Mean-values

There are so many results and conjectures having to do with the mean-values or moments of L-functions in families that it is difficult to mention all of them. So we will stick with some of the more familiar ones.

Moments of $\zeta(1/2+it)$. Let

$$I_k(T)=\frac{1}{T} \int_0^T \vert\zeta(1/2+it)\vert^{2k}~dt.$$

Then asymptotic formulae are known for $I_1$ and $I_2$. A good estimation for $I_3$ seems to be far out of reach of today's technology (maybe we need to know more about automorphic forms on GL3?) but would be a very important milestone. Conjectures based on number theory are known for $k=3,4$. Conjectures based on random matrix theory are known for all real $k>-1/2$.

Moments of $L(1/2,\chi_d)$. Let $\chi_d$ be a real, primitive, quadratic character to the modulus $\vert d\vert$ (i.e. a Kronecker symbol). Let

$$S(D)=\sum_{\vert d\vert\le D} L(1/2,\chi_d)^k.$$

Then asymptotics are known for $k=1,2,3$. A conjecture based on number theory is known for $k=4$. Conjectures based on random matrix theory are known for all real $k>-3/2$. The fourth moment is just slightly out of reach of current technology.

Moments of automorphic L-functions. Let $f$ be a normalized newform of weight $k$ and level 1 and let $L_f(s)$ be the associated $L$-function (with critical strip $0<\sigma<1$.) Let

$$O_\lambda(K)=\sum_{k\le K}\sum_{f\in S_k} L_f(1/2)^\lambda.$$

Then asymptotic formulae are known for $\lambda = 1,2,3,4$. Conjectures based on random matrix theory are known for all real $\lambda > -1/2$. Is the fifth moment doable?

Quadratic twists of automorphic L-functions. Let $f$ be a fixed newform (for example the newform associated with a given elliptic curve). Let $L_f(s)$ be the associated L-function and $L_f(s,\chi_d)$ the twist by the primitive quadratic character $\chi_d$. It is possible to evaluate the first moment

$$\sum_{\vert d\vert\le D} L(1/2,\chi_d)$$

but can one do the second moment

$$\sum_{\vert d\vert\le D} L(1/2,\chi_d)^2?$$

Again, this is just at the edge of what can be done by today's techniques and is a problem worthy of study.

Special families. Recently the bound

$$\sum_{f\in S_k^*(q)} L_f(1/2,\chi_q)^3 <<q^{1+\epsilon}$$

has been obtained, where $S_k^*(q)$ denotes the set of newforms of weight $k$ and level $q$ (no character), and $\chi_q$ is the real character to the modulus $q$ where $q$ is odd and squarefree. This estimate has been used to bound fourier coefficients of half-integral weight forms and (its analogue for Maass forms has been used) to bound $L(1/2+it, \chi_q)$. Can one strengthen this estimate to give an asymptotic formula (perhaps for $q$ restricted to prime values)? Are there other situations where the underlying arithmetic is so fortuitous so as to give such a strong bound?

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