What proportion of integers $n \leq N$ may be expressed as $x^2 + dy^2$ for some $d \leq \Delta $, with $x,y$ integers? Writing $\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$ for some $\alpha \in (-\infty , \infty )$, we show that the answer is $\Phi (\alpha ) + o(1)$, where $\Phi $ is the Gaussian distribution function $\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$.

A consequence of this is a phase transition: Almost none of the integers $n \leq N$ can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 - \varepsilon }$, but almost all of them can be represented by $x^2 + dy^2$ with $d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$.

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

We obtain lower bounds of the correct order of magnitude for the 2kth moment of the Riemann zeta function for all k≥1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the rational number k. Our new bounds are continuous in k, and thus extend also to the case when k is irrational. The method is a refinement of an approach of Rudnick and Soundararajan, and applies also to moments of L-functions in families.