Dirichlet series An extension of the area of convergence can be obtained by rearranging the original series. The series : \zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty \left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right) converges for , while : \zeta(s) =\frac{1}{s-1}\sum_{n=1}^\infty\frac{n(n+1)}{2}\left(\frac{2n+3+s}{(n+1)^{s+2}}-\frac{2n-1-s}{n^{s+2}}\right) converge even for . In this way, the area of convergence can be extended to for any negative integer . The recurrence connection is clearly visible from the expression valid for enabling further expansion by integration by parts. : \begin{aligned} \zeta(s)= & 1+\frac{1}{s-1}-\frac{s}{2 !}[\zeta(s+1)-1] \\ - & \frac{s(s+1)}{3 !}[\zeta(s+2)-1] \\ & -\frac{s(s+1)(s+2)}{3 !} \sum_{n=1}^{\infty} \int_0^1 \frac{t^3 d t}{(n+t)^{s+3}}. \end{aligned} This recurrence leads to this other series development that uses the
rising factorial and is valid for the entire complex plane
Mellin-type integrals The
Mellin transform of a function is defined as : \int_0^\infty f(x)x^s\, \frac{\mathrm{d}x}{x} in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of is greater than one, we have : \Gamma(s)\zeta(s) =\int_0^\infty\frac{x^{s-1}}{e^x-1} \,\mathrm{d}x \quad and \quad\Gamma(s)\zeta(s) =\frac1{2s}\int_0^\infty\frac{x^{s}}{\cosh(x)-1} \,\mathrm{d}x , where denotes the
gamma function. By modifying the
contour, Riemann showed that : 2\sin(\pi s)\Gamma(s)\zeta(s) =i\oint_H \frac{(-x)^{s-1}}{e^x-1}\,\mathrm{d}x for all (where denotes the
Hankel contour). We can also find expressions which relate to prime numbers and the
prime number theorem. If is the
prime-counting function, then : \ln \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,\mathrm{d}x, for values with . A similar Mellin transform involves the Riemann function , which counts prime powers with a weight of , so that : J(x) = \sum \frac{\pi\left(x^\frac{1}{n}\right)}{n}. Now : \ln \zeta(s) = s\int_0^\infty J(x)x^{-s-1}\,\mathrm{d}x. These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's
prime-counting function is easier to work with, and can be recovered from it by
Möbius inversion.
Theta functions The Riemann zeta function can be given by a Mellin transform : 2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \int_0^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t, in terms of
Jacobi's theta function : \theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}. However, this integral only converges if the real part of is greater than , but it can be regularized. This gives the following expression for the zeta function, which is well defined for all except and : : \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(it)-t^{-\frac12}\right)t^{\frac{s}{2}-1}\,\mathrm{d}t + \frac{1}{2}\int_1^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t.
Laurent series The Riemann zeta function is
meromorphic with a single
pole of order one at . It can therefore be expanded as a
Laurent series about ; the series development is then : \zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{\gamma_n}{n!}(1-s)^n. The constants here are called the
Stieltjes constants and can be defined by the
limit : \gamma_n = \lim_{m \rightarrow \infty}{\left(\left(\sum_{k = 1}^m \frac{(\ln k)^n}{k}\right) - \frac{(\ln m)^{n+1}}{n+1}\right)}. The constant term is the
Euler–Mascheroni constant.
Integral For all , , the integral relation (cf.
Abel–Plana formula) : \zeta(s) = \frac{ 1 }{s - 1} + \frac{1}{2} + 2 \int_0^{\infty} \frac{\sin(s\arctan t) }{ \left(1 + t^2 \right)^{s/2} \left(e^{2\pi t} - 1\right)\ }\ \operatorname{d}t holds true, which may be used for a numerical evaluation of the zeta function.
Hadamard product On the basis of
Weierstrass's factorization theorem,
Hadamard gave the
infinite product expansion : \zeta(s) = \frac{e^{\left(\log(2\pi)-1-\frac{\gamma}{2}\right)s}}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^\frac{s}{\rho}, where the product is over the non-trivial zeros of and the letter again denotes the
Euler–Mascheroni constant. A simpler
infinite product expansion is : \zeta(s) = \pi^\frac{s}{2} \frac{\prod_\rho \left(1 - \frac{s}{\rho} \right)}{2(s-1)\Gamma\left(1+\frac{s}{2}\right)}. This form clearly displays the simple pole at , the trivial zeros at ... due to the gamma function term in the denominator, and the non-trivial zeros at . (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form and should be combined.)
