1
vote
0answers
46 views

Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
5
votes
0answers
101 views

Asymptotic expansion of $\zeta(s)$

It is well known that $$ \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
1
vote
0answers
65 views

How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...
1
vote
0answers
131 views

Approximation of distribution of $\pi_k(n)$ using $\zeta(s)$

Let $\pi_k(n) $ be the number of numbers with k prime factors (repetitions included) less than or equal to n. If we take the sums: $z_1(s) = \sum_{n= 1}^\infty \frac{1}{(p_{1,n})^s},~ z_2(s) = \sum ...
0
votes
1answer
145 views

Simple clarification - deduction using big-O notation

A set of lecture notes I'm reading on Halasz's theorem makes the following statement in a proof, which I can't quite follow - I was hoping someone might be able to clear up what I'm missing: ...
8
votes
1answer
177 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...