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The identity $\zeta (s)=\eta (s)/(1-2^{1-s})$ continues the Riemann zeta function meromorphically to $\Re s>0$. Apparent singularities at nonreal zeros of $1-2^{1-s}$ are removable, as one sees by replacing $2$ with an integer $k$ for which $1-k^{1-s}\ne 0$. At $s=1$, $\eta (1)=\log 2$ and $1-2^{1-s}\sim (s-1)\log 2$, so $\zeta$ has a simple pole of residue one.

For every integer $n\ge 1$, the Riemann zeta function satisfies $\zeta (1-n)=(-1{)}^{n-1}{B}_n/n$. One proof applies meromorphic continuation of a Mellin transform from an asymptotic expansion to $(e^y-1{)}^{-1}$, whose transform is the Bose integral $\Gamma (s)\zeta (s)$, and divides residues by the residues of the Gamma function. The formula includes $\zeta (0)=-1/2$.

The critical strip for the Riemann zeta function is $0<\Re s<1$.

The Riemann zeta function satisfies ${\int }_0^T|\zeta (1/2+it){|}^{4}dt\ll T(\log T{)}^4$ for $T\ge 2$. Squaring a short Dirichlet polynomial produces the divisor function; the divisor-square summatory bound controls the off-diagonal and diagonal terms.

Abel summation applied to the counting function $\lfloor w\rfloor$ gives the displayed identity for $\Re s>1$ and every $x>0$. Since $0\le \{w\}<1$, the integral is locally uniformly convergent and holomorphic for $\Re s>0$. This supplies a meromorphic continuation of the Riemann zeta function to that half-plane, with its sole pole at one and residue one. Keeping the fractional endpoint term makes the formula valid at noninteger cutoffs.

For $r=n+1/2$, $n\ge 0$, the convergent frequency zeta function, expressed through the Riemann zeta function, is ${\sum }_r{r}^{-s}=(2^s-1){\zeta }_R(s)$ when $\mathrm{Re}s>1$. Its analytic continuation at $s=-1$ gives $(1/2-1)(-1/12)=1/24$. A common exponential frequency cutoff gives ${\sum }_{r}r{e}^{-\epsilon r}={\epsilon }^{-2}+1/24+O({\epsilon }^2)$, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum $-1/12$.

For $s=\sigma +it$, $\sigma >0$, and $x\ge |t|/\pi$, the displayed approximation holds away from the pole. Apply the Van der Corput sum-integral lemma to $-t\log w/(2\pi )$ on $w\ge x$, where its derivative has modulus at most one half. Abel summation with $w^{-\sigma }$ converts the bounded unweighted discrepancy to $O(x^{-\sigma })$. Locally uniform convergence of this weighted discrepancy extends the identity from $\sigma >1$ to $\sigma >0$. It turns estimates for finite exponential sums into estimates for the Riemann zeta function.

The inducing primitive Dirichlet character has conductor of a Dirichlet character $q_0$. Increasing the modulus removes Euler factors only at new prime divisors. The correction creates zeros on the imaginary axis; raising the exponents of primes already dividing the conductor of a Dirichlet character changes no Euler factor. Primitive gamma factors continue to use $q_0$. For real-even characters this includes principal primitive conductor of a Dirichlet character one, with its Riemann zeta function pole treated separately.

The logarithmic derivative of a nonzero differentiable function $f$ is $f^{\prime }/f$. For $\Re s>1$, the Euler product for the Riemann zeta function gives

A nontrivial zero of the Riemann zeta function lies in the critical strip $0<\Re s<1$. The functional equation of the Riemann zeta function reflects such zeros across the critical line $\Re s=1/2$.