7
votes
1answer
402 views

Can the Brun-Titchmarsh theorem be improved when the modulus is smooth?

For $q,a$ relatively prime, let $\pi(x,q,a)$ denote the number of primes less than $x$ which are congruent to $a$ modulo $q$. The Brun-Titchmarsh theorem states that $$\pi(x,q,a)\leq ...
7
votes
1answer
410 views

The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} ...
1
vote
1answer
486 views

Conjecture:if $i<j$,then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$

p[i] is the i-th prime. $\pi(x)$ is prime counting function. Firstly, I think that this Prime gap inequality holds true, $ p[i+1] - p[i] <= i $ Prove:for any i>0, we can always find distinct ...
4
votes
2answers
587 views

Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello! Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...
0
votes
0answers
301 views

Divisor function inequality

I have been reading a paper on the Goldbach conjecture found at http://people.exeter.ac.uk/pt224/Goldbach.pdf. At one point, the author (Paul Truman), states: Let $z=N^{1/8}$, then $$\sum_{w\leq ...