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Questions tagged [analytic-number-theory]

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

2
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1answer
24 views

An Inequality Involving Subsets of $\mathbb{N}$ of Positive Density

Given a set $V$ of non-negative integers, let: $$V\left(t\right)\overset{\textrm{def}}{=}\left\{ v\in V:0\leq v\leq t\right\} ,\textrm{ }\forall t\in\mathbb{R}$$ and let $\left|V\left(t\right)\right|$ ...
2
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0answers
32 views

The probability that $s$ integers selected according to the ideal soliton distribution are relatively $r$-prime

Fix integers $r,s\geq1$, not both 1. Definition. We say that integers $a_{1},\ldots,a_{s}$ are relatively $r$-prime if their greatest common divisor has no perfect $r$th power factors > 1. (When $r=...
2
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1answer
35 views

Doubt in Apostol's book theorem 8.20

I'm following the proof of the Fourier expansion of promiitive Dirichlet character from theorem 8.20 in Apostol's Introduction to Analytic Number Theory. Here is the complete proof: I do not see the ...
3
votes
1answer
48 views

Asymptotics for average of Fourier coefficients of cusp form

Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function $$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$ of a weight $k$ ...
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0answers
16 views

How to bound $\sum_m e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}$

I have to estimate the following sum $$ \sum_{m=0}^{\frac{2}{\log 2}\log\log\log T}e^{\mathcal{O}(k2^{m/2})-2^{3m/4}}. $$ I would like to show that this sum is $$ \ll_k 1 $$ and if possible that it ...
0
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1answer
44 views

$N$ birds are distributed on a telephone wire

$N$ birds are distributed on a telephone wire that can fit a maximum of $2N$ birds. The spacings between birds form a sequence $S$. The minimum space between birds is $1$ unit. The sequence is ordered ...
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0answers
34 views

Is there an example of a non-trivial function with a “nice” closed form expression that can be expressed as a Dirichlet series

Most examples of functions that can be expanded in Dirichlet series in my textbook involve the zeta function. I was wondering how the class of functions that can expanded in a Dirichlet series look ...
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0answers
25 views

complex integral with logarithmic derivative of $\zeta$

I want to prove that for any $x\geq 2$ we have $$ \begin{split} -\frac{\zeta^{\prime}}{\zeta}(s)&=\sum_{n\leq x}\frac{\Lambda(n)}{n^s}\frac{\log(x/n)}{\log x}+\frac{1}{\log x}\left(\frac{\zeta^{\...
0
votes
1answer
41 views

Confused as to how this step in a number theory proof is performed

How does this step $$D(q)=\sum_{n=1}^\infty d(n)q^n$$ Become this step? \begin{align} D(q) &=\sum_{n=1}^\infty\sum_{m|n}mq^n=\sum_{m=1}^\infty\sum_{m|n}mq^n \\ &=\sum_{m=1}^\infty\...
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0answers
51 views

Size of prime factors of $\text{gcd}(n,\phi (n))$

At this MathOverflow post there is an interesting comment below the O.P.'s question. The comment is due to Greg Martin and it basically reads that Allmost all integers have the property that $\text{...
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0answers
41 views

Can the prime number theorem be given in terms of d(x) or σ(x)?

Given according to Apostol's Introduction to Analytic Number Theorem p79, these relations satisfy equivalence to the Prime Number Theorem: $\lim_{x\to \infty}\frac{\pi(x)\ln(x)}{x}=1$ $\lim_{x\to \...
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0answers
25 views

Is this zeta related function surjective? [closed]

Let $\psi:[1,\infty[\to ]0,\infty[$ defined as $\psi(t)=|\zeta(1+it)|$, $t\geq 1$. Is $\psi$ surjective? Or there exist $t_0>0$ and $t_1>1$ for which $\psi:[t_1,\infty[\to ]t_0,\infty[$ is ...
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0answers
17 views

Recurrence relation for partition function for pentagonal numbers.

I know the following theorems. Theorem 1 $:$ For $|x|<1$ we have $$\prod\limits_{k=1}^{\infty} \frac {1} {1-x^k} = 1 + \sum\limits_{k=1}^{\infty} p(k)x^k.$$ Theorem 2 $:$ For $|x|<1$ we have $...
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0answers
28 views

Clarification of the proof of Euler's identity regarding the generating function for partitions.

In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
3
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1answer
64 views

First Fourier coefficient of weight $k$ holomorphic cusp form.

