# real analysis – Infinite products for linear combinations of sines or cosines

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.