Questions tagged [analytic-number-theory]

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

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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 ...
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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 ...
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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/...
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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) ...
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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/~...
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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/~...
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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/...
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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 ...
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$\sum_{ n \leq N , p\mid n \Rightarrow p < N^{\epsilon} } 1 \gg_{\epsilon} N.$ [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 ...
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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 ...
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Proving that this infinite product is convergent

This question was asked in my assignment in number theory and I could not prove it. Question : Define the multiplicative function w(n) such that $w(p^k) =0$ for $k\geq 2$ and w(p)= { $\frac{p} { f(p)...
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Compute the Fourier expansion of adelic Eisenstein series associated to the classical holomorphic Eisenstein series.

For each place $v$ of $\mathbf{Q}$, define $\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$ by $$ \Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if $v<\infty$},\\...
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What is wrong with my proof on elementary prime gaps?

Question I want to prove that the rationals of the form $\displaystyle \frac{p}{q}$ where $p,q$ are primes are dense in $[0,\infty)$. The usual way to do this using the original form of PNT $\...
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On number of coprimes in a bounded interval - II

If you pick $k$-tuples of integers then you can be guaranteed that they are coprime with probability $\frac1{\zeta(k)}$. However if you fix $a\in\Bbb N$ and pick $k$-tuple in $[0,a]$ what is the ...
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sum of square of three squares function $r_3(n)$ II

From sum of square of cube function $r_3(n)$, I believed that, as $r_k(n)$ denotes the number of ways that $n$ is the sum of $k$ squares, $$\sum_{n\le X} r_3(n)^2\asymp X^{3/2}\log^{C}X,...(*)$$ but ...
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Brun-Titchmarsh for semiprimes (or semi-primes)

We know, by Brun-Titchmarsh theorem and sieve methods that $\pi(X+Y)-\pi(Y)\leq \frac{2X}{\log(X)}(1+o_X(1))$. Do we know something for semiprimes? Like $\pi_2(X+Y)-\pi_2(Y)\leq \frac{2X\log(\log(X))}{...
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4 votes
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Expressing $ \sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}} \left \lfloor x^{1/n}-1 \right \rfloor$ in terms of the nontrivial zeros of $\zeta(s)$

Let $\left \lfloor \cdot \right \rfloor$ be the floor function. Is there a way to express the function $A(x)$ given by : $$A(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}} \left \lfloor x^{1/n}-1 \right \...
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real quadratic fields with 2 ramified primes

I have observed that when considering real quadratic fields with class number 1 and 2 having 2 ramified primes, the first ramified prime was (almost always, like 98% of the time) congruent to $3 \bmod ...
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2 votes
0 answers
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 ...
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1 vote
1 answer
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On the abscissa of convergence of a Dirichlet series.

I am trying to find the abscissa of convergence of the Dirichlet series for the arithmetic function $|\mu(n)|$. I have managed to show that $$\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\...
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2 votes
2 answers
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Asymptotics of sum of Chebychev function

Show that $\sum_{n\leq x}\frac{θ(n)}{n^2}=\ln x+O(1)$ where $θ$ is the Chebychev function. (We are searching for a solution without the prime number theorem, just Chebychev bounds or something like ...
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A question on the relation between the complex zeroes of zeta function and the estimate of the error in PNT

I'm currently working on the following problem from Analytic Number Theory. Assume that $\psi(x)-x=\mathcal{O}(x^a)$, for some $1/2<a<1$, where $\psi$ is the Chebyshev function. I would like to ...
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2 votes
1 answer
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Evaluating an exponential sum along an arithmetic progression

I am familiar with the identity $$\sum_{1 \leq n \leq q}e^{2 \pi i n^2 / q} = \left( \frac{1+ (-i)^q}{1-i} \right)\sqrt{q}$$ I am wondering if there is a similar evaluation for the sum $$S(r):= \sum_{...
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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 $\sigma(m)$ ...
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4 votes
1 answer
148 views

One series converges iff the other converges

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)}=\!\...
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1 vote
1 answer
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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 $\...
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1 vote
1 answer
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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 ...
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1 vote
1 answer
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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(...
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2 votes
1 answer
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Approximate functional equation of Selberg Class L-functions

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&...
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8 votes
1 answer
141 views

Best bound for $|\sum \prod_{p\mid n}(1-\frac{1}{p^k})-\frac m{\zeta(k+1)}|$?

Follow from the previous post, we have \begin{align*} f_2(x):=\sum_{n\le x} \prod_{p\mid n}(1-\frac{1}{p^2})-\frac x{\zeta(3)}=O(1)\quad \text{as }x\to\infty.\tag{*} \end{align*} When I tried to plot $...
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1 vote
0 answers
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Partial sum estimate of sum of reciprocals of prime powers

I want to estimate the following sum $$ \sum_{p\leq x}\frac{1}{{p^n}}\tag{1} $$ Here n=2,3... We already know by Mertens theorem: $$\sum_{ p \text{ prime,} p \leq k } 1/p - \log{\log{k}} = B + E(k) \...
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0 votes
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Analytic number theory-Are these sums equal?

