THE ZETA-FUNCTION CONSTRUCTED FROM THE ZEROS OF THE BESSEL-FUNCTION

This paper studies the zeta function zeta(alpha)(s) = Sigma(n)(j(alpha n)/pi)(-s) built from the zeros j(alpha n) of the Bessel function J(a lpha)(z). The known first eight terms of the McMahon expansion j(alpha n) similar to (n + a)[1 - Sigma(p greater than or equal to 1)b(p)(a)( n + a)(-2p)] with a = (2 alpha - 1)/4 are used to construct an accurat e approximation to zeta(alpha)(s). The quality of this approximation i s investigated numerically by comparison with a known but (at least nu merically) little-studied integral formula for zeta(alpha)(s). Excelle nt numerical agreement is found for fixed alpha and variable (real) s, and for fixed s and variable alpha. Both formulae for zeta(alpha)(s) therefore seem to work well. Our approximation also accurately reprodu ces known special values of zeta(alpha)(s). Important properties of ze ta(alpha)(s) are investigated for the first time, including several of its zeros. In addition, some general theory is presented in two areas : (i) perturbed spectra and (ii) the interrelationship between functio ns like J(alpha)(z) representable as infinite products, and the zeta f unctions constructed from their infinite spectrum of zeros.