There is a well known infinite product both for $\phi(x)=\sin x$ and $\phi(x)=\cos x$. These are particular cases of the Weierstrass factorization theorem. What about $\phi(x)=a_1\cos b_1 x + a_2\cos b_2 x + a_3\cos b_3 x$, where all coefficients are real? More specifically, under what conditions on the coefficients $a_n,b_n$ do we have the simplified product $$\phi(x)=c\cdot \prod_{k=1}^\infty \Big(1-\frac{x}{\rho_k}\Big)$$

where the product is over all real and complex roots (some of them possibly multiple) ordered in the following way:

- Roots are ordered by increasing moduli
- Conjugate and opposite roots are grouped together

I am particularly interested in factoring these two expressions:

$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\log n)}{n^\sigma},\\ \phi_2(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\sin(t\log n)}{n^\sigma}. $$

The reason is because when and only when $\phi_1(\sigma,t)=\phi_2(\sigma,t)=0$, then $s=\sigma+it$ is a non-trivial zero of $\zeta(s)$. See here for details. I am interested to see how the roots of $\phi_1$ and $\phi_2$ are jointly distributed. According to the Riemann Hypothesis, they can never be equal unless $\sigma=\frac{1}{2}$. I am wondering if this fact is also true for other similar types of non-periodic trigonometric series, one involving cosines, and its sister involving sines.