View source for Chebyshev function

One of the two functions, of a positive argument $x$, defined as follows:
$$
\theta(x) = \sum_{p \le x} \log p\,,\ \ \ \psi(x) = \sum_{p^m \le x} \log p \ .
$$
The first sum is taken over all prime numbers $p \le x$, and the second over all positive integer powers $m$ of prime numbers $p$ such that $p^m \le x$. The function $\psi(x)$ can be expressed in terms of the [[Mangoldt function|Mangoldt function]]
$$
\psi(x) = \sum_{n \le x} \Lambda(n) \ .
$$
It follows from the definitions of $\theta(x)$ and $\psi(x)$ that $e^{\theta(x)}$ is equal to the product of all prime numbers $p \le x$, and that the quantity $e^{\psi(x)}$ is equal to the least common multiple of all positive integers $n \le x$. The functions $\theta(x)$ and $\psi(x)$ are related by the identity
$$
\psi(x) = \theta(x) + \theta(x^{1/2}) + \theta(x^{1/3}) + \cdots \ .
$$

These functions are also closely connected with the function
$$
\pi(x) = \sum_{p \le x} 1
$$

which expresses the number of the prime numbers $p \le x$.  The [[prime number theorem]] may be expressed in the form $\psi(x) \sim 1$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,   "Mémoire sur les nombres premiers"  ''J. Math. Pures Appl.'' , '''17'''  (1852)  pp. 366–390  (Oeuvres, Vol. 1, pp. 51–70)</TD></TR>
</table>



====Comments====
For properties of the Chebyshev functions $\theta(x)$ and $\psi(x)$ see [[#References|[a1]]], Chapt. 12.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Ivic,   "The Riemann zeta-function" , Wiley  (1985)</TD></TR>
</table>

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[[Category:Number theory]]