Difference between revisions of "Harmonic series"
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| − | A '''harmonic series''' is a form of the zeta function : | + | A '''harmonic series''' is a form of the [[zeta function]] : |
<math> \zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+... </math>. | <math> \zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+... </math>. | ||
| − | When <math>\ x</math> has a value less than or equal to one the function outputs infinity. Euler found that when <math>\ x=2</math>, the zeta function outputs <math>\frac{\pi^2}{6} </math>. | + | When <math>\ x</math> has a value less than or equal to one the function outputs infinity. [[Euler]] found that when <math>\ x=2</math>, the zeta function outputs <math>\frac{\pi^2}{6} </math>. |
| − | Euler also realized that since every number is the multiplication of some order of | + | Euler also realized that since every number is the multiplication of some order of [[prime]]s, then the zeta function is equal to <math>(1+\frac{1}{2^x}+\frac{1}{4^x}+...)(1+\frac{1}{3^x}+\frac{1}{9^x}+...)...(1+\frac{1}{p^x}+\frac{1}{(p^2)^x}+...)...</math> |
| − | Riemann found that when complex | + | Riemann found that when [[complex number]]s are the input to the zeta function, the resulting graph is that which aids in the finding of the exact value of <math>\ \pi (n)</math> or the number of primes less than or equal to <math>\ n</math>. |
== How to solve == | == How to solve == | ||
Revision as of 11:47, 24 June 2006
A harmonic series is a form of the zeta function :
.
When 
has a value less than or equal to one the function outputs infinity. Euler found that when 
, the zeta function outputs 
.
Euler also realized that since every number is the multiplication of some order of primes, then the zeta function is equal to 
Riemann found that when complex numbers are the input to the zeta function, the resulting graph is that which aids in the finding of the exact value of 
or the number of primes less than or equal to 
.
