A Disciplined Reconstruction of the Explicit Formula for ψ0(x)

This paper presents a complete derivation of the classical explicit formula for the Perron-averaged Chebyshev function \psi_0(x):

\psi_0(x) = x - \lim_{T\to\infty} \sum_{|\operatorname{Im}\rho|<T} \frac{m_\rho x^\rho}{\rho} - \log(2\pi) - \frac12\log(1-x^{-2}),

where \rho ranges over the nontrivial zeros of the Riemann zeta function, counted with multiplicity, and the zero sum is interpreted as a symmetric limit.

The derivation proceeds from the Euler product for \zeta(s), the logarithmic derivative -\zeta'(s)/\zeta(s), Perron inversion, contour shifting, and explicit residue computation. Particular care is given to the Perron midpoint convention at prime powers, the contribution of the pole at s=1, the nontrivial zeros, the regularity of -\zeta'/\zeta at s=0, the trivial zeros at negative even integers, and the conditional nature of the zero sum.

The mathematical significance of the work lies in reconstructing one of the central bridges in analytic number theory: the direct relationship between prime-power counting and the analytic structure of the zeta function. The formula shows explicitly how the main term x, the oscillatory contribution from nontrivial zeros, the constant -\log(2\pi), and the trivial-zero correction together determine \psi_0(x). This relationship is foundational to understanding why the Riemann Hypothesis is equivalent to sharp control of the error term in prime-counting functions.

Although the explicit formula is classical, this paper emphasizes a fully transparent derivation, careful treatment of signs and residues, and explicit separation between derived steps and cited analytic estimates. It is intended as a rigorous expository resource for readers studying the connection between the zeta function, prime distribution, Perron inversion, and contour methods.

Why this is significant mathematically

The formula is significant because it makes precise the idea that the distribution of primes is encoded in the zeros and singularities of \zeta(s). The pole at s=1 gives the leading term x, the nontrivial zeros produce oscillations in prime counting, the point s=0 gives the constant term, and the trivial zeros give the logarithmic correction. This is one of the core mechanisms behind the analytic theory of primes and the connection between the Riemann Hypothesis and error bounds for \psi(x)-x.

Keywords

• Riemann zeta function, explicit formula, Chebyshev function, Perron inversion, analytic number theory, Riemann Hypothesis, von Mangoldt function, zeta zeros, prime number theorem, contour integration