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

2
votes
1answer
For $\kappa >1$ and $t,X\geq 1$ $$\sum _{n\leq X}a_n=\frac {1}{2\pi i}\int _{\kappa \pm iT}\frac {\mathcal F(s)X^sds}{s}+\mathcal O\left (x^\kappa \sum _{=1}^\infty \frac {1}{n^\kappa (1+T|\log (X/n)| …
asked Nov 29 '19 by tomos
5
votes
1answer
Is it possible to show (the trivial statement) $\sum _{n\leq x}1=x+\mathcal O\left (1\right )$ using Perron's formula? For $c$ a little bigger than $1$ and $1>c'>0$, a quantitative form of Perron's …
asked Jan 25 '19 by tomos
6
votes
1answer
Take $x>0$ large, $t\in \mathbb R$, $q\in \mathbb N$ and a non-principal character $\chi $ mod $q$. If you want, take $t\leq x$. How do I bound \[ \sum _{n\leq x}\frac {\chi (n)}{n^{it}}?\] My gue …
asked Mar 2 '19 by tomos
2
votes
1answer
Hi, Could anyone explain to me how A and B are the same/different/equivalent? A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport) http://en.wikipedia.org/wik …
asked Dec 26 '12 by tomos
2
votes
1answer
Hi, Could anyone tell me in what sense the following is an "asymptotic formula": Theorem 1 from http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/10290 …
asked Dec 26 '12 by tomos
7
votes
2answers
Hi, I have a question about the Hardy-Littlewood method. Writing $R_s(n)$ for the number of ways to write $n$ as a sum of $s$ $k$-th powers and $f(\alpha )$ for the sum $\sum _{m=1}^Ne(\alpha m^k)$, …
asked Jan 26 '13 by tomos
1
vote
1answer
$\DeclareMathOperator\gcd{gcd}$Take $q\in \mathbb N$ and $X>0$ ($q$ not necessarily smaller than $X$). A sum such as $$\sum_{d\leq X}(q,d)$$ is easily seen to be $\ll q^\epsilon (X+q)$ so that the gc …
asked Sep 22 '20 by tomos
4
votes
Unless I'm overlooking something (which is very very possible...) I think you can just use the Moebius function in the form \[ \sum _{d|n}\mu (d)=\left \{ \begin {array}{ll}1&\text { if }n=1\\ 0&\text …
answered Dec 19 '21 by tomos
3
votes
0answers
I find quite a few results about the binary additive divisor problem, that is evaluating \[ \sum _{n\leq x}d(n)d(n+h)\] for certain ranges of $h$. Are there any known results about the same count rest …
asked Dec 11 '21 by tomos
1
vote
I'd have a lousy suggestion, but it's too long for a comment so here goes. Maybe it's the wrong way to think with this problem, but my first idea would be to look for an approximate functional equa …
answered Dec 30 '19 by tomos
4
votes
0answers
Denote by $S(c;n,m)$ Kloosterman's sum. Take $X>0$ and take $n,m\in \mathbb Z$ smaller than a small power of $X$ in modulus. It is known that essentially \[ \sum _{c\sim X}\frac {S(c;n,m)}{c}\ll X^{ …
asked Jan 6 by tomos
3
votes
0answers
Take a set $\mathcal P$ of primes and denote by $\langle \mathcal P\rangle $ the set of all natural numbers composed of primes from $\mathcal P$. If \[ \sum _{p\in \mathcal P}\frac {1}{p}\] converges …
asked Oct 27 '21 by tomos