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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Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?

The Legendre Chi function $\chi_2(z)$ is define as $$ \chi_2(z) = \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^2} $$ for $-1 \le z \le 1$. But $z > 1$ the series diverges. For real value of $z$ is ...
Nilotpal Sinha's user avatar
1 vote
2 answers
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Additive characters over a Number Field

Given any non-zero integral ideal $\mathfrak{b}$ of $K$, an additive character modulo $\mathfrak{b}$ is defined to be a non-zero function $\sigma$ on $\mathfrak{o}/\mathfrak{b}$ which satisfies $$ \...
zero2infinity's user avatar
1 vote
0 answers
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Density of squares using large sieves

I am reading Serre's Lectures on the Mordell-Weil Theorem, where he specifically talks about a Large Sieve inequality and proceeds to give an example. Theorem. (Section 12.1) Let $K$ be a number ...
Batrachotoxin's user avatar
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24 views

How can I show that $\dfrac{1}{\sin t}-\dfrac{1}{t}$ is bounded for $0<t\leq\dfrac{3}{4}\pi$? [duplicate]

How can I show that $\dfrac{1}{\sin t}-\dfrac{1}{t}$ is bounded for $0<t\leq\dfrac{3}{4}\pi$ ? I could calculate that $\dfrac{d}{dt}\Big(\dfrac{1}{\sin t}-\dfrac{1}{t}\Big)=\dfrac{t^2\cos t-\sin^2t}...
mathstudent's user avatar
1 vote
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Explicit upper bound on $\pi(n)$ (Weak version of PNT)

Chebyshev first proved that there exist constants $a$, $b$ such that $$ a \frac{n}{\log n} < \pi(n) < b \frac{n}{ \log n}.$$ The proof is well understood, and relies on elementary techniques. ...
Navvye's user avatar
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I can't prove that the next dilichlet kernel converges on the boundary. [closed]

I can't prove $\displaystyle\int_{0}^1 \Big(y-[y]-\dfrac{1}{2}\Big)\dfrac{\sin(\pi(2m+1)(y-x))}{\sin(\pi(y-x))}dy-\displaystyle\int_{0}^1 \Big(y-[y]-\dfrac{1}{2}\Big)\dfrac{\sin(\pi(2m+1)(y-x))}{\pi(y-...
mathstudent's user avatar
2 votes
1 answer
63 views

Meromorphic continuation of Euler product

Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$ Longer version: I realise I have some misconceptions about ...
tomos's user avatar
  • 1,682
1 vote
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Convergence of Riemann zeta function [duplicate]

I am wondering whether the series $$\zeta(s) = \sum_{n=1}^\infty n^{-s}$$ converges for $s$ with $\mathsf{Re}(s)=1$ and $\mathsf{Im}(s) \neq 0$. Note that I use the representation of $\zeta$ as an ...
Leif Sabellek's user avatar
1 vote
1 answer
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a small doubt in the proof of the quantitative form of the prime number theorem

I have been studying the proof of the prime number theorem in the quantitative form as in Theorem 6.9 of Montgomery & Vaughan's book "Multiplicative Number Theory, which focuses on proving ...
Josh's user avatar
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Exercise 1 Section 9.2. Montgomery/Vaughan's Multiplicative NT

I am trying to show that for any integer $a$, $$e(a/q) = \sum_{d|q, d|a} \dfrac{1}{ϕ(q/d)} \sum_{χ \ (mod \ q/d)} χ(a/d) τ(χ).$$ First I considered the case $(a,q)=1$ and the mentioned equality holds. ...
Ali's user avatar
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1 answer
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On $- \dfrac{\zeta'}{\zeta} (1-c+it) - \dfrac{\zeta'}{\zeta} (c-it)$

In Gonek's paper the following is claimed: $$ - \dfrac{\zeta'}{\zeta} (1-c+it) = \dfrac{\zeta'}{\zeta} (c-it) + \log \dfrac{t}{2 \pi} + O(\dfrac{1}{t}) \tag{$*$} $$ According to Titchmarsh's book Ch4,...
Ali's user avatar
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1 answer
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Show that $| \sum_{0 < \gamma \le T} x^{\rho} | \gg \sum_{0 < \gamma \le T} x^{\beta}$.

