direuler

Function: direuler
Section: number_theoretical
C-Name: direuler0
Prototype: V=GGEDG
Help: direuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr
 from p=a to p=b, limited to b terms. Expr should be a polynomial or rational
 function in p and X, and X is understood to mean p^(-s). If c is present,
 output only the first c terms.
Wrapper: (,,G)
Description:
  (gen,gen,closure,?gen):gen direuler(${3 cookie}, ${3 wrapper}, $1, $2, $4)
Doc: computes the \idx{Dirichlet series} attached to the
 \idx{Euler product} of expression \var{expr} as $p$ ranges through the primes
 from $a$
 to $b$. \var{expr} must be a polynomial or rational function in another
 variable than $p$ (say $X$) and $\var{expr}(X)$ is understood as the local
 factor $\var{expr}(p^{-s})$.

 The series is output as a vector of coefficients. If $c$ is omitted, output
 the first $b$ coefficients of the series; otherwise, output the first $c$
 coefficients. The following command computes the \teb{sigma} function,
 attached to $\zeta(s)\zeta(s-1)$:
 \bprog
 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)))
 %1 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18]

 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 5) \\ fewer terms
 %2 = [1, 3, 4, 7, 6]
 @eprog\noindent Setting $c < b$ is useless (the same effect would be
 achieved by setting $b = c)$. If $c > b$, the computed coefficients are
 ``missing'' Euler factors:
 \bprog
 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 15) \\ more terms, no longer = sigma !
 %3 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 0, 28, 0, 24, 24]
 @eprog

 \synt{direuler}{void *E, GEN (*eval)(void*,GEN), GEN a, GEN b}