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

$phi(x)=a_1cos b_1 x + a_2cos b_2 x + a_3cos b_3 x$, where all coefficients are real? More specifically, under what conditions on the coefficients $a_n,b_n$ do we have the simplified product

$$phi(x)=ccdot prod_{k=1}^infty Big(1-frac{x}{rho_k}Big)$$

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(tlog n)}{n^sigma},\

phi_2(sigma, t) = sum_{n=1}^infty (-1)^{n+1}frac{sin(tlog 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.