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A function f is said to be an entire modular form of weight k if it satisfies 1. f is analytic in the upper half-plane H, 2. f((atau+b)/(ctau+d))=(ctau+d)^kf(tau) whenever [a ...

An Artin L-function over the rationals Q encodes in a generating function information about how an irreducible monic polynomial over Z factors when reduced modulo each prime. ...

If the total group of the canonical series is divided into two parts, the difference between the number of points in each part and the double of the dimension of the complete ...

product_(k=1)^(n)(1+yq^k) = sum_(m=0)^(n)y^mq^(m(m+1)/2)[n; m]_q (1) = sum_(m=0)^(n)y^mq^(m(m+1)/2)((q)_n)/((q)_m(q)_(n-m)), (2) where [n; m]_q is a q-binomial coefficient.

If (1-z)^(a+b-c)_2F_1(2a,2b;2c;z)=sum_(n=0)^inftya_nz^n, then where (a)_n is a Pochhammer symbol and _2F_1(a,b;c;z) is a hypergeometric function.

The Dedekind eta function is defined over the upper half-plane H={tau:I[tau]>0} by eta(tau) = q^_^(1/24)(q^_)_infty (1) = q^_^(1/24)product_(k=1)^(infty)(1-q^_^k) (2) = ...

Let s_b(n) be the sum of the base-b digits of n, and epsilon(n)=(-1)^(s_2(n)) the Thue-Morse sequence, then product_(n=0)^infty((2n+1)/(2n+2))^(epsilon(n))=1/2sqrt(2).

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 ...

A special case of the Artin L-function for the polynomial x^2+1. It is given by L(s)=product_(p odd prime)1/(1-chi^-(p)p^(-s)), (1) where chi^-(p) = {1 for p=1 (mod 4); -1 ...

Let f(theta) be Lebesgue integrable and let f(r,theta)=1/(2pi)int_(-pi)^pif(t)(1-r^2)/(1-2rcos(t-theta)+r^2)dt (1) be the corresponding Poisson integral. Then almost ...