On the non‐homogeneous quadratic Bessel zeta function

Abstract

This article studies the non‐homogeneous quadratic Bessel zeta function ζRB(s, v, a), defined as the sum of the squares of the positive zeros of the Bessel function Jv(z) plus a positive constant. In particular, explicit formulas for the main associated zeta invariants, namely, poles and residua ζRB(0, v, a) and ζRB(0, v, a), are given.

Vacuum expectation value of twist fields

“…However, this consideration cannot be extended to the case of the Bessel zeta functions, so one has to resort to other techniques [27]. As we will advocate in the next section, it more efficient to deal with the resolvent rather than the heat kernel that is traditionally used in the calculation of operator determinants by means of zeta function method [28].…”

“…The remaining integral performed over the contour C − . To evaluate this last term, we rescale the integration variable σ → σ/t and expand the integrand for small t. We then can deform C − contour to large values of |σ| and, thus, rely only on the asymptotic behavior of V(σ, m + α) [29,30,27],…”

Zeta invariants for Dirichlet series

“…We could now use this representation to compute both the value at s = 0 and the derivative, if we could provide an expansion for T (λ, s) for large λ [Spreafico 2004]. The problem is that more care is necessary due to the possible appearance of a singularity in the sum over n. The unique possible pole in such a sum comes from a term behaving like 1/n in the expansion of t n for large n uniformly in λ, so we can overcome this problem as follows.…”

“…Consequently, it is hard to find general results, and different techniques have been introduced to deal with the specific cases. See, for example, [Bombieri and Perelli 2001;Carletti and Monti Bragadin 1994a;1994b;Quine and Choi 1996;Eie 1990;Spreafico 2003;2004] for simple series or series that can be reduced to simple series. For what concerns multiple series, homogeneous linear series of Dirichlet type, defined (for Re(s) > k) by the sum…”

“…Notice also the work of Actor [1990], which gives a formula for ζ (s; a) as a power series whose coefficients are (infinite) sums of Riemann zeta functions and elementary functions and the related works of Matsumoto [1998], where asymptotic expansions are given for nonhomogeneous linear series ζ 1 (s; a, 1, q) = ∞ n,k=1 (an + k + q) −s , for large a, and observe that a formula for ζ 1 (0; a, 1) was given in [Spreafico 2004], using the Plana Theorem. Next, the case of the double homogeneous quadratic series of Dirichlet type, defined (for Re(s) > 1) by the sum…”