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We present a common ground for infinite sums, unordered sums, Riemann integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
The Riemann hypothesis is equivalent to the $varpi$-form of the prime number theorem as $varpi(x) =O(xsp{1/2} logsp{2} x)$, where $varpi(x) =sumsb{nle x} bigl(Lambda(n) -1big)$ with the sum running through the set of all natural integers. Let ${maths
If we cannot obtain all terms of a series, or if we cannot sum up a series, we have to turn to the partial sum approximation which approximate a function by the first several terms of the series. However, the partial sum approximation often does not
Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_
Let $sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$ sum_{d|n}frac{1+2,(-1)^{d}}{d}=sum_{r=1}^{n}frac{(-1)^{r}}{r}, bi
This paper proves that there does not exist a polynomial-time algorithm to the the subset sum problem. As this problem is in NP, the result implies that the class P of problems admitting polynomial-time algorithms does not equal the class NP of probl