two sides at the discontinuities of That added detail is used because the function may then be defined by an inverse Mellin transform. Formally, we may define by where the variable in each sum ranges over all primes within the specified limits. We may also write where is the von Mangoldt function and The Möbius inversion formula then gives where is the Möbius function. Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function , and using the Perron formula we have Chebyshev's function The Chebyshev function weights primes or prime powers by : For , and Formulas for prime-counting functions Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulas. We have the following expression for ψ: where Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula. For we have a more complicated formula Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The integral is equal to the series over the trivial zeros: The first term li(x) is the usual logarithmic integral function; the expression li(xρ) in the second term should be considered as Ei(ρ log x), where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals. Thus, Möbius inversion formula gives us valid for x > 1, where is Riemann's R-function and is the Möbius function. The latter series for it is known as Gram series. Because for all , this series converges for all positive x by comparison with the series for . The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as and not .