Intereting Posts

What does smooth curve mean?
Proving that a map is a weak homotopy equivalence
$n^5-n$ is divisible by $10$?
How to use $\binom a k = \frac{\alpha(a-1)(a-2)\cdots(a-k+1)}{k(k-1)(k-2)\cdots 1}$ to check that ${-1\choose 0}=1$?
Complete classification of the groups for which converse of Lagrange's Theorem holds
Find the Roots of $(x+1)(x+3)(x+5)(x+7) + 15 = 0$
Any closed subset of $\mathbb C$ is the set of limit points of some sequence
Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$
A game of guessing a chosen number
How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1\over n } }{ n } $?
Image of Möbius transformation
Analytic versus Analytical Sets
Concise proof that every common divisor divides GCD without Bezout's identity?
Is every submodule of a projective module projective?
Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.

As the title suggests, I want to find the asymptotic behaviour of this sum as $x\rightarrow \infty$, I tried by summation by parts but didn’t succeed I also tried using the asymptotic behvaiour of the sum

$$\sum_{p\leq x} \frac{1}{p} \sim_{x \to \infty} \log \log x$$

i.e squaring both sides gives me:

- Is there a way to show that $\sqrt{p_{n}} < n$?
- What is the analytic continuation of the Riemann Zeta Function
- Why does zeta have infinitely many zeros in the critical strip?
- Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?
- Intervals that are free of primes
- Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

$$\sum_{p\leq x} \frac{1}{p^2} + \sum_{\substack{q,p\leq x\\p\neq q}} \frac{1}{pq} \sim_{x \to \infty} \log^2(\log x)$$

But then, how do I estimate the second term in the LHS?

Thanks in advance.

- Is 'every exponential grows faster than every polynomial?' always true?
- Series about Euler-Maclaurin formula
- Strange Recurrence: What is it asymptotic to?
- What is the relationship between GRH and Goldbach Conjecture?
- Solve $\epsilon x^3-x+1=0$
- Divisor function asymptotics
- Zeros of the second derivative of the modular $j$-function
- A $\frac{1}{3}$ Conjecture?
- Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
- Find the leading order uniform approximation when the conditions are not $0<x<1$

The prime zeta function $P(s)$, for $\text{Real}(s) > 1$, is defined as

$$P(s) = \sum_{\overset{p=1}{p \text{ is prime}}}^{\infty} \dfrac1{p^s}$$

The sum converges for $\text{Real}(s) > 1$, similar to the $\zeta$-function. Your sum is $P(2)$ and is approximately $0.4522474200410654985065\ldots$.

There are no “nice” values for $P(s)$ where $s \in \mathbb{Z}^+ \backslash \{1\}$. A very crude argument why there are no “nice” values for $P(s)$ is due to the fact that the function, $$g(n) = \dfrac{\mathbb{I}_{n \text{ is a prime}}}{n^s}$$ is not a “nice” arithmetic function in the usual sense i.e. it is not even multiplicative for instance.

Using an estimate of the difference between the prime counting function, $\pi(n)$ and the logarithmic integral function, $\mathrm{li}(x)$, we can estimate the tail of the series.

Let $\Delta(n)=\pi(n)-\mathrm{li}(n)$. Without assuming the Riemann Hypothesis, we have that for any $m$

$$

\Delta(n)=O\left(\frac{\raise{2pt}n}{\log(n)^m}\right)\tag{1}

$$

Summing by parts and using $(1)$ with $m=2$, we get

$$

\begin{align}

\sum_{k=n}^\infty\frac{1}{k^2}(\Delta(k)-\Delta(k-1))

&=-\frac{\Delta(n-1)}{n^2}+\sum_{k=n}^\infty\Delta(k)\left(\frac{1}{k^2}-\frac{1}{(k+1)^2}\right)\\

&=O\left(\frac{1}{n\log(n)^2}\right)\tag{2}

\end{align}

$$

Using $(2)$, we get

$$

\begin{align}

\sum_{p\ge n}\frac{1}{p^2}

&=\sum_{k=n}^\infty\frac{1}{k^2}(\pi(k)-\pi(k-1))\\

&=\sum_{k=n}^\infty\frac{1}{k^2}(\mathrm{li}(k)-\mathrm{li}(k-1))+O\left(\frac{1}{n\log(n)^2}\right)\\

&=\int_n^\infty\frac{\mathrm{d}x}{x^2\log(x)}+O\left(\frac{1}{n\log(n)^2}\right)\\

&=\int_n^\infty\frac{\log(x)+1}{x^2\log(x)^2}\mathrm{d}x+O\left(\frac{1}{n\log(n)^2}\right)\\

&=\frac{1}{n\log(n)}+O\left(\frac{1}{n\log(n)^2}\right)\tag{3}

\end{align}

$$

- What is the sum of $\sum\limits_{i=1}^{n}ip^i$?
- Irreducible elements in $\mathbb{Z}$ and is it a Euclidean domain?
- Permutations to satisfy a challenging restriction
- Proof that $\frac{(2n)!}{2^n}$ is integer
- A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?
- Are there books introducing to Complex Analysis for people with algebraic background?
- Mean value theorem for the second derivative, when the first derivative is zero at endpoints
- Spectrum of infinite product of rings
- What are or where can I find style guidelines for writing math?
- units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic
- If $H$ is a subgroup of $\mathbb Q$ then $\mathbb Q/H$ is infinite
- Show that $a^3+b^5=7^{7^{7^7}}$ has no solutions with $a,b\in \mathbb Z.$
- How to prove that if the one sided limits are equal the general limit is that value?
- showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
- What is the significance of the graph isomorphism problem?