Provisional Bessel K decomposition conjecture for the Riemann zeta function

Let $r\in\{1,2,3\}$, let $\zeta^{*}(s,r)$ be the completed zeta function defined in the paper, and let $a_r(n)$ and $b_r(n)$ be the coefficient sequences introduced by the theta-function expansions. For $j\geq 1$, define

$$c_j(s,r)=\sum_{d\mid j}a_r(d)b_r\left(\frac{j}{d}\right)\left(\frac{j}{d^2}\right)^{\frac{rs+2-r}{4}}.$$

Let $K_\nu$ denote the modified Bessel function of the second kind. The provisional Bessel K decomposition conjecture. For $r=1,2,3$,

$$\zeta^{*}(s,r)=-\left(\frac{\Psi_r(s)}{s}+\frac{\Psi_r(1-s)}{1-s}\right),$$

where

$$\Psi_r(s)=2\pi\sum_{j=1}^{\infty}c_j(s,r)K_{\frac{rs+2-r}{2}}(2\pi\sqrt{j}).$$

The result is provisional because the paper does not justify the interchange of the sum and integral used in its derivation; it is intended to provide Bessel expansions for the zeta functions under consideration.