Chebyshev function

One of the two functions, of a positive argument , defined as follows:

The first sum is taken over all prime numbers , and the second over all positive integer powers of prime numbers such that . The function can be expressed in terms of the Mangoldt function

It follows from the definitions of and that is equal to the product of all prime numbers , and that the quantity is equal to the least common multiple of all positive integers . The functions and are related by the identity

These functions are also closely connected with the function

which expresses the number of the prime numbers .

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 and see [a1], Chapt. 12.

References

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