bolton8l

2022-10-08

Proof that $lim(\vartheta (x)-x)\ne 0?$ ?

Farbwolkenw

Beginner2022-10-09Added 6 answers

$\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x\text{}}\mathrm{log}p-x)$ does not exist. We can prove it by contradiction.

If $\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x\text{}}\mathrm{log}p-x)$ exists, then

$\begin{array}{r}0=\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le (x+1)}\mathrm{log}p-(x+1))-\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x}\mathrm{log}p-x)=\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p-1\end{array}$

It means that $\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p=1$

If there is a prime in $(x,x+1]$, then $\sum _{x<p\le (x+1)}\mathrm{log}p>\mathrm{log}x$, if there is no prime in $(x,x+1]$, then $\sum _{x<p\le (x+1)}\mathrm{log}p=0$, so $\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p$ cannot exist, it contradicts with the previous, so $\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x\text{}}\mathrm{log}p-x)$ cannot exist.

If $\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x\text{}}\mathrm{log}p-x)$ exists, then

$\begin{array}{r}0=\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le (x+1)}\mathrm{log}p-(x+1))-\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x}\mathrm{log}p-x)=\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p-1\end{array}$

It means that $\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p=1$

If there is a prime in $(x,x+1]$, then $\sum _{x<p\le (x+1)}\mathrm{log}p>\mathrm{log}x$, if there is no prime in $(x,x+1]$, then $\sum _{x<p\le (x+1)}\mathrm{log}p=0$, so $\underset{x\to \mathrm{\infty}}{lim}\text{}\sum _{xp\le (x+1)}\mathrm{log}p$ cannot exist, it contradicts with the previous, so $\underset{x\to \mathrm{\infty}}{lim}(\text{}\sum _{p\le x\text{}}\mathrm{log}p-x)$ cannot exist.

What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders?

How do I find the y-intercept of a parabola?

What are the vertices of $9{x}^{2}+16{y}^{2}=144$?

How to determine the rate of change of a function?

Why are the tangents for 90 and 270 degrees undefined?

How to find the center and radius of the circle ${x}^{2}+{y}^{2}-6x+8y=0$?

What is multiplicative inverse of a number?

How to find the continuity of a function on a closed interval?

How do I find the tangent line of a function?

How to find vertical asymptotes using limits?

How to find the center and radius of the circle ${x}^{2}-12x+{y}^{2}+4y+15=0$?

Let f be a function so that (below). Which must be true?

I. f is continuous at x=2

II. f is differentiable at x=2

III. The derivative of f is continuous at x=2

(A) I (B) II (C) I and II (D) I and III (E) II and IIIHow to find the center and radius of the circle given ${x}^{2}+{y}^{2}+8x-6y=0$?

How to find the center and radius of the circle ${x}^{2}+{y}^{2}+4x-8y+4=0$?

How to identify the center and radius of the circle ${(x+3)}^{2}+{(y-8)}^{2}=16$?