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Specific choices of give different types of Riemann sums: . If = for all i, the method is the left rule [2] [3] and gives a left Riemann sum.; If = for all i, the method is the right rule [2] [3] and gives a right Riemann sum.
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.
The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite ...
The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. 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.
The sum over prime powers then gets extra factors of χ(p m), and the terms Φ(1) and Φ(0) disappear because the L-series has no poles. More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke L-series. The sum over primes then gets replaced by a sum over ...
One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. In a left-hand Riemann sum, t i = x i for all i, and in a right-hand Riemann sum, t i = x i + 1 for all i. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.
assuming that both terms on the right-hand side exist and are finite. Abel's summation formula can be generalized to the case where ϕ {\displaystyle \phi } is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral :
Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably.