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math_grl
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If [tex]f[/tex] is completely multiplicative, then [tex]\sum_{d \mid n} f(d)[/tex] is completely multiplicative is not true. There must be an easy counterexample for this yet I cannot come up with one.
math_grl said:I was hoping for more an explicit function, f, such that F is not completely multiplicative.
"Completely multiplicative" refers to a mathematical property where a function or operation preserves multiplication when applied to two or more numbers. In other words, if a function is completely multiplicative, the result of multiplying two numbers and then applying the function to the product will be the same as applying the function to each individual number and then multiplying the results.
A common example of a completely multiplicative function is the Euler totient function, also known as the phi function. This function calculates the number of positive integers less than or equal to a given number that are relatively prime to that number. It is completely multiplicative because the totient of a product of two numbers is equal to the product of the totients of each individual number.
Prime numbers are completely multiplicative because they only have two factors: 1 and the number itself. This means that when a prime number is multiplied by any other number, the result will always have only two factors, making it another prime number. For example, 7 is a prime number and 7*2=14, which is also a prime number. This property is important in number theory and is used in various mathematical proofs.
No, not every function is completely multiplicative. A function that is not completely multiplicative may still preserve multiplication for some numbers, but it will not hold true for all numbers. For example, the function f(x) = x+1 is not completely multiplicative because f(2*3) = 7, but f(2)*f(3) = 6.
While both terms involve the concept of multiplication, "completely multiplicative" is a stronger property than just being "multiplicative." A function is considered multiplicative if it preserves multiplication for any two numbers, but it may not hold true for more than two numbers. In contrast, a completely multiplicative function preserves multiplication for any number of numbers, making it a more powerful property.