Mean values of long Dirichlet polynomials with higher divisor coefficients

Averages of long Dirichlet polynomials with modular coefficients

“…Results of the same nature have been proven also for other families, in particular by Baluyot and Turnage‐Butterbaugh [5] for the unitary family of Dirichlet L ‐functions of primitive characters and by Conrey and Rodgers [17] for the symplectic case of quadratic Dirichlet L ‐functions; both these results have been proven under GLH. We also mention [20], where Hamieh and Ng study the (continuous) family of ζ in the t ‐aspect, assuming the expected asymptotic formula for correlations of divisor sums. A motivation for all these results lies in the Conrey–Keating heuristic [16], which suggests how to ‘patch together’ the 1‐swap terms in order to construct the general ℓ‐swap term (see [4] for a complete overview on the topic).…”

Spectral moment formulae for $$GL(3)\times GL(2)$$ L-functions III: the twisted case

“…On the other hand, Young [59] applied the spectral method in conjunction with the delta method to a generalized binary additive problem. His strategy was favourable in establishing the conjecture of [20] for the moment M 4 ( p) and turned out to be useful in many other instances, see [4,5,36,37,54,57,61]. An entirely different sequence of transformations led to the same formula (1.1) once again.…”

Moments of zeta and correlations of divisor‐sums: Stratification and Vandermonde integrals

“…Another way we refine the approach of [14] is in making some of their technical arguments more precise. This includes expressing Fourier transforms of test functions in terms of the gamma function (Lemma 3.1), as has been done in [20,21], and [24].…”

“…The problem is so intractable that, up until recently, there had been no viable guess for the exact value of $$c_k$$. The approach of using correlations of divisor sums leads to conjectures for $$c_3$$ and $$c_4$$ [8, 9], and the process has recently been examined in close detail and made more precise by Ng [24], Hamieh and Ng [20], and Ng, Shen, and Wong [25]. This approach seems to fail, however, to give a reasonable guess for $$c_k$$ when $$k\geqslant 5$$ [9].…”

“…This approach has been used by Gonek and the second author [9] to recover the random matrix theory prediction for the sixth and the eighth moments of zeta, and has been further developed by Keating and the second author [12] to heuristically evaluate (1.1) when $$T^{1+\varepsilon }< X< T^{2-\varepsilon }$$. Their ideas were made precise by Hamieh and Ng [20], who assumed an asymptotic formula for correlations of divisor sums that has a specific error term. Under this assumption, Hamieh and Ng proved that the 0‐ and 1‐swap terms predicted by the recipe are indeed the main terms in the asymptotic formula for (1.1).…”