Split the area into vertical slices then use the Riemann sum formula: A = sum f(x)*Delta x, where A is the area under the curve, f(x) is the height of each rectangle (or the average of the two heights for a trapezoid), and Delta x is the width of each rectangle or trapezoid....
A definite integral is the limit of a Riemann sum. A definite integral represents the area under a curve given by some function, within an interval that is traditionally denoted by [a,b]. More formally, for a function f that is defined over an interval [a,b], the definite integral of...
Riemann Zeta FunctionAnalytic Number TheoryAn asymptotic formula is obtained for the sum of terms σ it ( n )σ - it ( N - n ) ( t is real) over 0 0. As a consequence, Porter's result on a power scale for the average number of steps in the Euclidean algorithm is improved....
We prove the Kloosterman-Spectral sum formula for PSL(2,Z[i])\PSL(2,C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. This sum formula allows the extension of the spectral theory of Kloosterman...
Riemann Sum: It is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation. Let {eq}f (x) {/eq} be a function that is continuous on interval {eq}a < x ...
Dedekind-Vasyunin-cotangent sumfractional part functionRiemann hypothesisRecently, several works are done on the generalized Dedekind￢asyunin sum where and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum which is a ...
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Let's scale X to Y={X-\lambda\over\sqrt\lambda} so that \mathbb{E}[Y]=0,\mathrm{Var}(Y)=1, we want to prove that the distribution of Y converges to normal distribution N(0,1) when \lambda\to\infty. For x\in\mathbb{N}, let y={x-\lambda\over\sqrt\lambda}, then \begin...
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Euler, L. "De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum dissertatio altera." Miscellanea Berolinensia 7, 172-192, 1743.Euler, L. Introductio in Analysin...
is calculated by means of this formula, the relative error is less thanewln–1 and thus approaches 0 asnincreases without bound. Whenn= 10, for example, the formula yieldsn!= 3,598,700, whereas the exact value of 10! is 3,628,800. In this case, the relative error is less than 1...