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The problem of finding the connection between a continuous function f on the boundary partialR of a region R with a harmonic function taking on the value f on partialR. In ...

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.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 ...

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 ...

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 ...

The Dirichlet eta function is the function eta(s) defined by eta(s) = sum_(k=1)^(infty)((-1)^(k-1))/(k^s) (1) = (1-2^(1-s))zeta(s), (2) where zeta(s) is the Riemann zeta ...

Let the divisor function d(n) be the number of divisors of n (including n itself). For a prime p, d(p)=2. In general, sum_(k=1)^nd(k)=nlnn+(2gamma-1)n+O(n^theta), where gamma ...

If g is continuous and mu,nu>0, then int_0^t(t-xi)^(mu-1)dxiint_0^xi(xi-x)^(nu-1)g(xi,x)dx =int_0^tdxint_x^t(t-xi)^(mu-1)(xi-x)^(nu-1)g(xi,x)dxi.

Let |sum_(n=1)^pa_n|<K, (1) where K is independent of p. Then if f_n>=f_(n+1)>0 and lim_(n->infty)f_n=0, (2) it follows that sum_(n=1)^inftya_nf_n (3) converges.

If, in an interval of x, sum_(r=1)^(n)a_r(x) is uniformly bounded with respect to n and x, and {v_r} is a sequence of positive non-increasing quantities tending to zero, then ...