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 $$ \psi(x) = \sum_{n \le x} \Lambda(n) \ . $$ 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

[1]	P.L. Chebyshev, "Mémoire sur les nombres premiers" J. Math. Pures Appl. , 17 (1852) pp. 366–390 (Oeuvres, Vol. 1, pp. 51–70)

Comments

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

References

[a1]	A. Ivic, "The Riemann zeta-function" , Wiley (1985)