[Math] What are the modular forms which “naturally” corresponds to Riemann Zeta function or Dirichlet L function

Question

We know that, if we take the coefficients of the Dirichlet series of Riemann zeta function or Dirichlet L functions and use those coefficients in q-expansion, we will not get modular forms.

But is there any other ways which can "naturally" generate modular forms from
Riemann zeta function or Dirichlet L functions ?

Answer 1

There is a simple reason why you will never get the coefficients of the Dirichlet series $\zeta(s)$ or $L(s, \chi)$ as the q-expansion coefficients of a modular form. The reason is that these $L$-series have Euler products where the term for a prime $p$ is $1 / $(linear polynomial in $p^{-s}$), while modular forms always give you $1 / $(quadratic polynomial in $p^{-s}$).

Of course, there's two easy ways to make quadratic polynomials out of linear polynomials:

• You can multiply two linear polynomials together. This leads you to the conclusion that products like $\zeta(s)^2$ or $L(s, \chi_1) L(s, \chi_2)$ should correspond to modular forms; and this is indeed the case, at least half of the time (there is a parity condition you have to impose). The corresponding modular forms are called Eisenstein series. An example is given in user1952009's answer, which constructs a modular form corresponding to the product $\zeta(s) \zeta(s + 1 - 2k)$. (The fact that $1-2k$ is odd is an instance of the parity condition: $\zeta(s) \zeta(s + 2)$ doesn't correspond to a modular form.)
• You can square the variable in your linear polynomial to get a quadratic polynomial. This leads you to the conclusion that e.g. $\zeta(2s)$ should correspond to a modular form (because $1 - p^{-2s}$ is a quadratic polynomial in $p^{-s}$). The corresponding modular forms are called $\theta$-series. In fact, Riemann used this in his original 1859 paper where he introduced the Riemann hypothesis: his proof of the functional equation of the Riemann zeta function relies on the fact (due to Jacobi) that the $\theta$-series $\tfrac{1}{2} + \sum_{n \ge 1} q^{n^2}$ is a modular form of weight 1/2.