Geodesic
On a~conjecture on sums of multiplicative functions
Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 387-403

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We consider the following conjecture, which was made in 1970 by B. V. Levin and A. S. Fainleib; if $f\in W$, $f(p)\leqslant g(p)$, $\sum_{p\leqslant x}g(p)\ln p\sim\tau_gx$, and (2) is fulfilled with $\tau_f\ne0$, then (1) holds. We prove that this conjecture holds if $\tau_f\cdot\tau_g >0$. In the case $\tau_f\cdot\tau_g\leqslant0$ we construct a counterexample to the conjecture. The asymptotic behavior of the sum of values of the function is found by an analytic method. 
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     author = {S. T. Tulyaganov},
     title = {On a~conjecture on sums of multiplicative functions},
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     number = {2},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_71_2_a7/}
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S. T. Tulyaganov. On a~conjecture on sums of multiplicative functions. Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 387-403. http://geodesic.mathdoc.fr/item/SM_1992_71_2_a7/