In the explicit formula for $\psi_0(x)$ used in the PNT proof : $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$

In particular the part $$f(x)=\sum_{\rho} \frac{x^{\rho}}{\rho}$$ , I am confused about some things.

I thought about truncated sums and conditional sums but I was still confused.

My problems are these :

1) it seems $f(x)$ converges using the formula here :


although it is not trivial (to show) imho.

However on the wiki page


the error term is given as

Here the sum over zeroes should again be taken in increasing order of imaginary part

$\sum_\rho\dfrac{x^\rho}{\rho} = \lim_{T \rightarrow \infty} S(x,T) \ $


$S(x,T) = \sum_{\rho:|\Im \rho| \le T} \dfrac{x^\rho}{\rho} \ .$

The error involved in truncating the sum to $S(x,T)$ is of order

$ x^2 \dfrac{\log^2 T}{T} + \log x.$

But this formula seems to large ? For small $T$ we get about $x^2$ ? That does not make sense does it ?

2) It seems the function is differentiable ??? In the error formula above we see that if $T$ gets large we still have an error of about $C log(x)$. SO (?) the formula is considered as an asymtotic with error of log type and the differentiable function is then just the approximation to the true nondifferentiable function ?

3) Although the function is differentiable apparantly , I again wonder if the first derivative actually converges ??

4) If the RH is true then $$\psi_0(x) = x + O(ln(x)^2 \sqrt x)$$ But I do not see how that follows from the above. In particular the formula for the error involving $T$ does not seem to fit ??

5) Analogue (generalized) question to 1) 2) 3) ; how about twice differentiable ? And does the second derivative converge ? How about the $n$-th derivative ?

6) $\psi_0(x)$ should be discontinu at $x$ if $x$ is a prime. I do not see that from the formula , I assume it relates to the previous questions about differentiability and the error term. In fact I think this question is answered if the previous ones are.

7) Does this formula not give the possibility to prove the second hardy-littlewood conjecture when assuming RH and Cramer's prime gap conjecture ?

  • $\begingroup$ What proof are you reading? Do you have some link/reference to it? $\endgroup$ – DonAntonio Sep 18 '13 at 22:11
  • $\begingroup$ @DonAntonio I do not have a link/reference. And Im not reading a book. I only got the sketchy proof from wiki. Most other parts of the PNT proof are very clear to me. $\endgroup$ – mick Sep 18 '13 at 22:58

1) This is an EXACT formula, so be careful with truncating infinite sums! It does not make sense to study it for "small" $T$ values. Keep in mind that Riemann proved it working basically with Fourier analysis (read the first chapter of the Edward's book), so what you want to do is something similar to stop a Fourier series of some stepwise function to the first terms.

2) The correct error term (see Davenport) is $$ (\log x) \min\left\{1,\frac{x}{T<x>} \right\}\;, $$ so when $T\rightarrow \infty$...

3) Restrict the function $\psi_0(x)$ to natural values, it jumps whenever $x$ is a prime power, so...

4) That's the PNT! Read Chapter 18 of Davenport.

5) As in 3)...

6) Exactly!

7) Don't understand your point here...

  • $\begingroup$ +1 for the correct error term. Although I do not know for certain what $<x>$ is meant to be. Nearest integer to $x$ ?? $\endgroup$ – mick Oct 7 '13 at 18:25
  • $\begingroup$ You need to adress my comment , otherwise I cannot understand and accept. $\endgroup$ – mick Dec 8 '13 at 22:59
  • 1
    $\begingroup$ I wrote the reference to understand my comment (Davenport, page 107). $<x>$ is the minimum distance from $x$ to the nearest prime power $p^k$. $\endgroup$ – PITTALUGA Dec 10 '13 at 9:16

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