254A, Supplement 3: The Gamma function and the functional equation (optional)

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In Notes 2, the Riemann zeta function $latex {zeta}&fg=000000$ (and more generally, the Dirichlet $latex {L}&fg=000000$-functions $latex {L(cdot,chi)}&fg=000000$) were extended meromorphically into the region $latex {{ s: hbox{Re}(s) > 0 }}&fg=000000$ in and to the right of the critical strip. This is a sufficient amount of meromorphic continuation for many applications in analytic number theory, such as establishing the prime number theorem and its variants. The zeroes of the zeta function in the critical strip $latex {{ s: 0 < hbox{Re}(s) < 1 }}&fg=000000$ are known as the non-trivial zeroes of $latex {zeta}&fg=000000$, and thanks to the truncated explicit formulae developed in Notes 2, they control the asymptotic distribution of the primes (up to small errors).

The $latex {zeta}&fg=000000$ function obeys the trivial functional equation

$latex displaystyle zeta(overline{s}) = overline{zeta(s)} (1)&fg=000000$

for all $latex {s}&fg=000000$ in its domain of definition. Indeed, as $latex {zeta(s)}&fg=000000$ is real-valued when $latex…

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