Globally convergent series A globally convergent series for the zeta function, valid for all complex numbers except for some integer , was conjectured by
Konrad Knopp in 1926 Hasse also proved the globally converging series : \zeta(s)=\frac 1{s-1}\sum_{n=0}^\infty \frac 1{n+1}\sum_{k=0}^n\binom {n}{k}\frac{(-1)^k}{(k+1)^{s-1}} in the same publication. has found that a similar, equivalent series was published by
Joseph Ser in 1926. In 1997 K. Maślanka gave another globally convergent (except ) series for the Riemann zeta function: : \zeta (s)=\frac{1}{s-1}\sum_{k=0}^\infty \biggl(\prod_{i=1}^{k} (i-\frac{s}{2})\biggl) \frac{A_{k}}{k!}= \frac{1}{s-1} \sum_{k=0}^\infty \biggl(1-\frac{s}{2}\biggl)_{k} \frac{A_{k}}{k!} where real coefficients A_k are given by: : A_k=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}(2j+1)\zeta (2j+2)=\sum_{j=0}^{k}\binom{k}{j}\frac{B_{2j+2}\pi ^{2j+2}}{\left(2\right) _{j}\left( \frac{1}{2}\right) _{j}} Here are the Bernoulli numbers and denotes the Pochhammer symbol. Note that this representation of the zeta function is essentially an interpolation with nodes, where the nodes are points , i.e. exactly those where the zeta values are precisely known, as Euler showed. An elegant and very short proof of this representation of the zeta function, based on
Carlson's theorem, was presented by Philippe Flajolet in 2006. The asymptotic behavior of the coefficients A_{k} is rather curious: for growing k values, we observe regular oscillations with a nearly exponentially decreasing amplitude and slowly decreasing frequency (roughly as k^{-2/3}). Using the saddle point method, we can show that : A_{k}\sim \frac{4\pi ^{3/2}}{\sqrt{3\kappa }}\exp \biggl( -\frac{3\kappa }{2}+\frac{\pi ^{2}}{4\kappa }\biggl) \cos \biggl( \frac{4\pi }{3}-\frac{3\sqrt{3} \kappa }{2}+\frac{\sqrt{3}\pi ^{2}}{4\kappa }\biggl) where \kappa stands for: : \kappa :=\sqrt[3]{\pi ^{2}k} (see for details). On the basis of this representation, in 2003 Luis Báez-Duarte provided a new criterion for the Riemann hypothesis. Namely, if we define the coefficients as : c_{k}:=\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}\frac{1}{\zeta (2j+2)} then the Riemann hypothesis is equivalent to : c_{k}=\mathcal{O}\left( k^{-3/4+\varepsilon }\right) \qquad (\forall\varepsilon >0)
Rapidly convergent series Peter Borwein developed an algorithm that applies
Chebyshev polynomials to the
Dirichlet eta function to produce a
very rapidly convergent series suitable for high precision numerical calculations.
Series representation at positive integers via the primorial : \zeta(k)=\frac{2^k}{2^k-1}+\sum_{r=2}^\infty\frac{(p_{r-1}\#)^k}{J_k(p_r\#)}\qquad k=2,3,\ldots. Here is the
primorial sequence and is
Jordan's totient function.
Series representation by the incomplete poly-Bernoulli numbers The function can be represented, for , by the infinite series : \zeta(s)=\sum_{n=0}^\infty B_{n,\ge2}^{(s)}\frac{(W_k(-1))^n}{n!}, where , is the th branch of the
Lambert -function, and is an incomplete poly-Bernoulli number.
Mellin transform of the Engel map The function is iterated to find the coefficients appearing in
Engel expansions. The
Mellin transform of the map g(x) is related to the Riemann zeta function by the formula : \begin{align} \int_0^1 g (x) x^{s - 1} \, dx & = \sum_{n = 1}^\infty \int_{\frac{1}{n + 1}}^{\frac{1}{n}} (x (n + 1) - 1) x^{s - 1} \, d x\\[6pt] & = \sum_{n = 1}^\infty \frac{n^{- s} (s - 1) + (n + 1)^{- s - 1} (n^2 + 2 n + 1) + n^{- s - 1} s - n^{1 - s}}{(s + 1) s (n + 1)}\\[6pt] & = \frac{\zeta (s + 1)}{s + 1} - \frac{1}{s (s + 1)} \end{align}
Stochastic representations The
Brownian motion and Riemann zeta function are connected through the
moment-generating functions of
stochastic processes derived from the
Brownian motion. == Numerical algorithms ==