I am studying analytic number theory by myself and I came across with two posts that seem to confuse me . Let $S(x)=\sum_{p\le x,\; q\le x,\; pq\gt x}\frac{1}{pq}$, where p and q are primes. Find the ...
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Show that $\sum a(n)/\log n$ converges if and only if $\sum a(p)$ converges. [duplicate]

I am studying analytic number theory at the time and I came across with this. I was wondering if there is a somehow smart way to prove the following without using prime number theorem(but probably use ...
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0 votes
0 answers
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Sum Over the Group of Dirichlet Character

I have the following question: Suppose $x=q^{o(1)}$, I want to prove the following estimate: $$\frac{1}{\phi(q)}\sum_{p_1, p_2\leq x}\sum_{\chi mod q}Re(\chi(p_1))Re(\chi(p_2))=\frac{1}{2}\pi(x)+o(1) \...
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Question about regularised product of primes:

We know "super-regularized"( the term coined by authors of the paper: http://cds.cern.ch/record/630829/files/sis-2003-264.pdf) product of primes $4π²$ i.e. $$\infty \# = \prod_{k=1}^\infty ...
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3 votes
1 answer
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Counting numbers up to $n$ whose prime factorizations have exactly $k$ prime factors with exponent $1$

Question. Let $N_k(n)$ count how many numbers $1\le x\le n$ for which $x$ has exactly $k$ unitary prime divisors, or equivalently $x$'s prime factorization has exactly $k$ primes with exponent $1$. ...
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On the behaviour for the quotient involving Fermat numbers of $\frac{\psi(F_m)}{F_m}$ where $\psi(x)$ denotes the Dedekind psi function

In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. This is an important arithmetic fuction in several subjects of mathematics. As reference I add the Wikipedia ...
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Exponential sums with the divisor function

Can anyone explain the first $\ll $ on line 8 on page 188 of "Jutila: On exponential sums involving the divisor function" for me? (https://eudml.org/doc/152693) Specifically, I think he is ...
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0 votes
1 answer
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On the minimum integer not coprime to numbers in an interval

Consider the set of positive integers $[a,a+1,\ldots, a+k]$ (i.e. the interval $[a,a+k]$). What can we say about the smallest positive integer $N$ such that $\gcd(a+i,N)>1$ for all $0\leq i\leq k$? ...
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2 votes
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Different ways to average an arithmetic function

Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$: Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$ Factorized: $\displaystyle\...
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1 vote
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A question related to Selberg's sieve theory

I am currently reading upon Selberg's sieve and following is an argument that I came across: Multiplying both sides of $\lambda_{\delta}=\delta \sum_{\substack{d<z \\ \delta|d}}\frac{\mu(d/\delta)...
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James Maynard's "Small Gap Between Primes"

Let $\mathcal{C}$ be the collection of function f such that f is a Lebesgue integrable function on $[0, 1]^2$ such that it vanishes outside the set $\{ (x, y) \in [0, 1]^2 | x + y \leq 1 \}$, and ...
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2 votes
1 answer
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Analytic continuation of this function to |z|>1?

Consider the following function: $\sum_{n=1}^\infty \sum_{s | n} s \mu(s) z^{n^2}$, where $\mu(s)$ is the Mobius function. This converges for $|z|<1$. Does there exist an analytic continuation to $|...
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9 votes
2 answers
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Asymptotic on $\max\{d(n): 1\leq n\leq N\}$.

Question: Let $d(n)=\sum_{d|n}1$, be the divisor function. Estimate the asymptotic behaver $$D(N):=\max\{d(n):1\leq n\leq N\}$$ when $N$ is large. I know $$\sum_{n=1}^N d(n)=\sum_{n=1}^N [\frac{N}{n}]...
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2 votes
1 answer
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Estimating reducible monic polynomials of degree n with integer coefficients of height of atmost N

This question is from my assignment in Sieve Theory and I am struck on it. I am following following notes on Sieve Theory: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html Question: For a ...
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2 votes
1 answer
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Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$

I'm investigating the behavior of the following function as $x\to \infty$: $$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$ where $J_k(n)$ is the Jordan's totient function $$J_k(n):=n^k\prod_{p\mid n}(1-...
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0 votes
0 answers
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A question regarding number of farey sequences [duplicate]

I have taken a course in number theory this semester and we are learning about Farey Sequences. This question was left as an exercise in the class and I was unable to solve it. So, I am looking for ...
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  • 1,344
0 votes
1 answer
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Sum of infinite series, Are both of these series equal? 1/2+1/3+1/4...

from an old Numberphile video they explain that the sum of all natural numbers is equal to -1/12, 1+2+3+4+5+...= -1/12. Obviously it diverges, but the -1/12 is meant to be a meaningful representation ...
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  • 3
0 votes
1 answer
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Asymptotics of the sum of squares of Von Mangoldt function values.

This question was asked in my quiz of number theory and I was not able to make any progress in it. Question: Show that $$\sum_{ n\leq x}{\Lambda(n)}^2 = x\log x- x+o(x),$$ where $\Lambda(n)$ is the ...
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3 votes
1 answer
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Show that number of primes $p\leq x$ so that $2p+1$ is also a prime...

This question is from assignment 4 of the following sieve theory course: http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html Question : Show that the number of primes $p\leq x$ so that $2p+1$ ...
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