Geodesic
On Differences of Multiplicative Functions and Solutions of the Equation $n-\varphi(n)=c$
Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 608-615

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The following general problem is studied: Given a positive integer $c$ and two multiplicative functions $f$ and $g$, it is required to determine for what values of $n$ the equality $f(n)-g(n)=c$ holds. It is proved that, under certain constraints on the functions $f$ and $g$ and the solutions (in particular, under the constraint $f(n)>g(n)$ for $n>1$), this equation has at most $c^{1-\epsilon}$ solutions. For the equation $n-\varphi(n)=c$, it is proved that the number of solutions equals $$ G(c+1)+O(c^{3/4+o(1)}), $$ where $G(k)$ is the number of ways in which $k$ can be represented as a sum of two primes. This result is based on an assertion concerning configurations of points and straight lines.
Keywords: multiplicative functions
Mots-clés : Euler totient function. 
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     author = {A. S. Semchankau},
     title = {On {Differences} of {Multiplicative} {Functions} and {Solutions} of the {Equation} $n-\varphi(n)=c$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {608--615},
     publisher = {mathdoc},
     volume = {109},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a11/}
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A. S. Semchankau. On Differences of Multiplicative Functions and Solutions of the Equation $n-\varphi(n)=c$. Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 608-615. http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a11/