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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.

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57 views

Proof claimed that almost all zeroes of the Riemann zeta function lie on the critical line

There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that $$\liminf_{t\to\infty} \frac{...
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17 views

Asymptotic Expression for the number of solutions to linear Diophantine equation.

Consider the the general linear diophantine equation $$\sum_{i=1}^{k}a_ix_i =n $$ with $a_i\geq 1, n\geq 1$ and $x_i\geq 0.$ Then the generating function that counts the number of solutions to this ...
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2answers
150 views

Closed form for $\sum\limits_{n=2}^{\infty}\frac1{n^3-1}$

I am investigating (just for fun) the sum $$S=\sum_{n=2}^{\infty}\frac1{n^3-1}$$ Wolfram Alpha gives me the 'value' $$S=-\frac13\sum_{\{\omega\,:\,\omega^3+6\omega^2+12\omega+7=0\}}\frac{\psi_{0}(-\...
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1answer
25 views

Understanding the error term in the Siegel‒Walfisz theorem

I'd like to inquire as to the nature of the error term in the Siegel‒Walfisz theorem. I understand that it takes the form $$ O\left( x \exp\left( -C \sqrt{\ln x} \right) \right). $$ Yet this function ...
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0answers
48 views

Truncated sum of divisors bound

I am interested in an upper bound for $$ \sum_{\substack{d|N\\ d>A}}\frac{1}{d^3},$$ in particular, I can get the above to be $$\sum_{\substack{d|N\\ d>A}}\frac{1}{d^3}\ll \frac{\text{exp}\...
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2answers
43 views

Bad approximation, alternative definiton

I read Complex dynamics of Carleson and Gamelin. They state without a proof the following $\theta$ is called bad approximate if there exists $c>0$ and $\mu<\infty$ such that $$ \Big\vert \...
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22 views

Goldbach's conjecture and convergence of a Dirichlet series

Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) : =\inf\{r>0, (n-r,n+r)\in\mathbb{P}^{2}\} $. The assumption of GC implies $ r_{0}(n)<n $. Let's now consider the series $ G(s) : =\...
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1answer
50 views

Let $f(x)$ be defined for all rational $x$ in $0\leq x\leq 1$

Let $f(x)$ be defined for all rational $x$ in $0\leq x\leq 1$ $$F(n)=\sum_{k=1}^n f\bigg(\frac kn\bigg), \quad F^* (n)=\sum_{k=1\\(k,n)=1}^nf\bigg(\frac kn\bigg).$$ Prove that $$F*=\mu * F$$ where $*$ ...
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1answer
58 views

Laurent Series of $\ln(|x|)$?

I read in this question that no Laurent series exists for $\ln(x)$, as Patch's answer shows: "The problem with $\log z$ in the complex plane is that it is a "multi-valued function", so we must ...
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1answer
53 views

Why we need to have $S(x)=O(x)$?

Currently, I'm working on problem 4.21 in Apostol's Analytic number theory and I solved that in different ways. The original problem is as follows: Given two real-valued functions $S(x)$ and $T(x)$ ...
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2answers
61 views

On the definition of Liouville number

Definition: (from Wikipedia) In number theory, a Liouville number is a real number $x$ with the property that, for every positive integer $n$, there exist integers $p$ and $q$ with $q > 1$ and ...
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61 views

What are the minor and major arcs in the Circle method (Hardy & Littlewood)?

I was wondering if somebody would be so kind as to graphically show what are the minor and major arcs in the Circle method? In this example, extract below, I've attempted a simple diagram to show ...
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23 views

Why is number of real characters mod $q$ a multiplicative function?

