Proof that lim ( ϑ ( x ) − x ) ≠ 0 ? ?

Question

Proof that   lim  (  ϑ  (  x  )  −  x  )  ≠  0  ? ?

Farbwolkenw · Accepted Answer

lim          x      →      ∞        (         ∑          p      ≤      x               log  ⁡  p  −  x  ) does not exist. We can prove it by contradiction.If       lim          x      →      ∞        (         ∑          p      ≤      x               log  ⁡  p  −  x  ) exists, then                    0        =                  lim                      x            →            ∞                          (                           ∑                      p            ≤            (            x            +            1            )                          log        ⁡        p        −        (        x        +        1        )        )        −                  lim                      x            →            ∞                          (                           ∑                      p            ≤            x                          log        ⁡        p        −        x        )        =                  lim                      x            →            ∞                                             ∑                      x            &amp;lt;            p            ≤            (            x            +            1            )                          log        ⁡        p        −        1            It means that       lim          x      →      ∞               ∑          x      &amp;lt;      p      ≤      (      x      +      1      )        log  ⁡  p  =  1If there is a prime in   (  x  ,  x  +  1  ], then       ∑          x      &amp;lt;      p      ≤      (      x      +      1      )        log  ⁡  p  &amp;gt;  log  ⁡  x, if there is no prime in   (  x  ,  x  +  1  ], then       ∑          x      &amp;lt;      p      ≤      (      x      +      1      )        log  ⁡  p  =  0, so       lim          x      →      ∞               ∑          x      &amp;lt;      p      ≤      (      x      +      1      )        log  ⁡  p cannot exist, it contradicts with the previous, so       lim          x      →      ∞        (         ∑          p      ≤      x               log  ⁡  p  −  x  ) cannot exist.

Proof that lim(v(x)-x) ne 0 ?

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