Let $f$ a weight $k$ holomorphic Hecke cusp form with $\|f\|^2=\langle f,f\rangle=1$ with fourier expansion $$f(z)=\sum_{r\geq 1}a_f(r)e(rz)$$ Let $\displaystyle\lambda_f(r)=\frac{a_f(r)r^{(-k+1)/2}}{...
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0answers
34 views

Is there a way to encode the prime counting function in the unit square?

Here's what I tried: Let $$\Psi(x)=\int_0^x e^{\frac{1}{\ln(t)}} dt $$ and consider a map that associates the logarithmic integral with $\Psi(x)$: $f: Li(x) \mapsto \Psi(x).$ See this question ...
4
votes
1answer
94 views

How to extend Euler's identity regarding partition on the unit disk?

Theorem (Euler) $:$ For $|x|<1$ we have $$\prod\limits_{m=1}^{\infty} \frac {1} {1-x^m} = \sum\limits_{n=0}^{\infty} p(n) x^n,$$ where $p(n)$ denotes the number of partitions of $n$ for $...
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0answers
64 views

Plot $g(x)=e^{\pi'(x)}$

Assuming the Riemann Hypothesis, $$\pi(x)=R(x)-\sum_\rho R(x^\rho),$$ where $\pi(x)$ is the prime counting function and $R(x)$ is the Riemann prime counting function. What does a plot of $g(x)=e^{\...
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2answers
47 views
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0answers
53 views

Maynard-Tao vs GPY sieve weights for bounded gaps between primes

I'm trying to understand why the Maynard-Tao weights allow one to obtain bounded gaps between primes whereas GPY fail The usual response is the additional flexibility of allowing the sieve weights to ...
1
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1answer
58 views

Real values of $\frac{\zeta(2 s)}{\zeta(s)}$

If $\frac{\zeta(2 s)}{\zeta(s)}$ is a real number, then must $s$ be real ?
4
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2answers
83 views

Asymptotic expansion of $Li^{-1}$ and zeros of $F(s)$ and $G(s)$

If you downvote please leave some constructive feedback. I would like to compare and visualize/gain insight about the zeros of two functions, $F(s)$ and $G(s).$ $\pi(m)$ is the prime counting ...
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0answers
40 views

Stirling's formula for normalized $\Gamma$

Let $$ H(s)=\frac{1}{2}s(1-s)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right). $$ Using Stirling's approximation for the Gamma function I would like to prove that $$ \frac{H(1/2+it)\overline{H}(1/2+it+iu)}{\...
1
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0answers
52 views

Generalising the Dirichlet L-Series

An MSE user and myself were discussing the Dirichlet L-function from one of their posts when we thought of a generalisation to the Riemann Hypothesis. The Dirichlet L-series is defined by $$L(s,\chi)=...
4
votes
1answer
231 views

Riemann Zeta function with the prime counting function in place of $n$

Interested in the following function: $$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}=\sum_{n=1}^\infty \frac{\lambda_n}{n^s}, $$ where $\pi(n)$ is the prime counting function. Was thinking about: ...
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0answers
31 views

Meaning of absolute constant

In my analytic number theory course, we discussed Dirichlet Series of Dirichlet Characters. A question I was asked was to show that: If $\chi_1, \chi_2$ are two Dirichlet Characters of modulo $q_1, ...
1
vote
1answer
32 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
4
votes
2answers
73 views

Estimating $\sum\limits_{d\mid n}{d+a\choose b}$

Is there any way of estimating a sum like $$\sum_{d\mid n}{d+a\choose b},$$ for positive integers $a$ and $b$? For example, in the OEIS we find that $$\begin{align*} \sum_{d\mid n}{d+1\choose 2} &...
4
votes
1answer
52 views

Density of primes of the form $x^2+my^2$

I was playing with numbers and have the nice conjectures: Let $m$ be a fixed positive integer, and $\pi(N)$ denote the numbers of primes not exceeding $N$ and $\pi_m(N)$ denote the number of prime ...
3
votes
1answer
97 views

Sums of the form $\sum\limits_{a_1=1}^n\sum\limits_{a_2=1}^{a_1}\cdots \sum\limits_{a_r=1}^{a_{r-1}}{[\gcd(a_1,a_2,\dots ,a_r)=1]}$

