Let \(f\) be a Weierstrass canonical product of noninteger order \(\rho\), assume the zeroes of \(f\) are located on the negative ray, and denote by \(n(r)\) the number of the zeros in the disc \(\{ z : |z| \leq r \}\). It was proved by the second author [On asymptotic properties of entire functions with real negative zeros, Zap. Mat. Otd. Fiz-Mat. Fak. KHGU i KHMO 28, 23--32 (1961)] that \[ \liminf_{r \to \infty} \frac{\log |f(re^{i\theta})|}{n(r)} \leq \frac{\pi \cos \theta \rho}{\sin \theta \rho} \leq \limsup_{r \to \infty} \frac{\log |f(re^{i\theta})|}{n(r)}, \quad \theta \in (-\pi,\pi), \] both inequalities being sharp. The goal of the paper under review is to extend the above result to subharmonic functions in \(\mathbb{R}^n\), \(n \geq 2\). To this end, the authors consider the function \[ I(\rho,n,\theta ) = \int_0^\infty \left(y^{2-n} - (1 + y^2 +2y\cos \theta)^{(2 - n)/n} \right)y^{\rho + n - 3} \,dy, \quad 0 0, 0, \dots , 0\}\). For \(q= 0,1,2,\dots \) let \[ K_q^{(n)}(x,y) = - \left( |x|^2 + |y|^2 - 2 |x||y| \cos (\widehat{x,y})\right)^{\frac{2 - n}{2}}+ |y|^{2-n} \sum_{j=0}^q \left(\frac{|x|}{|y|}\right)G_j^{(n)}(\cos (\widehat{x,y})) , \] where \(G_j^{(n)}(w)\) are Gegenbauer polynomials with the generating function \((1 + z^2 - 2zw)^{(2-n)/2}\). If \(\mu\) is a locally finite measure in \(\mathbb{R}^n\) and \(n(r) : = \frac{\mu(\{ x : |x| \leq r\})}{r^{2-n}}\), \(r \geq 0\), it is known that if \(\int_1^\infty \frac{n(r)}{r^{q + n}}dr < \infty\) then the integral \[ v(x) = \int_{|y| \geq 1}K_q^{(n)}(x,y)\,d\mu(y) \tag{1} \] converges and \(v(x)\) is a subharmonic function in \(\mathbb{R}^n\) whose order coincides with that of the function \(n(r)\). A function of the form (1) is called a Weierstrass canonical integral with genus \(q\). The main result of the paper states that if \(\mu\) is a locally finite measure in \(\mathbb{R}^n\), \(n \geq 2\), supported on the ray \(l_{-}\) and such that the function \(n(r)\) has noninteger order \(\rho\), and \(v\) is a Weierstrass canonical integral in \(\mathbb{R}^n\) of genus \(q = [\rho]\), then we have \[ \liminf_{r \to \infty} \frac{v(r\xi)}{n(r)} \leq I(\rho,n, (\widehat{\xi,l_+})) \leq \limsup_{r \to \infty} \frac{v(r\xi)}{n(r)}, \quad \xi \in S_n \setminus l_{-}. \] Moreover, both inequalities are sharp. The key point in the proof is an integral representation of \(I(\rho,n,\theta)\) in terms of a certain function depending on the Gegenbauer polynomial \(G_j^{(n)}(w)\).