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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18 views

Assume Elliott-Halberstam conjecture, then prove a ANT inequality.

Here is Elliott-Halberstam conjecture. $$ \sum_{q\leqslant x^{1-\varepsilon}}{\underset{y\leqslant x}{\max}\underset{\begin{array}{c} 1\leqslant a\leqslant q\\ \left( a,q \right) =1\\ \end{array}}{\...
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20 views

Resources for asymptotic analysis in analytic number theory?

Currently, I'm looking for some good resources on the methods of asymptotic analysis that can be applied in analytic number theory. So far, I've found books on asymptotic analysis (e.g. the book by de ...
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34 views

Why is $\lim_{x\rightarrow \infty} \sum_{x<n\leq\infty}a_{n}n^{-s}= 0$ a sufficient condition for convergence?

Assume the following is true $$\left|\sum_{x_{1}<n\leq x_{2}}\frac{a_{n}}{n^{s}}\right| \leq Kx_{1}^{-\sigma}$$ where $s=\sigma+it$ and $a_n$ are complex, and all other variables are real, and $\...
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77 views

Closed form of $\sum_{z=1}^{\infty}\dfrac{z^n}{\text{exp}(kz^m) - 1}$

Recently while working on some problems, I came across this beautiful infinite series: $$\sum_{z=1}^{\infty}\dfrac{z^n}{\text{exp}(kz^m) - 1}$$ where $m,n \in \mathbb{N} $. Does there exist a closed ...
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73 views

Prove that the sequence {$π(2n+1)!$} is equidistributed in the interval (0,1). [closed]

Let $n\in\mathbb{N}$. From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
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1answer
100 views

Using Abel's Identity to evaluate $\sum_1 ^N n^3$ [closed]

An exercise on the internet suggested it was informative to evaluate $\sum_1 ^N n^3$ using Abel's identity (Apostol Theorem 4.2), also known as Abel's summation formula (link). $$\sum_{x_{1}<n\leq ...
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97 views
+100

Amount of numbers that won't satisfy a list of residue classes

I've been working on functions of the type Px+a where P is a prime number, x is a positive integer and a is an integer in the residue class of P. Now consider a list of primes (2....P) with different ...
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17 views

Why is a theory of newforms in half-weight difficult? Why are $U$-operators used instead of $V$-operators?

Integer weight case: Modular forms in $S_{2k}(N) := S_{2k}(\Gamma_{0}(N))$ can come from lower levels, and we'd like to know when that happens. This is where we call a modular form $f$ "old" ...
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59 views

Why is Abel's Identity (Apostol Theorem 4.2) valid for complex functions?

Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below) $$ \sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt $$ to prove a result about complex ...
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23 views

Simple proof for horizontal zero density estimate for the Riemann-Zeta-function

Let $N_c(T)$ denote the cardinality of the set $\{\rho | \zeta(\rho)=0 \text{ and } \Re \rho > c \text{ and } 0 \leq \Im \rho \leq T\}$. I am trying to find a prove for the fact $$N_c(T) = o(T)$$ ...
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1answer
62 views

Examples of Theorems of the form “If a set could contain a prime, it will contain a prime”.

One of my favourite theorems in mathematics is Dirichlet's theorem on arithmetic progressions, which states that every set of arithmetic progressions $\{a, a+d, a+2d, \dots \}$ will contain a prime as ...
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19 views

Explicit computation of the factor in the interpolation formula for p-adic Rankin-Selberg L functions

I'm currently trying to understand the appendix of this article by David Loeffler https://arxiv.org/pdf/1704.04049.pdf, which consists in an explicit computation of the factor allowing the ...
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60 views

Approximation for the number of unordered $k$-factorizations of positive integer $n$

I am trying to find an approximation formula for the number of multiplicative partitions of $n$ with $k$ parts. I found that an approximation formula for the number of multiplicative partitions with ...
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1answer
44 views

$F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ s.t $F(s_k)=G(s_k)$ show that $f(n)=g(n)$

Let $F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ be two Dirichlet series which are absolutely convergent for $\Re(s)>a$ for some $a\in \Bbb R$. If there ...
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70 views

What is $\sum_{n=2}^{\infty} \frac{d(n)}{n(n-1)} $?

While trying to evaluate a multiple rational zeta series, the following sum came up: $$\sum_{n=2}^{\infty} \frac{d(n)}{n(n-1)} = \sum_{n=2}^{\infty} \Big{(} \frac{d(n)}{n-1} - \frac{d(n)}{n} \Big{)}. \...
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49 views

Applications of Mertens' Theorems [closed]

I'm doing research on Mertens' theorems for a presentation on the Euler-Mascheroni constant $\gamma$. My target audience is other 1st year bachelor students, so I don't plan to go in to the proof of ...
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1answer
95 views

what Is the derivative of logarithmic integral function? [closed]

I happened to learn about the logarithmic integral function. I was trying to find its derivative and most of my attempts went futile. So I want the derivative of $\operatorname{Li}(x)$. I tried to do ...
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1answer
113 views

Series sum of sin log and cos log [closed]

I was solving a series based question and I am now totally confused on how to proceed further to find the analytical values of $a$ and $b$ (where $a,b\in \mathbb{R}$) such that: $$ \sum_{k=2}^{\infty}...
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2answers
43 views

Elementary method to prove that $|\overline B_{2n}(x)|\le|B_{2n}|$?

