:

11N37 Asymptotic results on arithmetic functions
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