Restricted Goldbach Sums Modulo 2^k and the Shifted-Prime Singular Series

Source abstract

For an odd residue class $a$ modulo $q = 2^{k}$ we study the weighted number of representations $R_{a,q}(N) = \sum_{p_{1}+p_{2}=N,\ p_{1}\equiv a\,(q)} \log p_{1}\log p_{2}$ of an even integer $N$ as a sum of two primes, the first lying in the class $a \bmod q$. We prove, by the circle method with an additive-character decomposition of the restricted exponential sum, that $\sum_{N\le X} |R_{a,q}(N)-M_{a,q}(N)|^{2} \ll_{A,q} X^{3}(\log X)^{-A}$ for every $A>0$, where $M_{a,q}(N) = \frac{2C_{2}}{\varphi(q)}\mathfrak{S}_{0}(N)N$ and $\mathfrak{S}_{0}$ is the classical binary singular-series factor; the local factor at the prime $2$ is computed exactly. Consequently the exceptional set of even integers not representable in this restricted way has size $O_{A}(X(\log X)^{-A})$, and $R_{a,q}(N) = M_{a,q}(N)(1+O((\log N)^{-A}))$ outside a set of the same size. The same method gives a pointwise asymptotic formula for the ternary problem in which one prime lies in the class $a \bmod 2^{k}$, and hence positivity for all sufficiently large odd integers. For general moduli with odd prime factors we prove that a local obstruction forces a positive proportion of even integers to have essentially no restricted representation, state the corresponding local-density main term as a conjecture, and test it numerically for eleven pairs $(q,a)$ covering nine distinct moduli. In the second part we study the amplitude $\mathfrak{S}_{0}(p+h)$ of the main term along shifted primes. We prove, by an elementary argument based on the prime number theorem in progressions and the Brun–Titchmarsh inequality, that $\mathfrak{S}_{0}(p+h)$ has a limit distribution equal to the law of an explicit random Euler product $Y_{h}$, that all moments converge to the corresponding Euler products $A_{m}(h)$, that the moment generating function of $\log Y_{h}$ is entire with $\log \mathbb{E}Y_{h}^{s} \le s\log\log s + O(s)$, that the upper tail of $Y_{h}$ is doubly exponentially thin and this is sharp on a doubly logarithmic scale, and that $Y_{h}$ has no atoms. Certified enclosures of $A_{m}(h)$ and a numerical comparison with the primes up to $10^{7}$ are given. A final section reports a computational study of a shifted-sum digraph on the first primes, including an analysis showing that its negative in/out-degree correlation is almost entirely explained jointly by the size of $p$ and by $\mathfrak{S}_{0}(p+1)$.