I am currently trying to estimate the error term in Euler-Maclaurin summation formula and need to establish an upper bound for the periodic Bernoulli polynomial $\overline B_{2n}(x)=B_{2n}(x-\lfloor x\...
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1answer
56 views

Partial summation of $d(n)/(n-1) $

In the answers to this question, it is established that $$\sum_{n\leq x}\frac{d(n)}{n}=\frac{1}{2}(\log(x))^{2}+2\gamma\log (x)+\gamma^{2}-2\gamma_{1}+O\left(x^{-1/2}\right).$$ A related result can be ...
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1answer
67 views

Average order of $\phi(n)$

Theorem 3.7 of the book Analytic Number Theory by Apostol states: $$\sum_{n\le x} \phi(n)= \frac{3}{\pi^2} x^2 + O(x\log x)$$ and then it claims : Hence the average order of $\phi(n)$ is $\frac{3n}{\...
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60 views

Average Number of Small Divisors

I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
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1answer
93 views

Analogy to Legendre's formula for $N!$

Suppose $A$ is a product of primes $p\le x$. For any $m$ let $A(m)$ be the largest such $A$ dividing $m$. Proposition 1. For any $N$ sufficiently large, $$\prod_{i=1}^NA(i) < \prod_{p\le x}p^{\frac{...
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1answer
62 views

A question about proof of Ternary Goldbach Conjecture.

Let's recall: Von Mangoldt function $\Lambda$ is a following function: $$\Lambda: \mathbb{N}\rightarrow \mathbb{R}$$ $$ \Lambda(n) = \left\{ \begin{array}{ll} \log p & \textrm{if $n = p^k$ for ...
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16 views

How big is the number of representations of integers as a sum of prime numbers?

Let $r_k(N) :=|\{(p_1,p_2,...,p_k)\in\mathbb{P}^k:p_1+p_2+...+p_k = N\}|$ It is known that for $2n-1\ge 7$ we have $r_3(2n-1)>0$ I look for some increasing differentiable function $f:\mathbb{R}\...
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1answer
64 views

Asymptotic growth of summation of $\sum_{j\leq t}' 2^{\omega(j)}$ (restricting sum to a certain subset of $j$)

The following question came up in a thesis discussion I had with a student (undergraduate). I am a number theory researcher, but not within analytic number theory. One challenge I have faced is not ...
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2answers
125 views

Show that, $(-1)^{\mu(1)}+(-1)^{\mu(2)}+…+(-1)^{\mu(n)}<0$ and proof on conjecture of OEIS A209802

Following is an experimental math claim. We denote $\mu(a)$ as Möbius function Let $$F(a)=\sum_{i=1}^{a}(-1)^{\mu(i)}.$$ Can it be shown that for every positive integer $a$, $F(a)<0$? Table $$\...
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36 views

number of ways, n can be written as a product of k integers [duplicate]

For $k\ge2$, let $d_k(n)$ denote the number of ways of writing $n$ as a product of $k$ positive integers (so that $d_2(n) = d(n)$ where $d(n)$ counts the number of positive integral divisors of $n$). ...
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55 views

Dirichlet series of Bernoulli polynomials

I am interested in the Dirichlet series $$ \mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s} $$ where $B_k(x)$ is the $k$th Bernoulli polynomial and $\Re(s) > k + 1$. This converges for all $k$ ...
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1answer
118 views

Non trivial zeros of Riemann zeta function

The non trivial zeros of Riemann zeta function , x$\zeta(s)$ lies in the critical strip $0<\Re(s)<1$ Riemann Hypothesis states that all the zeros of Riemann zeta function, $\zeta(s)$ lies on ...
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46 views

Estimate the number of certain fractions

I am able to estimate the quantity $$ N(x) = \sum_{\substack{(a,b)=1\\1\leq ab\leq x}}1 = \sum_{n\leq x} \sum_{\substack{(a,b)=1\\ab=n}} 1. $$ For this, I started observing that the inner sum is $2^{\...
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31 views

Engel-form of Fibonacci sequences series

What is the exact (closed form) value of the expression, where $f_1=f_2=1$ and $f_{n+1}=f_n+f_{n-1}\forall n\ge 2$,$$\dfrac{1}{f_1}+\dfrac{1}{f_1f_2}+\dfrac{1}{f_1f_2f_3}+\cdots+\dfrac{1}{f_1f_2\cdots ...
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1answer
32 views

Asymptotics on certain number of polynomials

I want to count number of polynomials $f(x) = (ax - b) (cx - d)$ with $a, b, c, d \in\mathbb{Z}$ such that the each coefficients of $f(x)$ lies in $\{-L, -L+1, \ldots, -1, 0, 1, \ldots, L-1, L\}$ and $...
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37 views