In Gonek's paper it is claimed without proof the following: Let $\rho = \beta + i \gamma $ be zeros of Riemann zeta function and $\beta_T = \max_{0 < \gamma \le T} \beta $. Then we deduce from ...
Ali's user avatar
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3 votes
0 answers
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A natural intractable geometrically inspired double-sum

Consider the following geometric setup: at every $(m,n)$ with $m$, $n$ both even, place a unit circle. Then we invert this entire setup through the unit circle at the origin. How much of the area of ...
Robert's user avatar
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1 vote
0 answers
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Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
tomos's user avatar
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1 vote
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disk of convergence of composition of p-adic functions

I watched a video that sketched out the regions of convergence for both the p-adic logarithm and the p-adic exponential functions. I thought about all this and asked myself: How would I find the ...
John Zimmerman's user avatar
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0 answers
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analytically continue functions on independent planes in $\Bbb R^3$

I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on ...
John Zimmerman's user avatar
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Estimate the order of restricted number partitions

There are $k$ integers $m_l,1\leq l\leq k $(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$. I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$. I came ...
Trinifold's user avatar
2 votes
0 answers
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Introductory books in Random Matrix Theory for Riemann zeta function

In many papers when studying Riemann zeta function (like Alternate Hypothesis, or pair correlation conjecture) I faced with terms like "gaussian unitary ensemble" or "random matrix ...
Ali's user avatar
  • 193
2 votes
1 answer
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Proving Euler product related to Riemann zeta function

Let $\omega(n)$ denote the number of prime factors of a positive integer $n$. Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}\end{equation} My ...
alidixon222's user avatar
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70 views

Applications, Generalisations and developments of Green-Tao Theorem after 2018

The well-known Green-Tao Theorem is definitely one of the most striking results among different area of Mathematics such as: Number Theory, Combinatorics, Graph Theory, Ergodic Theory,... etc. https://...
Neil hawking's user avatar
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Is $\phi_0$ equivalent to the Riemann hypothesis?

This is an extension (and more distilled version) of Extension of PDE's to critical strip, with new information. I am fairly sure that my constructions are an alternate description of the De Brujn ...
John Zimmerman's user avatar
1 vote
0 answers
42 views

How to correct the error between $\log(x!)\approx x\sum_{n\leq x}\delta(n)\frac{\log(n)}{n}$, where $\delta(n)$ is the density of primes near x?

Well assuming that the Prime Number Theorem is true, when substituting $\delta(x)$, the density of primes near $x$—which I am being vague of what it means 'cause I don't have enough foundation about ...
Mina Basilious's user avatar
6 votes
4 answers
279 views

Limit of lacunar power series in $1^-$.

Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series $$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$ Can any real number in $[0,1]$ be ...
Tuvasbien's user avatar
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17 votes
3 answers
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One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational

I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
Max's user avatar
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1 answer
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Probabilistic prime number theorem

The Cramér random model for the primes is a random subset ${{\mathcal P}}$ of the natural numbers with ${1 \not \in {\mathcal P}}, {2 \in {\mathcal P}}$, and the events ${n \in {\mathcal P}}$ for ${n=...
shark's user avatar
  • 833
2 votes
1 answer
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Value of a Sum linked with Beta Dirichlet Function and Zeta Function

Recently, I tried to calculate this double sum: $$ F(s) = \sum_{(a,b) \in \mathbb{Z}^2 \backslash (0,0) } \frac{1}{(a^2 + b^2)^s}$$ For $ s \in \mathbb{C}, Re(s) > 1$ I think i found the value of ...
Azoth's user avatar
  • 31
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0 answers
31 views

