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Approximation of $\pi(x)$ by $\Psi(x)$  
 
  Authors: Mehdi Hassani,  
  Keywords: Primes, Harmonic series, Gamma function, Digamma function.  
  Date Received: 07/03/05  
  Date Accepted: 25/08/05  
  Subject Codes:

11A41, 26D15, 33B15.

  
  Editors: Jozsef Sandor,  
 
  Abstract:

In this paper we find some lower and upper bounds of the form $ frac{n}{H_n-c}$ for the function $ pi(n)$, in which $ H_n=sum_{k=1}^nfrac{1}{k}$. Then, we consider $ H(x)=Psi(x+1)+gamma$ as generalization of $ H_n$, such that $ Psi(x)=frac{d}{dx}logGamma(x)$ and $ gamma$ is Euler constant; this extension has been introduced for the first time by J. Sándor and it helps us to find some lower and upper bounds of the form $ frac{x}{Psi(x)-c} $ for the function $ pi(x)$ and using these bounds, we show that $ Psi(p_n)simlog n$, when $ nrightarrowinfty$ is equivalent with the Prime Number Theorem. ;


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