The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.

Ultimately, I am interested in analytically continuing the function $$ \eta _a(s):=\sum _{n=1}^\infty \frac{1}{(n^2+a^2)^s}, $$ where $a$ is a non-negative real number, and calculating $\eta _a$ and its derivatives (at least the first derivative) at the origin: $\eta _a(0),\eta _a'(0),\ldots $.

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ , is a mathematical function of a complex variable defined as () = = = + + + for ⁡ >, and its analytic continuation elsewhere.

$\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}$ (the standard Zeta function) provided the sum converges. $\zeta(0)=-1/2$ Thus, $1+1+1+...=-1/2$ ? How can this possibly be true? I guess I'm under the impression that $\sum 1$ diverges.

Zeros of the Riemann zeta function zeta(s) come in two different types. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, ..., and "nontrivial zeros" occur at certain values of t satisfying s=sigma+it (1) for s in the "critical strip" 0<sigma<1.

Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music…

The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem.

Nov 29, 2014 · This formula predicts correctly that ζ(0) = −1 2, and allows to calculate ζ(2k) as a rational multiple of π2k as well (in terms of Bernoulli numbers). Is there some simple explanation why the above prediction ζ(0) = −1 2 is valid? Perhaps there is a not so simple but still transparent explanation via Eisenstein series. Added.

Feb 9, 2018 · Recall that one of the for the Riemann zeta function in the critical strip is given by ζ ⁢ ( s ) = 1 s - 1 + 1 - s ⁢ ∫ 1 ∞ x - [ x ] x s + 1 ⁢ 𝑑 x , where [ x ] denotes the integer part of x .

May 9, 2018 · $$ \zeta'(0)=-1-\int_1^\infty{\overline B_1(x)\over x}\mathrm dx $$ To determine the value of the latter integral, consider applying Euler-Maclaurin formula to factorials: $$ \log N!=\left(N+\frac12\right)\log N-N+1+\int_1^N{\overline B_1(x)\over x}\mathrm dx $$