Let $R(q)$ be the number of real characters mod $q$. A character $\chi \mod q$ is called real if $\chi(a)\in\mathbb{R}$ for every $a\in \mathbb{Z}$, which means $\chi(a)\in\{-1,1\}$ for every $a\in\...
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1answer
57 views

Number of primes between $n$ and $2n$

What is a good lower bound on $\pi(2n)-\pi(n)$? Bertrand's postulate gives $1$. It is expected to be as I understand of form $\frac{c\cdot n}{\log n}$ from Prime Number Theorem. Does the ratio ...
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18 views

About the Möbius inversion formula and convolution product

In my analytic number theory book, I have the following exercise: Consider $F:[1,\infty)\rightarrow\mathbb{C}$ as any function and define $G(x):=\sum_{n\leq x} F(x/n)$ for $x\geq 1$. I have to prove ...
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111 views

Riemann $\zeta$ and Chebyshev's estimates

I would like to shred some light concerning the relations between properties of the Riemann $\zeta$ function and the weak version of the Prime Number Theorem proven by Chebyshev. More precisely, we ...
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1answer
59 views

Asymptotic formula for $\sum_{n\le x}\frac 1n$

Prove that $\displaystyle \sum_{n\le x}\frac 1n=\log (x+2)+O(1)$. We have, $\displaystyle \sum_{n\le x}\frac 1n=\log x +\gamma+O(1/x)$. Now if I can show that $\log(x+2)-\log x+O(1/x)=O(1)$ then I ...
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29 views

Some Trys related to Von Mangoldt function, What is $\Lambda * \Lambda$

Given Mangoldt function: $\Lambda(x)=\begin{cases}\ln p,&\text{if $x=p^k$}\\0,&\text{otherwise}\end{cases}$ I wonder what about $\Lambda* \Lambda$ where $*$ denotes the dirichlet ...
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Functional equation for $GL(3)\times GL(2)\times GL(1)$ L-functions

For two Maass forms $$f(z)=\sum_{n\neq 0}a(n)\sqrt{2\pi y}K_{v_1-\frac{1}{2}}(2\pi|n|y)e^{2\pi inx}$$ and$$g(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m=1}^{\...
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2answers
40 views

$\lim_{n \to \infty} \dfrac{e^{c \sqrt{\ln n . \ln \ln n}}}{n^{\epsilon}}=0$?

The general rule is discussed here but that doesn't solve my problem. I want to prove that $$\lim_{n \to \infty} \dfrac{e^{c \sqrt{\ln n . \ln \ln n}}}{n^{\epsilon}}=0$$ where $c>0$ a fixed ...
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21 views

Simplifying an Expression for an Arithmetic Function

Question "Evaluate" or express $g_k(n)=\sum_{d\mid n,\,(d,k)=1}\mu(d)$ (where $k\in\mathbb{N}$ is fixed) in terms of elementary arithmetic functions. My attempt Using the fundamental property ...
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1answer
67 views

Why is the Euler product expected to play a role in a solution of the Riemann Hypothesis?

The Riemann Hypothesis is the statement that the Riemann zeta function $\zeta(s)$ does not vanish for $1/2<\Re(s)<1$. $\zeta(s)$ can also be expressed by the Euler product over primes $$\zeta(s)=...
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1answer
31 views

Is $e^{\frac{\vartheta(x)}{x}}\ll_{\varepsilon}x^{\varepsilon} $?

The question is in the title : let $ \vartheta(x) : =\displaystyle{\sum_{p\leqslant x}\log p} $ denote the first Chebyshev function. Is it true that $ \forall\varepsilon>0,e^{\frac{\vartheta(x)}{...
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0answers
43 views

range of sum in “the polynomial $X^2+Y^4$ captures its prime”

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes". At page 997, just below equation (12.7) we start estimating the following sum $$ V(f,g)=\sum_d f(...
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2answers
136 views

Let $g(k)$ be the greatest odd divisor of $k$ show that $ 0< \sum_{k=1}^n \frac {g(k)}{k} - \frac {2n}{3} \lt \frac 23$

Prove that for all positive intergers $n$, $$ 0< \sum_{k=1}^n \frac {g(k)}{k} - \frac {2n}{3} < \frac {2}{3}$$ Where $g(k)$ denotes the greatest odd divisor of $k$. Here's my try: ...
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33 views

a question about the notation in the book “Opera de Cribro”

When I study the book "Opera De Cribro" by John Friedlander, Henryk Iwaniec-(2010), in Sections 1.2 and 1.3, I confused with notation used there. In page 3 it is defined: $$\cal{A} = (a_n) , n\le x$$,...
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22 views

