High moments of the Riemann zeta-function

Shifted convolution and the Titchmarsh divisor problem over 𝔽q[t]

“…This conjecture appears in the work of Ivić [20] and Conrey & Gonek [5], and from their work, with some effort, we can explicitly write the conjectural leading coefficient for the desired polynomial. The conjecture over Z states that…”

“…The higher divisor problem k ≥ 3 is also of importance, in particular, in relation to computing the moments of the Riemann ζ -function on the critical line [5,6]. It is conjectured that D k (x; h) ∼ xP 2(k−1) (log x; h) as x → ∞, (1.5) where P 2(k−1) (u; h) is a polynomial in u of degree 2(k − 1), whose coefficients depend on h (and k).…”

“…But even a conjectural description of the polynomials P 2(k−1) (u; h) is difficult to obtain (see §7, [5,6]). A variant of the problem about the autocorrelation of the divisor function is to determine an asymptotic for the more general sum given by (1.7)…”

The Riemann zeta-function and moment conjectures from Random Matrix Theory

“…In view of the estimate of B a l a s u b r a m a n i a n & R a m a c h a n d r a (9) it follows that the contribution of the integral over 2T, 2T + α L is negligible. Thus, we can replace (19) by…”

“…Consequently, the arithmetic information a(k), appearing in the heuristics of C o n r e y & G h o s h , has to be inserted in an ad hoc way. Recently, C o n r e y , K e a t i n g et al modified the standard Random Matrix Theory-model in order to incorporate also the arithmetic information a(k) (see G o n e k [9]); this leads directly to:…”