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mathslover
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Please help me in solving the problem,
find the sum
Sum{r=2 to infinity} (von Mangoldt(r)-1)/r
Your help is appreciated.
find the sum
Sum{r=2 to infinity} (von Mangoldt(r)-1)/r
Your help is appreciated.
mathslover said:hi mhill,
can you prove that the series is divergent?
mathslover said:when n is a prime or prime power, the summation is okay.
but suppose when n=6, then the sum is (von Mangoldt(6) -1)/6 , which is = -1/6,
as n runs from 2 to infinity,can we settle the problem of convergency or divergency?
mathslover said:Is something missing ?
The von Mangoldt function is a mathematical function that is defined as the natural logarithm of a prime number, and 0 for all other numbers. It is denoted by Λ(n) and is named after mathematician Hans von Mangoldt. It is used in number theory to study the distribution of prime numbers.
The von Mangoldt function is used in a summation known as the summation involving von Mangoldt function. This summation is defined as the summation of the von Mangoldt function over all positive integers up to a given number n. It is commonly used in number theory to study the behavior and properties of prime numbers.
The summation involving von Mangoldt function has many important applications in number theory. It is used to prove the Prime Number Theorem, which states that the number of prime numbers less than a given number n is approximately equal to n/ln(n). It is also used in the Riemann hypothesis, a famous unsolved problem in mathematics.
The prime counting function is defined as the number of prime numbers less than or equal to a given number n. It is denoted by π(n). The summation involving von Mangoldt function is closely related to the prime counting function, as it is used in the proof of the Prime Number Theorem, which gives an approximation for π(n).
Yes, there are several open problems related to the summation involving von Mangoldt function. One of the most famous is the Riemann hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2. Another open problem is the Goldbach conjecture, which states that every even number greater than 2 can be expressed as the sum of two prime numbers.