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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.
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Defining the two Wirtinger derivatives as = (), ¯ = (+), the Cauchy–Riemann equations can then be written as a single equation ¯ =, and the complex derivative of in that case is =. In this form, the Cauchy–Riemann equations can be interpreted as the statement that a complex function f {\textstyle f} of a complex variable z {\textstyle z ...
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 ...
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.
The above equation no longer applies for these extended values of , for which the corresponding summation would diverge. For example, the full zeta function exists at s = − 1 {\displaystyle s=-1} (and is therefore finite there), but the corresponding series would be 1 + 2 + 3 + … , {\textstyle 1+2+3+\ldots \,,} whose partial sums would grow ...
The nth partial sum is given by a simple formula: ... and its sum is the Riemann zeta ... applying the Ramanujan sums of known series to find the sums of related ...
Abel's summation formula can be generalized to the case where is only assumed to be continuous if the integral is interpreted as a Riemann–Stieltjes integral: ∑ x < n ≤ y a n ϕ ( n ) = A ( y ) ϕ ( y ) − A ( x ) ϕ ( x ) − ∫ x y A ( u ) d ϕ ( u ) . {\displaystyle \sum _{x<n\leq y}a_{n}\phi (n)=A(y)\phi (y)-A(x)\phi (x)-\int _{x ...