The integral of $\exp(i \lambda S(t))$

In the paper [CMP 1 794 265] by Keating and Snaith, the authors conjecture (see equation (100)) that

$$\int_0^T e(2\lambda S(t))~dt \sim T(\log T)^{-\lambda^2} G(1-\lambda) G(1+\lambda) b(\lambda)$$

where $b(\lambda)$ may be expressed as an absolutely convergent product over primes and G is the Barnes double gamma function.

If $\lambda$ is an integer, then the main term is 0. If $\lambda$ is not an integer, then no instance of this remarkable formula has been proven (or even conjectured) before.

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