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

For $\kappa >1$ and $t,X\geq 1$ $$\sum _{n\leq X}a_n=\frac {1}{2\pi i}\int _{\kappa \pm iT}\frac {\mathcal F(s)X^sds}{s}+\mathcal O\left (x^\kappa \sum _{=1}^\infty \frac {1}{n^\kappa (1+T|\log (X/n)| … asked Nov 29 '19 by tomos 1answer Is it possible to show (the trivial statement) \sum _{n\leq x}1=x+\mathcal O\left (1\right ) using Perron's formula? For c a little bigger than 1 and 1>c'>0, a quantitative form of Perron's … asked Jan 25 '19 by tomos 1answer Take x>0 large, t\in \mathbb R, q\in \mathbb N and a non-principal character \chi  mod q. If you want, take t\leq x. How do I bound $\sum _{n\leq x}\frac {\chi (n)}{n^{it}}?$ My gue … asked Mar 2 '19 by tomos 1answer Hi, Could anyone explain to me how A and B are the same/different/equivalent? A = The Siegel-Walfisz Theorem as stated in Wikipedia (this is the statement in Davenport) http://en.wikipedia.org/wik … asked Dec 26 '12 by tomos 1answer Hi, Could anyone tell me in what sense the following is an "asymptotic formula": Theorem 1 from http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.mmj/10290 … asked Dec 26 '12 by tomos 2answers Hi, I have a question about the Hardy-Littlewood method. Writing R_s(n) for the number of ways to write n as a sum of s k-th powers and f(\alpha ) for the sum \sum _{m=1}^Ne(\alpha m^k), … asked Jan 26 '13 by tomos 1answer \DeclareMathOperator\gcd{gcd}Take q\in \mathbb N and X>0 (q not necessarily smaller than X). A sum such as$$\sum_{d\leq X}(q,d) is easily seen to be $\ll q^\epsilon (X+q)$ so that the gc …
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 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 …
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 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 …