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An average order of σ(n), the sum of divisors of n, is nπ 2 / 6; An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;
Fast2Sum is often used implicitly in other algorithms such as compensated summation algorithms; [1] Kahan's summation algorithm was published first in 1965, [3] and Fast2Sum was later factored out of it by Dekker in 1971 for double-double arithmetic algorithms. [4]
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function . The various studies of the behaviour of the divisor function are sometimes called divisor problems .
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula c q ( n ) = ∑ 1 ≤ a ≤ q ( a , q ) = 1 e 2 π i a q n , {\displaystyle c_{q}(n)=\sum _{1\leq a\leq q \atop (a,q)=1}e^{2\pi i{\tfrac {a}{q}}n},}
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
In number theory, the gcd-sum function, [1] also called Pillai's arithmetical function, [1] is defined for every by P ( n ) = ∑ k = 1 n gcd ( k , n ) {\displaystyle P(n)=\sum _{k=1}^{n}\gcd(k,n)} or equivalently [ 1 ]
Whenever the sum of the current element in the first array and the current element in the second array is more than T, the algorithm moves to the next element in the first array. If it is less than T, the algorithm moves to the next element in the second array. If two elements that sum to T are found, it stops. (The sub-problem for two elements ...
While not derived as a Riemann sum, taking the average of the left and right Riemann sums is the trapezoidal rule and gives a trapezoidal sum. It is one of the simplest of a very general way of approximating integrals using weighted averages. This is followed in complexity by Simpson's rule and Newton–Cotes formulas.