Summation involving von Mangoldt function

1. Aug 3, 2008

mathslover

find the sum
Sum{r=2 to infinity} (von Mangoldt(r)-1)/r

2. Aug 11, 2008

mhill

do you mean $$\sum _{n=2}^{\infty} \frac{ \Lambda (n) -1}{n}$$ ??

i think is divergent

3. Aug 12, 2008

mathslover

hi mhill,
can you prove that the series is divergent?

-Ng

4. Aug 12, 2008

CRGreathouse

$$\frac1n \sum_{k=1}^n\Lambda(k)=1+o(1/\log n)$$

so your series seems to be something like

$$\sum\frac{1}{n\log n}\approx\log\log n$$

Obviously this is very heuristic here.

5. Aug 12, 2008

CRGreathouse

OK, it diverges.

$$\sum_{n=2}^{\infty} \frac{\Lambda(n) -1}{n}=\sum_p\sum_{k=1}^\infty$$\frac{\log p-1}{p}+\frac{\log p-1}{p^2}+\cdots$$=\sum_p\frac{\log p-1}{p-1}$$
and we all know that
$$\sum_p\frac1p=+\infty$$

6. Aug 13, 2008

mathslover

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?

-Ng

7. Aug 13, 2008

CRGreathouse

My post addressed the case where n runs from 2 to infinity, which diverges.

8. Aug 15, 2008

mathslover

Hi CRGreatHouse,

In your post 1673, the summation on LHS runs from n=2 to infinity, (n=2,3,4,5,6,7,8,...)

But the summation on RHS runs over all primes.(p=2,3,5,7,...)

From the definition of von Mangoldt function,when n=6,10,12,14,15,18,... , the summand

became (-1/n) whenever n is not equal to any prime or prime power.

Is something missing ?

-Ng

9. Aug 15, 2008

CRGreathouse

Right, right... yeah, I calculated it for numerator $\Lambda$ first, forgetting about the -1 term, and when I added it back in forgot that part.

But wouldn't that also suggest divergence (in the other direction), since the prime powers are density 0, the reciprocal primes vary as log log n, and the reciprocal integers vary as log n?

Numerical experimentation would be nice here.

10. Aug 16, 2008

mathslover

I have tried numerical calculation and the sum seems to converge to ~ -1.16

Can we approach the problem from Zeta function?

-Ng