Coefficient of Maass cusp forms are bounded

Let $$\phi(z)=\sum_{\gamma\in U_2(\mathbb{Z})\backslash SL(2,\mathbb{Z})} \,\,\,\,\,\sum_{m_1=1}^{\infty}\,\,\sum_{m_2\neq 0}a(m,n)W_{\text{Jacquet}}\left(\begin{pmatrix} |m_1m_2| & & \\ &...
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0answers
13 views

Means of powers of the zeta function

It is well known that the Lindel\"of Hypothesis is equivalent to the statement that $$\frac 1T\int_0^T|\zeta(1/2=it)|^{2k} =O(T^\epsilon)$$ for all positive integers $k$ and all positive real $\...
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1answer
477 views

Why couldn't Baez-Duarte prove the Riemann Hypothesis?

Define \begin{equation} I_n=\int_{0}^{1/n} |U s_{n}(x)|^2 \mathrm{d}x \end{equation} where $Us_{n}(x)=\frac{1}{x}\sum_{j=1}^{n} \frac{\mu(j)}{j}\rho(jx), \mu$ denotes the Mobius function and $\rho(y)...
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29 views

On Ramanujan's proof of Bertrand's postulate (using stirling formula)

I'm trying to understand Ramanujan's proof of Bertrand's postulate, but I don't get the step in which it says But is easy to see that $\log\Gamma(x) - 2\log\Gamma(\frac{1}{2}x + \frac{1}{2}) \...
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1answer
83 views

A line integral involving $\zeta(s)$

Consider the line integral $$I=\int_{1/2 -i\infty}^{1/2 + i\infty} \frac{\log((s-1)\zeta(s))}{s} \mathrm{d}s-\int_{1/2 -i \infty}^{1/2 + i\infty} \frac{i\arg \zeta (s)}{s}\mathrm{d}s$$ where $\zeta$ ...
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1answer
28 views

How correct an estimate for twin primes obtained from the Rosser-Iwaniec sieve and Prime number theorem for arithmetic progressions could be?

Let $P$ be the set of primes $p$, $x \geq D \geq z^2 \geq 2$, and let $A⊂[1,x]$ be a set of integers. Suppose $A_{d}=|A| \frac{v(d)}{d}+R_{d}$ for square free d with $v$ being multiplicative and $v(p)$...
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1answer
21 views

Reformulating Theta Function Symmetry as a Modular Form

If $\theta$ is the Jacobi theta function $\theta(\tau) = \sum e^{\pi i n^2 \tau}$, then $\theta$ satisfies the Modular symmetries $\theta(\tau + 2) = \theta(\tau)$ and $\theta(-1/\tau) = \sqrt{-i \tau}...
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1answer
52 views

No elliptic curves over $\mathbb{Q}$ with everywhere good reduction

I'm trying to prove that there aren't any elliptic curves $E$ over $\mathbb{Q}$ with everywhere good reduction. I first suppose that $\Delta = \pm 1$ and am trying to reduce the quantities for $c_{4}, ...
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0answers
33 views

Might there be a Skewes number for semiprimes?

Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), ...
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1answer
48 views

On the behavior of $|\zeta(1/2 + it)|$ for $t\in(-14,14)$

Denote by $\zeta$ the Riemann zeta function. Since $\zeta(1/2 + it)\neq 0$ for $|t|\leq 14$, it seems to follow that $|\zeta(1/2 + it)|$ is either strictly decreasing or strictly increasing on $(-14,...
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1answer
60 views

Is this zeta-type function meromorphic?

