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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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

$\DeclareMathOperator\gcd{gcd}$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

**0**answers

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

**0**answers

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

**0**answers

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