The mean-value conjectures

For convenience we state examples of the three main conjectures for moments in families. There are no open questions in this article (except how to prove these conjectures!). For simplicity, we state the conjectures only for integral moments.

Unitary

Let

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

Conjecture.

$$I_k(t)\sim a_k \prod_{j=0}^{k-1}\frac{j!}{(j+k)!}(\log T)^{k^2}$$

where

$$a_k= \prod_p(1-1/p)^{(k-1)^2} \sum_{j=0}^{k-1}\binom{k-1}{j}^2p^{-j}.$$

Symplectic

Let $\chi_d$ be a real, primitive, quadratic character to the modulus $\vert d\vert$ (i.e. a Kronecker symbol) and let

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

where $D^*=\sum_{\vert d\vert\le D}1$.

Conjecture.

$$S_k(D)\sim a_k \prod_{j=1}^{k}\frac{j!}{(2j)!} (\log D)^{k(k+1)/2}$$

where

$$a_k= \prod_p \frac{(1-1/p)^{k(k+1)/2}}{(1+1/p)}\left(\frac{(1-1/\sqrt{p})^{-k}+(1+1/\sqrt{p})^{-k}}{2}+\frac 1p\right)$$

Orthogonal

Let $f$ be a normalized newform of weight $2$ and level $q$ (where $q$ is prime) (we write $f\in F(q)$ and let $L_f(s)$ be the associated $L$-function (with critical strip $0<\sigma<1$.) Let

$$O_k(q)=\frac{1}{q^*}\sum_{f\in F(q)} L_f(1/2)^k$$

where $q^*=\sum_{f\in F(q)} 1.$

Conjecture.

$$O_k(q)\sim a_k 2^{k-1}\prod_{j=1}^{k-1}\frac{j!}{(2j)!} (\log q)^{k(k-1)/2}$$

where $a_k$ is a product over primes (which can be worked out for any $k$, but for which we don't have a simple closed form expression);

$$a_1= \zeta(2),$$

$$a_2=\zeta(2)^2\prod_p (1+1/p)^2,$$

$$a_3=\zeta(2)^3\prod_p(1-1/p)(1+1/p+4/p^2+1/p^3+1/p^4),$$

and so on.

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