Riemann zeta function and analytic continuation

In the original half-plane, partial summation expresses zeta as $s\int_1^\infty\lfloor x\rfloor x^{-s-1}\,dx$. Split the floor function into the variable and minus its fractional part. The integral of the variable evaluates explicitly, leaving a bounded numerator in the remaining improper integral. This remainder converges absolutely and locally uniformly when $\operatorname{Re}s>0$, so it defines a holomorphic function there. Agreement where the original series converges proves that this is its unique continuation.

The rational term has a simple pole at one with residue one, while the remainder is holomorphic near that point. This identifies both the location and the order of the singularity. It also illustrates the difference between the convergence domain of a representation and the analytic domain of a function: the floor-remainder representation reaches further left than the series, but does not yet reach the entire complex plane. A new identity will extend it further without redefining any previously determined value.

The gamma function and a power of pi supply the factors needed for reflection across the central line. Define the completed expression $\Lambda_\zeta(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$; the subscript distinguishes it from the arithmetic von Mangoldt function. It satisfies $\Lambda_\zeta(s)=\Lambda_\zeta(1-s)$ as a meromorphic identity and has simple poles at zero and one. Multiplying by the polynomial that cancels those poles yields an entire function with the same reflection symmetry.

One route to this equation takes a Mellin transform of a Gaussian theta sum and uses its transformation under reciprocal scaling. The Gaussian decay permits justified integration in an initial half-plane; splitting the integral and using the transformation reveals the reflected argument. This sketches the mechanism, rather than deriving Poisson summation here. An equivalent form is $\zeta(s)=2^s\pi^{s-1}\sin(\pi s/2)\Gamma(1-s)\zeta(1-s)$, interpreted meromorphically. At apparent singularities one takes limits, rather than multiplying an undefined infinity by zero.

At each negative even integer the sine factor in the second functional-equation form vanishes while the remaining factors are finite and nonzero. These are the trivial zeros. At negative odd integers, the reflected zeta value lies in its absolutely convergent domain, so the equation provides a concrete continued value using ordinary gamma values. Such computations depend on an analytic identity whose proof starts in a justified domain, not on an algebraic rearrangement of divergent sums of positive integers.

Conjugation gives $\zeta(\overline{s})=\overline{\zeta(s)}$, since this identity holds in the series domain and continues uniquely. Combined with reflection for the completed function, it organizes the nontrivial zeros in the strip between zero and one into symmetric configurations, counted with multiplicity. The statement that these zeros all lie on the central line is the Riemann hypothesis; results conditional on that statement must be labeled conditional. Neither analytic continuation itself nor the functional equation forces every zero to occupy that line.