Additive uniqueness of $\mathtt{PRIMES}-1$ for multiplicative functions

$k$-additive uniqueness of the set of squares for multiplicative functions

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are involved, then such $f$ is uniquely determined. That is, if a multiplicative function $f$ satisfies \[ f(a_1^2 + a_2^2 + \dotsb + a_k^2) = f(a_1^2) + f(a_2^2) + \dotsb + f(a_k^2) \] for arbitrary positive integers $a_i$, then $f$ is the identity function. In this sense, we call the set of all posotive squares a \emph{$k$-additive uniqueness set} for multiplicativ…

斐波那契数不是乘法函数的加法唯一性集合

The Fibonacci numbers are not an additive uniqueness set for multiplicative functions

本文通过构造反例，否证了 Spiro 关于斐波那契数构成乘法函数加法唯一性集合的猜想，并给出 AI 辅助搜索与验证方法。

Prime numbers as a uniqueness set of the parallelogram equation via the Goldbach's conjecture

Multiplicative arithmetic functions satisfying the parallelogram functional equation on prime numbers are investigated. It is derived that the unique solution is a quadratic function by the Goldbach's conjecture.

On multiplicative functions which are additive on positive cubes

Let $k \geq 3$. If a multiplicative function $f$ satisfies \[ f(a_1^3 + a_2^3 + \cdots + a_k^3) = f(a_1^3) + f(a_2^3) + \cdots + f(a_k^3) \] for all $a_1, a_2, \ldots, a_k \in \mathbb{N}$, then $f$ is the identity function. The set of positive cubes is said to be a $k$-additive uniqueness set for multiplicative functions. But, the condition for $k=2$ can be satisfied by infinitely many multiplicative functions. Besides, if $k \geq 3$ and a multiplicative function $g$ satisfies \[ g(a_1^3 + a_2^3 + \cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + \cdots + g(a_k)^3 \] for all $a_1, a_2, \ldots, a_k \in \…

关于乘性函数的若干分类问题

On some classification problems of multiplicative functions

本文刻画乘性函数的 Toeplitz 性、唯一 Furstenberg 系统与 Besicovitch 殆周期性之间的关系，并由修正 Elliott 猜想推出 Frantzikinakis-Host 唯一性猜想。