Bound for exponential

Let $\displaystyle f(x)= e^{-Ae^x} e^{Bx} \left( 1+ \mathcal{O} \left( e^{-x} \right)\right)$, where $A$ and $B$ are constants. Claim: $|f(x)| = \mathcal{O}\left( \exp(-e^x)\right)$ as $x \rightarrow ...
Richard D's user avatar
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3 votes
0 answers
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A question on Beukers proof of irrationality of $\zeta(3)$

I am reading the paper A note on the Irrationality of ζ(2) and ζ(3) by F. Beukers. In equation $(7)$, we have $$I_n=\int_{(0,1)^3}\frac{x^n(1-x)^ny^n(1-y)^nw^n(1-w)^n}{(1-(1-xy)w)^{n+1}} dx dy dw \tag{...
Max's user avatar
  • 546
4 votes
0 answers
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A problem regarding lcm of the first few consecutive natural numbers.

Let $n$ be a positive integer, and $f(2n+1)$ be equal to $lcm[1,2,\ldots,2n+1]$. I am supposed to show that $\log(f(2n+1))\ge 2n\log 2$ by using the fact that $e^{f(2n+1)}\int\limits_{0}^{1} x^n(1-x)^...
ShyamalSayak's user avatar
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0 answers
77 views

Number theoretic partial differential equation

Consider the equation $$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$ where $\...
John Zimmerman's user avatar
4 votes
1 answer
269 views

Newman's Short Proof of Prime Number Theorem

I'm going through the paper of D. Zagier on Short Proof of Prime Number Theorem. There it says in V that $\Phi(s)=\int_1^\infty \frac{d\vartheta(x)}{x^s}$ . Can someone please explain in details why ...
Sagnik Dutta's user avatar
0 votes
2 answers
48 views

Counting odd smooth numbers

Let $P(n)$ be the largest prime factor of $n$, and let $\Psi(x,B) = |\{ n \mid n \leq x \wedge P(n) \leq B\}|$. (This is a well-studied function in analytic number theory.) Define $\Psi'(x,B) = | \{ n ...
user432944's user avatar
2 votes
1 answer
37 views

Number of primitive Dirichlet characters of certain order and of bounded conductor

Writing $q(\chi)$ for the conductor of a Dirichlet character $\chi$, one can show using Mobius inversion that $$\#\{\text{$\chi$ primitive Dirichlet characters}\,:\,q(\chi)\leq Q\}\sim cQ^2.$$ My ...
user avatar
0 votes
0 answers
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Proving a Dirichlet series relating to the zeta function [duplicate]

Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{d(n^2)}{n^s}=\frac{\zeta(s)^3}{\zeta(2s)},\end{equation}where $d(n)$ denotes the number of divisors of $n$. My solution: Observing that we have a ...
alidixon222's user avatar
0 votes
1 answer
64 views

Zeros of L function on the 0.5 line

Could someone please tell what results are known about the zeros of L function, $L(s,\chi)$ on the 1/2 line, where $\chi$ is a character mod $q$? Is there an upper bound for this count when we count ...
math is fun's user avatar
  • 1,142
2 votes
0 answers
29 views

Important Subgroups of Arithmetical Functions [closed]

I am taking a course in Analytic Number Theory. The main object of study is arithmetical functions. Moreover, if we look at the arithmetical functions which do not vanish at $1$, then they form a ...
ALNS's user avatar
  • 457
1 vote
2 answers
125 views

Dyadic sum of $\frac{x}{\ln x}$ (i.e. dyadic asymptotic for prime number theorem)

For $x\geq 10$, denote $j=j_x$ s.t. $4 \geq \frac{x}{2^j}\geq 2$. I want to prove that $$\sum_{i=0}^j \frac{x/2^i}{\log(x/2^i)} \sim \frac{2x}{\log x}$$ The reason I care is because the prime number ...
D.R.'s user avatar
  • 8,711
0 votes
1 answer
25 views

Zeros of L function on imaginary axis

Could someone please tell me what results are known about the zeros of L function, $L(s,\chi)$ on the imaginary axis, where $\chi$ is a character mod $q$? In particular, what is known about zeta ...
math is fun's user avatar
  • 1,142
2 votes
0 answers
70 views

Best explicit bound(s) for count of smooth numbers?

Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function in analytic number theory, and there is a large ...
user432944's user avatar
3 votes
0 answers
145 views

Sequence of non-extreme digits of power sequence must be uniformely distributed

Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits can be settle to the following: The last $k$ digits of $a^n$ are given by $a^n\...
Alma Arjuna's user avatar
  • 3,801
0 votes
0 answers
36 views

Conductor of a character

We know, a character $\chi$ mod $q$ is induced by a primitive character $\chi^{*}$ mod $q^{*}$. I have the following questions. If $\chi$ is non principal, can $\chi^{*}$ be principal? Let $\chi$ be ...
math is fun's user avatar
  • 1,142
2 votes
1 answer
92 views

Real non-trivial zeros of Riemann zeta function inside critical strip

Could someone please tell me what is known about the real non-trivial zeros of Riemann zeta function inside critical strip? Do we know there is none? I want to know what is known about the above ...
math is fun's user avatar
  • 1,142
0 votes
0 answers
40 views

Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two be derived?

After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations: with $M$ the KummerM confluent hypergeometric funcion and ...
Agno's user avatar
  • 3,181
0 votes
2 answers
106 views

Proper Way to Calculate Value of Riemann Zeta function?

I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in. I've been looking at one of the Analytic Continuations of the Zeta ...
Sbsty 's user avatar
0 votes
1 answer
45 views

A Question on the Bound of a Characteristic Function

In the proof of the Erdös-Kac theorem in section $III.4$ of $\mathit{Introduction\ to\ Analytic\ and\ Probabilistic\ Number\ Theory}$ by Gérald Tenenbaum, we need to bound the characteristic function $...
AHanley's user avatar
3 votes
0 answers
113 views

Visual proof of $\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1 $

Background Let $\zeta(\cdot)$ be the Riemann zeta function. I'm looking for a visual proof of the infinite series identity $$\sum_{n=2}^{\infty} \left( \zeta(n)-1 \right) = 1. \tag{1}\label{1}$$ This ...
Max Muller's user avatar
  • 7,048
1 vote
1 answer
70 views

Dirichlet series and Euler product

For a multiplicative function $f$, show that we have \begin{equation}\sum_{n=1}^{\infty}\frac{f(n)}{n^s}=\prod_p\left(\sum_{\nu=0}^{\infty}\frac{f(p^{\nu})}{p^{\nu s}}\right).\end{equation} My ...
turkey131's user avatar
  • 123
4 votes
0 answers
146 views

Probability that one random number among many has a unique prime factor

If I sample $N+1$ integers $x, x_1, \ldots, x_N$ uniformly and independently from $\{1, \ldots, M=2^k\}$, what is the probability that $x$ contains a prime divisor that does not divide any of the $\{...
user432944's user avatar
0 votes
1 answer
44 views

On the remainder term in selberg's proof of prime number theorem

Here is an explanation of selberg's proof of prime number theorem. Here by using mertens theorem they show that, $$\sum_{k≤n}\frac{R(k)}{k^2}=O(1)$$ And then proceed to show that, $$\left|\frac{R(y)}{...
RAHUL 's user avatar
  • 1,511
3 votes
1 answer
56 views

Question about the distribution of primes in Davenport's Multiplicative Number Theory

The following comes from Page 57 of Davenport's Multiplicative Number Theory. My question is: why is (6) being convergent necessary in deducing the result from (5)?
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