# Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line,

$$M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt,$$

where $$\chi$$ is a Dirichlet character and $$k > 0$$ is a fixed real number (though I mostly care about $$k \in \mathbb{N}$$).

Conjecturally, one expects that $$M_k(T;\chi) \sim c_k(\chi) T (\log T)^{k^2}$$ for fixed $$k,\chi$$ and $$T \to \infty$$.

I am looking for two results that probably exist in the literature, because analogous results exist for the Riemann zeta function, and the same tools should generalize, but I'm having trouble finding them on mathscinet or google scholar.

# Question 1

Does the sharp estimate $$M_k(T;\chi) \ll_{k,\chi} T (\log T)^{k^2}$$ conditionally on GRH for $$L(s,\chi)$$ appear in the literature?

This estimate was shown on RH for the Riemann zeta function by Harper. For $$L$$-functions, I found this paper of Milinovich and Turnage-Butterbaugh where they show almost sharp conditional upper bounds on products of automorphic $$L$$-functions. Specializing to a single Dirichlet $$L$$-function, their results give on GRH that $$M_k(T;\chi) \ll_{k,\chi,\epsilon} T (\log T)^{k^2 + \epsilon}$$ which is just shy of what I think can be proved.

# Question 2

In his paper on applying the Montgomery-Vaughan mean value theorem to the fourth moment of the Riemann zeta function, Ramachandra states that the methods of his paper can prove the following theorem:

"Theorem 4. Let $$\chi_1$$ and $$\chi_2$$ be two characters mod $$q_1$$ and $$q_2$$ respectively. Then for $$T \geq 2$$, the mean value

$$\frac{1}{T} \int_0^T \left|L\left(\frac{1}{2}+it,\chi_1\right)L\left(\frac{1}{2}+it,\chi_2\right)\right|^2\,dt$$

is $$C(\log T)^4 + O((\log T)^3)$$ or $$D(\log T)^2 + O(\log T)$$ according as $$\chi_1$$ and $$\chi_2$$ are equivalent characters or not. (Here $$C$$ and $$D$$ are constants depeneding on $$\chi_1$$ and $$\chi_2$$."

He doesn't provide a proof of this, but notes that the details of the proof will be published elsewhere.

My question here is: did Ramachandra ever publish the details of the proof? I couldn't find it.

I am especially interested in a reference with a precise value for $$C$$ and $$D$$ as functions of the characters, especially if this exists in the literature already.

• With regards to Question 1: I'm pretty sure a detailed proof of the sharp estimates under GRH aren't written down anywhere, but they should be a pretty straightforward modification of Harper's result. Jun 8, 2020 at 21:40
• Thanks - I was afraid that would be the case. I found a few analogous results for classes of $L$-functions associated with automorphic forms (for example, this), but since I don't know much automorphic forms I was having trouble telling whether this contains the result I wanted. I guess I'll put a pin on this and come back to it when I understand Harper's method. Jun 9, 2020 at 0:59

Since I discovered the answer to my Question 2 in the literature, I figured I would update this in case someone stumbles upon it in the future.

It does not appear to be the case that that Ramachandra published the details of his proof anywhere [though it's a straightforward modification of the earlier results in the same paper]. However, Topacogullari [1] has recently proved a full asymptotic formula with power savings in the error term. His result is also uniform in the conductors of the characters involved.

For primitive characters $$\chi,\chi_1,\chi_2$$, $$\chi_1 \neq \chi_2$$ with conductors $$q,q_1,q_2$$ respectively, the constants $$C(\chi)$$ and $$D(\chi_1,\chi_2)$$ are given by

$$C(\chi) = \frac{1}{2\pi^2} \frac{\varphi(q)^2}{q^2} \prod_{p \mid q} \left(1 - \frac{2}{p+1}\right),$$

and

$$D(\chi_1,\chi_2) = \frac{6}{\pi^2} |L(1,\overline{\chi_1}\chi_2)|^2 \frac{\varphi(q_1)\varphi(q_2)}{\varphi(q_1 q_2)} \prod_{p \mid q_1 q_2}\left(1 - \frac{1}{p+1}\right).$$

As Peter Humphries mentions in the comments to the OP, the answer to Question 1 is no, this result does not appear to have been written down yet.

[1] Topacogullari, Berke. "The fourth moment of individual Dirichlet L-functions on the critical line." Mathematische Zeitschrift (2020): 1-48.