1+Sum of primes^-1 * (-1)^(PI)

  • Thread starter Matt Benesi
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In summary, the formula "1+Sum of primes^-1 * (-1)^(PI)" is known as the Riemann zeta function, which is an important concept in number theory and complex analysis. It relates to the distribution of prime numbers and has connections to other areas of mathematics. The Riemann zeta function is undefined for 1, as the summation in the formula diverges. However, it can be evaluated for any complex number with real part greater than 1. The Riemann zeta function is closely related to the prime number theorem, and the Riemann hypothesis is a famous and unsolved conjecture that states all non-trivial zeros of the function lie on a specific line.
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Matt Benesi
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Is this series divergent and what is its name? If the series is not divergent, what is the constant named? Note that [itex]\pi[/itex] is the prime counting function, or number of primes.

[tex]1+\sum_{p=primes}^{\infty}\frac{(-1)^{\pi}}{p}={1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{11}...[/tex]

Thanks :D
 
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  • #3
Thanks.
 

What is the significance of the formula "1+Sum of primes^-1 * (-1)^(PI)"?

The formula "1+Sum of primes^-1 * (-1)^(PI)" is known as the Riemann zeta function, which is an important concept in number theory and complex analysis. It relates to the distribution of prime numbers and has connections to other areas of mathematics.

What is the value of the Riemann zeta function for 1?

The Riemann zeta function is undefined for 1, as the summation in the formula diverges.

Can the Riemann zeta function be evaluated for any other values?

Yes, the Riemann zeta function can be evaluated for any complex number with real part greater than 1. For values in this range, it has a well-defined value.

Is there a connection between the Riemann zeta function and the prime number theorem?

Yes, the Riemann zeta function is closely related to the prime number theorem, which states that the number of prime numbers less than a given number x is approximately x/ln(x). The prime number theorem is equivalent to the statement that the Riemann zeta function has no zeros on the line where the real part is equal to 1.

What is the Riemann hypothesis?

The Riemann hypothesis is a famous and unsolved conjecture in mathematics, which states that all non-trivial zeros of the Riemann zeta function lie on the line where the real part is equal to 1/2. This hypothesis has many important consequences and implications in number theory and has been a subject of much study and research for over a century.

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