Full moment conjecture. What are the lower order terms in the moment formulae for and for ? These are known in a few instances (see [ MR 88c:11049] Theorem 7.4 and [ MR 97e:11096] for the second and fourth moments of ) but not in general. The difficulty is that random matrix theory does not ``see'' the contribution of the arithmetic factor . Lower order terms will likely involve a mix of derivatives of and secondary terms from the moments of the characteristic polynomials of matrices. In general, a better understanding of how is modeled by a characteristic polynomial of a certain type of matrix is needed; how do the primes come into play? Perhaps we should think of as a partial Hadamard product over zeros multiplied by a partial Euler product. Perhaps these two parts behave independently, and the Hadamard product part can be modeled by random matrix theory. See the article on Explicit Formula for more discussion of this point.
Recently, Conrey, Farmer, Keating, Rubinstein, and Snaith and, independently by another method, Diaconu, Goldfeld, and Hoffstein, have developed a conjecture for these lower order terms.
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