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galoisjr
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It has been determined that the square of zeta can be written in terms of the divisor function.
[tex]{\zeta ^2}(s) = \sum\limits_{n = 1}^\infty {\frac{{d(n)}}{{{n^s}}}} [/tex]
Being a first semester student in complex variables, I have only recently started looking at zeta. But I have deduced an alternate and equivalent representation of the square of zeta. I must say that I am ignorant to the value of its importance, but I must say that in my opinion it is a beautiful series, and a very interesting representation.
From the Fundamental Theorem of Arithmetic, we know that each natural number n can be written as an unique product of primes. A consequence of this is that we can view the primes to be constant throughout the natural numbers, then we can write each n as an infinite product of primes in which the sequences of powers are unique to its respective value of n. In other words for each n,
[tex]n = p_0^{{\alpha _0}} \cdot \cdot \cdot p_n^{{\alpha _n}}[/tex]
Although I know that it is unconventional to assume that 1 is a prime number, but I do also know that there have been a handful great mathematicians in history who have done so. For the sake of this argument, we can further define p_0=1 and alpha_0=1. So for example, since all primes are assumed in this constant sense, 6={1,1,1,0,...}(where alpha_0=1, alpha_1=1, alpha_2=1, and for n>2 the sequence vanishes).
Using this in the square of zeta we can instead sum over the prime factorizations of numbers and we obtain
[tex]{\zeta ^2}(s) = \sum\limits_{n = 1}^\infty {\frac{{d(n)}}{{{n^s}}}} = 1 + \sum\limits_{n = 1}^\infty {\frac{{{\alpha _0} + {\alpha _n}}}{{p_n^{{\alpha _n}s}}} + \sum\limits_{n = 0}^\infty {\frac{{1 + {\alpha _0} + ... + {\alpha _n}}}{{{{\left( {p_0^{{\alpha _0}} \cdot \cdot \cdot p_n^{{\alpha _n}}} \right)}^s}}}} } [/tex]
In which the first term has denominators that are powers of a single prime and the second term has more than 1 prime in its prime factorization (excluding 1). We can see that for the first term with denominator p^k will have numerator k+1, and for the second term each denominator will have a unique prime factorization, and thus, the numerator will be the sum of the powers + 1.
[tex]{\zeta ^2}(s) = \sum\limits_{n = 1}^\infty {\frac{{d(n)}}{{{n^s}}}} [/tex]
Being a first semester student in complex variables, I have only recently started looking at zeta. But I have deduced an alternate and equivalent representation of the square of zeta. I must say that I am ignorant to the value of its importance, but I must say that in my opinion it is a beautiful series, and a very interesting representation.
From the Fundamental Theorem of Arithmetic, we know that each natural number n can be written as an unique product of primes. A consequence of this is that we can view the primes to be constant throughout the natural numbers, then we can write each n as an infinite product of primes in which the sequences of powers are unique to its respective value of n. In other words for each n,
[tex]n = p_0^{{\alpha _0}} \cdot \cdot \cdot p_n^{{\alpha _n}}[/tex]
Although I know that it is unconventional to assume that 1 is a prime number, but I do also know that there have been a handful great mathematicians in history who have done so. For the sake of this argument, we can further define p_0=1 and alpha_0=1. So for example, since all primes are assumed in this constant sense, 6={1,1,1,0,...}(where alpha_0=1, alpha_1=1, alpha_2=1, and for n>2 the sequence vanishes).
Using this in the square of zeta we can instead sum over the prime factorizations of numbers and we obtain
[tex]{\zeta ^2}(s) = \sum\limits_{n = 1}^\infty {\frac{{d(n)}}{{{n^s}}}} = 1 + \sum\limits_{n = 1}^\infty {\frac{{{\alpha _0} + {\alpha _n}}}{{p_n^{{\alpha _n}s}}} + \sum\limits_{n = 0}^\infty {\frac{{1 + {\alpha _0} + ... + {\alpha _n}}}{{{{\left( {p_0^{{\alpha _0}} \cdot \cdot \cdot p_n^{{\alpha _n}}} \right)}^s}}}} } [/tex]
In which the first term has denominators that are powers of a single prime and the second term has more than 1 prime in its prime factorization (excluding 1). We can see that for the first term with denominator p^k will have numerator k+1, and for the second term each denominator will have a unique prime factorization, and thus, the numerator will be the sum of the powers + 1.
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