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Given a sequence {a_n}_(n=1)^infty, a formal power series f(s) = sum_(n=1)^(infty)(a_n)/(n^s) (1) = a_1+(a_2)/(2^s)+(a_3)/(3^s)+... (2) is called the Dirichlet generating ...

A robust nonparametric test which is an alternative to the paired t-test. This test makes the basic assumption that there is information only in the signs of the differences ...

Given two paired sets X_i and Y_i of n measured values, the paired t-test determines whether they differ from each other in a significant way under the assumptions that the ...

Let sum_(k=1)^(infty)u_k be a series with positive terms, and let rho=lim_(k->infty)u_k^(1/k). 1. If rho<1, the series converges. 2. If rho>1 or rho=infty, the series ...

If a number fails Miller's primality test for some base a, it is not a prime. If the number passes, it may be a prime. A composite number passing Miller's test is called a ...

A.k.a. the pigeonhole principle. Given n boxes and m>n objects, at least one box must contain more than one object. This statement has important applications in number theory ...

A test used to determine the statistical significance of an observation. Two main types of error can occur: 1. A type I error occurs when a false negative result is obtained ...

Ramanujan's Dirichlet L-series is defined as f(s)=sum_(n=1)^infty(tau(n))/(n^s), (1) where tau(n) is the tau function. Note that the notation F(s) is sometimes used instead ...

Given a series of positive terms u_i and a sequence of positive constants {a_i}, use Kummer's test rho^'=lim_(n->infty)(a_n(u_n)/(u_(n+1))-a_(n+1)) (1) with a_n=n, giving ...

The Dirichlet beta function is defined by the sum beta(x) = sum_(n=0)^(infty)(-1)^n(2n+1)^(-x) (1) = 2^(-x)Phi(-1,x,1/2), (2) where Phi(z,s,a) is the Lerch transcendent. The ...