Background: Consider sums of the form $$\sum_{a_1=1}^n\sum_{a_2=1}^{a_1}\cdots \sum_{a_r=1}^{a_{r-1}}{[\gcd(a_1,a_2,\dots ,a_r)=1]},$$ with $[\gcd(a_1,a_2,\dots ,a_r)=1]$ being equal to $1$ if the ...
0
votes
1answer
25 views

Zeta function of the hypersurface of some homogeneous polynomial

Let $f(y)\in \Bbb Z_p[y_0,y_1,....,y_n]$ be a homogeneous polynomial. Let $N_s$ be the number of zeros of $f$ in $\Bbb P^n(F_{p^s})$. Here, $\Bbb P^n(F_{p^s})$ denotes the $n$-th projective space ...
4
votes
1answer
78 views

How to estimate $\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}$?

How to estimate $$\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}, \qquad\qquad(1)$$ where $p$, $q$ are prime numbers. We have the Mertens' formula $$ \sum_{p\leqslant x} \frac{1}{p} = \log\log ...
1
vote
1answer
31 views

Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
0
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2answers
60 views

Using Dirichlet's theorem to show existence of number coprime to $n$

I have the following question: Let $n$ be a positive integer and $d$ be divisor of $n$. Use Dirichlet's theorem to show that there exists an integer $k$, where $1\le k\le d-1$ such that the number $m:=...
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0answers
23 views

Convergence region of local and global zeta functions

Let $\chi = \prod_{v} \chi_{v}: \mathbb{A}^{\times}/F^{\times} \to \mathbb{C}^{\times}$ be a finite order Hecke character and let $\Phi = \prod_{v} \Phi_{v}$ be a Schwartz function on $\mathbb{A}$. ...
1
vote
1answer
34 views

Counting number of ideals in quadratic number field

Let $K$ be a quadratic number field and $R$ be its number ring, and if $a(n)$ denotes number of ideals of norm $n$, if $n$ is a prime number, then number of ideals of norm $n$ is $1+(d|n)$, where $d$ ...
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0answers
34 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
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0answers
22 views

Bound for sum of squares $r_{2k}(m)$ for $m \geq 1$

I happened to read in (Iwaniec-Kowalkski) Analytic Number Theory book that the Sum of Squares function satisfies the bound $r_{2k}(m) << m^{k-1+\epsilon}$ for $m \geq 1$. But $\epsilon$ is not ...
1
vote
1answer
71 views

The equation $\zeta(q)=0$ for $q$ a quaternion

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists ...
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2answers
33 views

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$.

Let $s_n$ denote the sum of the first $n$ primes. Prove that for each $n$ there exists an integer whose square lies between $s_n$ and $s_{n+1}$. I cannot give a proof to this, although I have try on ...
3
votes
1answer
57 views

Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
3
votes
1answer
46 views

the exponent of convergence of $\frac{p_{n+1}}{p_n}$ to $1$

Let $p_k$ be the $k$-th prime. Then $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$ -- this is well known. I was looking for more specific information: What is the exponent of convergence of $\frac{p_{n+...
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0answers
12 views

application of distribution of primes in arithmetic progressions

I try to understand an application of distribution of primes in arithmetic progressions Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$ So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23 ...
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0answers
63 views

On the proof that $\phi(n)/n$ has a limit law

In this question, $\mathbb{N}$ denotes the set of positive integers. Also, $\overline{\mathrm{d}}$, and $\mathrm{d}$ means upper natural density, and natural densitiy respectively. (They are the ...
2
votes
1answer
36 views

$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$

I wish to prove the following equality $$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$$ For some constant A Own work: Let $$A(x) = \sum_ {p \le x} ...
2
votes
2answers
42 views

$F(x) = L(1, \chi ) \log x + O(1)$

I wish to prove $$F(x) = L(1, \chi ) \log x + O(1)$$ when $A(n) = \sum_{d|n} \chi (d)$ and $F(x) = \sum_{n \leq x} \frac{A(n)}{n}$ I started of course by substituting $A(n)$ in $F(x)$, which becomes ...
0
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0answers
15 views

Effectiveness of Landau's Prime Ideal Theorem

Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same ...
2
votes
1answer
33 views

How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem: Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then $\lim_\limits{...
0
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0answers
50 views

“on average” in the Bombieri-Vinogradov theorem

TLDR: I don't understand the bit in bold, i.e. where in the formula is the average q? Thanks. The Bombieri-Vinogradov theorem states the following: For any $A > 0$ there exists a $B = B(A)$ such ...
0
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0answers
33 views

Bound on log integral

I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+...