EFFICIENT ESTIMATE OF THE REMAINDER FOR THE DIRICHLET FUNCTION eta(p) FOR p is an element of R+

摘要

For any k, n, q is an element of N, where n >= 2k + 1 >= 3, and p is an element of R+ the nth remainder r(n, p) of the Dirichlet eta function eta(p), known also as the alternating Riemann zeta function, eta(p) :- Sigma(infinity)(i=1)(-1)(i+1) 1/i(P), is given in the form rho(n, p) = Delta(q)(n, p) + delta(q)(k, p), where k and q are parameters controlling the magnitude of the error term delta(q) (k, p). The function Delta(q) (n, p) consists of inverted right perpendicular q/2 inverted left perpendicular + 1 simple summands and delta(q)(k, n, p) is estimated as vertical bar delta(q)(k, p)vertical bar < 5/3 p((q-1))/pi(q-1) 1/(2k)(p+q-1), where p((q-1)) is the rising Pochhammer product.