Search Results for ""

In his monumental treatise Disquisitiones Arithmeticae, Gauss conjectured that the class number h(-d) of an imaginary quadratic field with binary quadratic form discriminant ...

Hadjicostas's formula is a generalization of the unit square double integral gamma=int_0^1int_0^1(x-1)/((1-xy)ln(xy))dxdy (1) (Sondow 2003, 2005; Borwein et al. 2004, p. 49), ...

There are a number of formulas variously known as Hurwitz's formula. The first is zeta(1-s,a)=(Gamma(s))/((2pi)^s)[e^(-piis/2)F(a,s)+e^(piis/2)F(-a,s)], where zeta(z,a) is a ...

Legendre's constant is the number 1.08366 in Legendre's guess at the prime number theorem pi(n)=n/(lnn-A(n)) with lim_(n->infty)A(n) approx 1.08366. Legendre first published ...

Consider the sequence of partial sums defined by s_n=sum_(k=1)^n(-1)^kk^(1/k). (1) As can be seen in the plot above, the sequence has two limit points at -0.812140... and ...

Define q=e^(2piitau) (cf. the usual nome), where tau is in the upper half-plane. Then the modular discriminant is defined by ...

The transform inverting the sequence g(n)=sum_(d|n)f(d) (1) into f(n)=sum_(d|n)mu(d)g(n/d), (2) where the sums are over all possible integers d that divide n and mu(d) is the ...

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes ...

Voronin (1975) proved the remarkable analytical property of the Riemann zeta function zeta(s) that, roughly speaking, any nonvanishing analytic function can be approximated ...

The first of the Hardy-Littlewood conjectures. The k-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, ...