We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.

The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

In this paper we prove two results concerning Vinogradov’s three primes theorem with primes that can be called almost twin primes. First, for any $m$, every sufficiently large odd integer $N$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, the interval $[p_{i},p_{i}+H]$ contains at least $m$ primes, for some $H=H(m)$. Second, every sufficiently large integer $N\equiv 3~(\text{mod}~6)$ can be written as a sum of three primes $p_{1},p_{2}$ and $p_{3}$ such that, for each $i\in \{1,2,3\}$, $p_{i}+2$ has at most two prime factors.

For $n= 1, 2, 3, \ldots $ let ${S}_{n} $ be the sum of the first $n$ primes. We mainly show that the sequence ${a}_{n} = \sqrt[n]{{S}_{n} / n}~(n= 1, 2, 3, \ldots )$ is strictly decreasing, and moreover the sequence ${a}_{n+ 1} / {a}_{n} ~(n= 10, 11, \ldots )$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.

We give explicit formulas for the ${{L}_{4}}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials

$$f\left( z \right):=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n}{N} \right){{z}^{n}}.}$$

and

$${{f}_{t}}(z)\,=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n+t}{N} \right){{z}^{n}}.}$$

where $\left( \frac{.}{N} \right)$ is the Jacobi symbol.

Two cases of particular interest are when $N\,=\,pq$ is a product of two primes and $p\,=\,q\,+\,2$ or $p\,=\,q\,+\,4$. This extends work of Høholdt, Jensen and Jensen and of the authors.

This study arises from a number of conjectures of Erdős, Littlewood and others that concern the norms of polynomials with −1, 1 coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite $N$.

To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function $$ F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), $$ where $\alpha$ is real and $T\ge 2$, $\gamma$ and $\gamma '$ denote the imaginary parts of zeros of the Riemann zeta-function, and $w(u) = 4/(4 + u^2)$. Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for $F(\alpha)$ when $|\alpha|\le 1$, and made the conjecture that $F(\alpha) = 1 + o(1)$ as $T\to \infty$ for any bounded $\alpha$ with $|\alpha |\ge 1$. In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions, then for any $\epsilon >0$ we have $$ F(\alpha) \ge {3\over 2} - |\alpha| - \epsilon ,$$ uniformly for $1\le |\alpha| \le \frac32 -2\epsilon $ and all $T \ge T_0(\epsilon)$.

1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.