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Multiplicative functions which are additive on triangular numbers
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초록
Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$.
Abstract
Fix $k \ge 3$. If a multiplicative function $f$ satisfies \[ f(x_1+x_2+\dots+x_k) = f(x_1) + f(x_2) + \dots + f(x_k) \] for arbitrary positive triangular numbers $x_1, x_2, \dots, x_k$, then $f$ is the identity function. This extends Chung and Phong's work for $k=2$.
참고문헌 (6)
- P. V. Chung, "Multiplicative functions satisfying the equation f(m2+n 2)= f(m2)+f(n 2)", Math. Slovaca, 46(2-3), pp.165~171 (1996)
- P. V. Chung, "Additive uniqueness sets for multiplicative functions", Publ. Math. Debrecen, 55(3-4), pp.237~243 (1999)
- Artūras Dubickas, "On multiplicative functions which are additive on sums of primes", Aequationes mathematicae, 86(1-2), pp.81~89 (2013), Springer Science and Business Media LLC[DOI]
- Jin-Hui Fang, "A characterization of the identity function with equation f(p+q+r)=f(p)+f(q)+f(r)", Combinatorica, 31(6), pp.697~701 (2011), Springer Science and Business Media LLC[DOI]
- Poo-Sung Park, "On k-additive uniqueness of the set of squares for multiplicative functions", Aequationes mathematicae, 92(3), pp.487~495 (2018), Springer Science and Business Media LLC[DOI]
- Claudia A Spiro, "Additive uniqueness sets for arithmetic functions", Journal of Number Theory, 42(2), pp.232~246 (1992), Elsevier BV[DOI]
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