Approximation of summation of $\log x-\log n$

I was trying to give an asymptotic calculation for $\sum_{n\leq x}(\log x-\log n)$, and the proof provided by the instructor is that $$ \sum_{n\leq x}(\log x-\log n)=(x\log x + O(\log x))+ (x\log x+O(...
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12 views

On an inequality related to Selberg's sieve

I am reading section 9.3 of GTM206, which is about the derivation and application of Selberg's sieve, and I was stuck at proving $$ \sum_{n\le x}\left(\sum_{d|(n,P_z)}\lambda_d\right)^2\le\sum_{d_1,...
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51 views

Density of primes is $1/\log x$

The prime number theorem states that $\lim_{x \rightarrow \infty} \pi(x)/Li(x) = 1$, if we denote $\pi(x) := \sum_{p \le x} 1$ for the prime counting function and $Li(x) = \int_2^x \frac{ dt}{\log t}$ ...
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1answer
81 views

Can sufficient high degree polynomial sequences contain infinitely many primes?

I have a conjecture: For any integer $N$, there exist an positive integer $n > N$ such that there exist a degree-$n$ polynomial $P(x)$ satisfying:the sequence $\left\{P(n) \right\}$ contains ...
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67 views

Conformal-onto vs Onto map.

In Balazard, Saias and Yor paper they assume that f is in the Hardy space $ H^p\mathbb{(D)}$ where $\mathbb{D}=\{z\in \mathbb{C}\mid |z|<1\}$. Let, $f^*$ denote the function defined almost ...
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1answer
39 views

A curious question about holomorphicity

Holomorphic functions don't necessarily have antiderivatives/primitives. For example, $\frac{1}{z}$ is holomorphic on the punctured plane $V:= \mathbb{C}- \{ 0\},$ but it does not have any ...
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1answer
149 views

Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
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17 views

Lower bound for the multiplicative order of a fixed integer $a$ modulo $n$, as $n$ grows large

Let $a \geq 2$ be fixed. Is there any good lower bound known for the multiplicative order $\text{ord}_n(a)$ of $a$ modulo $n$ (with $n,a$ relatively prime) as $n$ grows large? Clearly $\text{ord}_n(a)$...
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43 views

Given a fixed integer $a > 1$, what is the amount of integers $1 \leq n \leq x$ such that the multiplicative order of $a \pmod{n}$ is small?

Let $x > 1$ be a large real variable and let $a > 1$ be a fixed integer. For an integer $n$ coprime to $a$, let $\text{ord}_n(a)$ denote the multiplicative order of $a$ modulo $n$. Fix an $0 <...
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61 views

If, $0<r<1$ then prove that $|(1-r^2)^3F(r)|$ is bounded. [closed]

Let, $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta(s)$ denotes the Riemann zeta function. Let, $H^{p}(D)$ , $p>0$ denote the Hardy space. Then, $f\in H^{1/3}(D).$ So there exists a ...
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46 views

Measure $\mu\neq 0$ then $\lim_{r \to 1,r<1} f(r)=0$

I am reading the paper of Balazard ,Saias and Yor. Let, $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta(s)$ denotes the Riemann zeta function. Denote by $$\exp\left[\int_{-\pi}^{\pi}\frac{e^...
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0answers
92 views

Confusion about factorial $\dfrac{(N!)^2}{(2N)!}$

I am confused with the expression $\dfrac{(N!)^2}{(2N)!}$. Suppose we write $(2N)!=p_1^{\alpha_1}p_2^{\alpha_2}\dots p_\ell^{\alpha_\ell}$ and by Hardy-Ramanujan theorem that states that almost all ...
2
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1answer
39 views

Functional equation for Rankin-Selberg L functions in the imprimitive case

If $f$ and $g$ are primitive modular forms of characters $\chi$ and $\psi$, such that $\chi, \psi$ and $\chi * \psi$ are all primitive, then we have an explicit functional equation. This is proven in ...
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1answer
149 views

Measure $\mu$ is 0.

I am reading the paper of Balazard ,Saias and Yor. Let, $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta(s)$ denotes the Riemann zeta function. Denote by $$\exp\left[\int_{-\pi}^{\pi}\frac{e^...
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1answer
49 views

How to show an integral representation for the logarithmic derivative of the Riemann zeta function?

Prove that the logarithmic derivative of the Riemann zeta function $$-\frac{\zeta^{\prime}(s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^s}=s\int_{1}^{\infty}\psi(x)x^{-s-1}\, d\mathrm{x},$$ ...
3
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1answer
61 views

Show that the Kloosterman Sum $S(a,b,p^k)$ vanishes under the given condition.

Let $a,b,k\in \mathbb{N}$, $k>1$. $p $ be an odd prime number and $p$ does not divide $a,b$. The Kloosterman sum is given by $$S(a,b,p^k)=\sum_{x\in (\mathbb{Z}/p^k \mathbb{Z})^\times}e\Big(\frac{a ...
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1answer
34 views

$O_q(1)$ notation

I cant find anywhere what this notation is supposed to mean: $$\frac{L'(s,\chi_0)}{L(s,\chi_0)} = -\frac{1}{s-1}+O_q(1)$$ Where $\chi_0$ is the principal character mod $q$ What is $O_q(1)$ here? Im ...

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