In An older question I asked : ( See A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH) ) —— Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ ...
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0answers
19 views

Abscissa of absolute convergence for a particular Dirichlet series

For $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdot\cdot\cdot p_k^{\alpha_k}$ we denote $\alpha(n)=\alpha_1\alpha_2\cdot\cdot\cdot\alpha_k$. Show that $F(s)=\sum_{n\geq 1}\frac{\alpha(n)}{n^s}$ is absolutely ...
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36 views

Question on page 432 of Iwaniec-Kowaleski's book

Suppose $f$ is any function supported on $[w/v,xv]$ which is continuous, bounded and piecewise monotonic. Then, why do we have $$\sum_{n \equiv \alpha \mod q} f(dn) (dn)^{-s}=\frac{F(s)}{dq} +O(|s| \...
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0answers
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Proving $\left|\sum_{n>x/m} \frac{ \chi(n)}{ n^{s}} \right| \leq 2q |s| (m/x)^{\sigma}$

$\text{Show}\:\displaystyle\left|\sum_{n>x/m} \frac{ \chi(n)}{ n^{s}} \right| \leq 2q |s| \left(\frac mx\right)^{\sigma}$ where $s=\sigma +it$. Here is what I tried: $\displaystyle\left|\sum_{n&...
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10 views

What does the following distribution look like?

I want to pick $x+\sqrt{-1}y$ at random with respect to the hyperbolic measure from the standard domain $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ for $SL_2(\mathbb Z)\backslash\mathbb H$. What does ...
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0answers
20 views

upper bound for $L(1,\chi^2)$

Suppose $\chi$ is a complex number. We know $\chi \bar{\chi}=\chi^2.$ Can anyone give me a reference for an upper bound for $L(1,\chi^2)?$ I know there exists upper bounds for $L(1,\chi),$ but I ...
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1answer
40 views

Unique mixed base number representations?

I would like to represent $\pi$ in an arbitary mixed base. Is this possible to do formulaically and uniquely, before knowing the base of the $n^{th}$ digit? For example, say the number is to be ...
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0answers
27 views

$\zeta(1+it)=0$ implies sum of prime if finite?

Can anyone explain why $\zeta(1+it)= \sum_{n=1}^{\infty} \frac{1}{n^{1+it}}=0$ will imply $\sum_{\text{p prime}} \frac{1-\Re(-p^{-it})}{p}< \infty?$ Here is what I have so far: We know $\zeta(s)$...
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0answers
32 views

Prove $π(x+y)- π(x) \ll \frac y{\log (\log (y))}$ using Legendre sieve such that $10 ≤ y ≤ x$

How to prove $$π(x+y)- π(x) \ll \frac y{\log (\log (y))}$$ using Legendre sieve such that $10 ≤ y ≤ x$?
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0answers
52 views

Riemann zeta zeros Fourier like divergent square wave. Can you complete this analogy?

The question is to complete this analogy: $$\left|Z(t)\right|=\left|\zeta \left(\frac{1}{2}+i t\right)\right| \tag{1}$$ is to: $$Z(t)=e^{i \vartheta (t)} \zeta \left(\frac{1}{2}+i t\right)...
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1answer
122 views

Question in Iwaniec-Kowaleski's book : bound for a twisted series

I am currently reading Iwaniec-Kowaleski's book on Analytic Number Theory. My question is on page 431. Here is what it says: For any $\chi \mod q$, we have the twisted series $K(s,\chi)=\sum_{n=1}^\...
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1answer
81 views

Show that the natural density is $1/2$.

Let $$A_b= \left\lbrace{p \in \mathbb{P}| \left(\frac{b}{p}\right)=1 } \right\rbrace $$ and $$ \nu(A_b)=\lim\limits_{x \to \infty} \frac{\#\lbrace {p \in A_b|p\le x}\rbrace}{\pi(x)}$$ the natural ...
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0answers
39 views

What are the Fourier Transform of Dirichlet L-function?

We know that the Fourier transform for Riemann $\Xi$ ${\displaystyle \Xi (t)=\xi ({\frac 12}+it) }$ where: $\xi (s)={\tfrac {1}{2}}s(s-1)\pi ^{{-s/2}}\Gamma \left({\tfrac {1}{2}}s\right)\zeta (s)...
4
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1answer
53 views

Convexity Bound of Rankin-Selberg L-Function

Let $f,g$ be primitive modularforms of arbitrary levels $N_1,N_2$, trivial nebentypus and same weight $k$. Let $L(f\otimes g,s)=\zeta(2s)\sum_{n\geq1}\frac{\lambda_f(n)\lambda_g(n)}{n^s}$ be the ...