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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

1 vote
0 answers
162 views

Is this averaged exponential sum over primes small infinitely often?

Do there exist infinitely many positive integers $N$ such that $$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N …
Linden's user avatar
  • 217
4 votes
1 answer
528 views

An Exponential Sum Restricted to Primes

Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$. Is the following estimate true, and, if so, how can it be proved? \[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) …
Linden's user avatar
  • 217
0 votes
2 answers
366 views

Counting Divisors in $\mathbb{Z}^n$

Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$. For every positive integer $n$, I'm looking for an algebra structure on $\mat …
Linden's user avatar
  • 217