# Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

3,525 questions
Filter by
Sorted by
Tagged with
37 views

### A question in a theorem related to bounding primes values of $n^2 +1$ [closed]

This question is in the proof of theorem 2.1 in Lecture 12 of the notes of Sieve Theory here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html I have checked in the notes of earlier lecture ...
68 views
+150

### Questions about Hooleys Approach in Artin's primitive root Conjecture

I know that I have asked 5 questions but they are all part of same proof. To each answer that answer 3 or more questions I will grant a bounty of 100 points and if someone answers all 5 questions I ...
1 vote
44 views

### 2 questions in the theory of Counting Perfect Squares

I have been reading sieve theory from notes of zeev rudnick here :http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and in page 2 of lecture 16(http://www.math.tau.ac.il/~rudnick/courses/...
17 views

### How to prove this deduction in the Analytic Large Sieve using Beurling - Selberg function

I have been studying sieve theory from the following notes : http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and I am struck on the following deduction in the proof of theorem 3.1 ( page 6) ...
24 views

### Some questions in the proof of Analytic Large Sieve

I am learning about the analytic large sieve from the lecture notes here:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html . I have some question in lecture 15:http://www.math.tau.ac.il/~...
16 views

### Questions in proof of Arithmetic Large Sieve

I am studying Arithmetic Large Sieve from following notes of Zeev Rudnick:http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html I have questions in lecture 14 here: http://www.math.tau.ac.il/~...
42 views

### A question in proof of Linnik's Theorem in Arithmetic Large Sieve

This question is from course notes in sieve theory and I am struck on this assertion in the proof of Linnik's theorem. Consider Page 4 of lecture 14 here: http://www.math.tau.ac.il/~rudnick/courses/...
27 views
+50

### How to estimate S(z) in Arithmetic Large Sieve

This question is part of a proof in course in Sieve Theory( http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html, precisely lecture 11 and 14)and I am not able to prove this particular ...
44 views

### $\sum_{ n \leq N , p\mid n \Rightarrow p < N^{\epsilon} } 1 \gg_{\epsilon} N.$ [closed]

I am reading class notes of number theory of a senior and I am struck an assertion of the proof. I have thought about it many times, so I am posting it here. It is a first course on Analytic Number ...
23 views

### How to prove that Vinogradow inequality implies that...

This question was asked in my assignment on Number Theory and I am struck on it. Define $n_p$ = min { $1 \leq m \leq \frac{(p+1) } {2} : ( m/p)=-1$}. Assuming polya vinogradow inequality prove ...
36 views

56 views

### Showing the density of primes around $n$ is approximately $1/\ln(n)$

The following shows the density of primes around $x$ is approximately $1/\ln(x)$. The argument, although based on approximation, benefits from being simple enough for those without advanced ...
1 vote
47 views

1 vote
45 views

### How should I evaluate the average order of $\sigma(n)$?

I was reading Apostol's book Introduction to Analytic Number Theory, and was trying to use the method introduced in this book to evaluate $\frac{1}{n}\sum_{1\leq m\leq n}\sigma(m),$ where $\sigma(m)$ ...
148 views

In this post Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this series converges $$\sum_{1<n\leq N}\frac{a_{n}}{\log\left(n\right)}=\!\... 1 vote 1 answer 39 views ### Find the fourier coefficient of given function I'ven been reading Dimitris Koukoulopouls' analytic number theory book. I'm stuck at p.63, the following part. Let f : \mathbb{R} \to \mathbb{C} be twice-continuously differentiable function of \... 1 vote 1 answer 58 views ### Estimating the n^{th} prime p_n in terms of n for n large From the prime number theorem we know that for n large , n=\pi(p_n)\sim\frac{p_n}{\log p_n} \implies \log n \sim \log(p_n)-\log\log p_n \ \ -(i) Now p_n \sim n\log p_n. Some calculations with ... 1 vote 1 answer 68 views ### Proving this identity using summation by parts This problem was left as an exercise from the notes which I am self studying and I am not able to get any ideas on how to solve it. Problem: Let$$\pi(x;q,a)=\#\{p \leq x : p=a \mod q\}$$and$$\psi(...
I was reading the paper "Integral moments of $L$-functions" by Conrey et al. They have used the following sharp cutoff version of the approximate functional equation for "Selberg class&...