Explicit formula

An identity connecting zeros of a zeta or L-function with weighted sums over primes or prime powers. It translates information about prime distribution into information about the function’s zeros, and conversely.

Riemann zeta function: The meromorphic function defined initially by the sum of reciprocal powers of positive integers, with a meromorphic continuation to the complex plane. Its zeros and prime-power terms form the classic setting for the explicit formula.
Prime number theorem: The theorem that the number of primes up to x is asymptotic to x divided by the natural logarithm of x. The explicit formula refines its average estimate by exposing oscillations from zeta zeros.
Riemann explicit formula: An identity relating a weighted count of prime powers to a sum over zeros of the Riemann zeta function. This is the classical instance connecting prime distribution with zeta zeros.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. Its truth would constrain the zero terms and sharpen error estimates for prime counts.
Zeros of the Riemann zeta function: The complex numbers at which the Riemann zeta function vanishes. The nontrivial zeros contribute oscillatory terms that encode fluctuations in prime counts.
Chebyshev function: The function that sums logarithmic weights over prime powers up to a given bound. Its weighted prime-power count is a standard arithmetic quantity in explicit formulas.
Weil explicit formula: A family of identities relating zeros of zeta and L-functions to sums over primes or prime ideals. It extends the classical relation through test functions and broader arithmetic settings.
Prime-counting function: The function counting primes no greater than a given real number. Explicit formulas describe its deviations from smooth approximations through zero contributions.
Von Mangoldt function: The arithmetic function equal to the logarithm of a prime on prime powers and zero elsewhere. It weights prime powers in the formula’s arithmetic sum.
Dirichlet series: A series whose terms are coefficients multiplied by complex powers of positive integers. The zeta function and its logarithmic derivative encode arithmetic data through Dirichlet series.

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