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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functional equation that relates $\psi(x,\chi)$ and $\psi(x^{-1},\overline{\chi})$.
On Davenport's Multiplicative Number Theory, Page $68$, It tried to derive a functional equation that relates $\psi(x,\chi)$ and $\psi(x^{-1},\overline{\chi})$. However I am stuck. I tried using the ...
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Asymptotics for $g(n) = \sum_{k = 1}^{n - 1} {\frac{{\log (1 + p_k)}}{p_k}}$?
Let $p_k$ be the $k$th prime.
Now define $g(n)$ as
$$g(n) = \sum_{k = 1}^{n - 1} {\frac{{\log (1 + p_k)}}{p_k}}$$
What are the asymptotics for this $g(n)$ ?
The related sum
$$ \sum_{k = 1}^{n - 1} {\...
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Error in working out identity $\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}=\sum_{m\ge 1}\sigma_a(m)\sigma_b(m)m^{-s}$
I'm working through S.J. Patterson's "An introduction to the theory of the Riemann Zeta-Function" for fun, and trying to solve the exercises. However, I am already experiencing some trouble ...
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Distribution of primes and growth rate of $\sigma(n)/n$ over colossally abundant numbers
I am reading this paper written by Jeffrey C. Lagarias "An Elementary Problem Equivalent to the Riemann Hypothesis".
On page 5, it says "Fluctuations in the distribution of primes will ...
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All about Riemann Zeta Function
I am trying to understand the Riemann Zeta Function. Here is what I think. I would request to add other important details to this.
The Riemann zeta function is defined to be
$$\zeta(s)=\sum_{n=1}^{\...
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Connection and overlap between Analytic and Algebraic Number Theory
I was browsing on Part III guide to courses and found out that the course named Algebraic Number Theory covers topics that I would not expect them to be covered in an algebraic course. I am talking ...
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How do I find job postings for analytic number theory?
My question is as in the title. I am aware of how to find websites advertising them in the UK or in the states (e.g. jobs.ac.uk and mathjobs.org seem to cover a lot) but I can't really find a way of ...
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Complexity of computing Chebyshev's $\vartheta(x)$
I would like to compute Chebyshev's function
$$
\vartheta(x) = \sum_{p\le x} \log p
$$
to within an error of $o(\log x).$ (The estimate $\vartheta(x) \sim x$ has sqrt-error or so even under RH.) What ...
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Take any number N greater than 2P, will there be always be a number that isn't a multiple of 2,3...P between N and N+2P
I was working on the Bertrand's postulate(which states that there's always a prime between any integer n and 2n), and I wonder if it won't only work for 1,2,3...2P for set of primes (2,3,5...P), but ...
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Reference request for the zeros of the zeta function
Levinson proved that at least one-third of the zeros of the Riemann zeta function lies on the critical line. Conrey later improved it to two-fifth. I would like some recommendations for references on ...
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Gaussian Sums of a Dirichlet's Character
In Davenport's chapter 9, They defined
$$\tau(\chi)=\sum_{m=1}^{q} \chi(m) e_q(m)$$
Further if $(n,q)=1$, then we have that $$\chi(n)\tau(\overline{\chi})=\sum_{h=1}^{q} \overline{\chi}(m) e_q(nh)$$ ...
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Finding the value of the square of the Gauss sum [closed]
While finding the value of the square of the Gauss sum $G^2$, at some point we make a substitution as follows:
$$G^2 = \sum_{m_1=1}^{q-1}\sum_{m_2=1}^{q-1} \left(\frac{m_1 m_2}{q}\right) e_q(m_1+m_2) =...
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What is the probability that the greatest prime factor of a sequence of uniformly distributed integers increases?
Let $f_k(n), n = 1,2,3...$ be a sequence of random integer uniformly distributed in $[2,k]$ for some fixed $k \ge 3$. Let $l_n$ be the largest prime factor of $f_k(n)$. What is the probability that $...
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Analytic Number Theory - distribution of $x^2$ vs. the distribution of $x^2 - 2y^2$
My question originates from Rational Points on Elliptic Curves, (Silverman & Tate), though has little to do with elliptic curves.
In chapter $V$: Integer Points on Cubic Curves, section $3$ it ...
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How we can show $\Lambda (s,\chi) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$?
We know that the functional equation for Riemann-Zeta is $\psi(s)=\frac{1}{2}s(s-1)\pi^{-1/2s}\Gamma(1/2s) \zeta(s)$ and $\psi(s) < \exp(C|s| \log |s|)$ as $|s| \rightarrow \infty$. Does it make ...
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Show that a peiodic series of primes modulo 5 converges
I am asked to show that a complex series converges or diverges. (Based on the question I am asked, I assume it converges, but I am not too sure.) I have broken it down into four components:
The basic ...
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How to make an estimate for this integral?
Suppose that function $f$ satisfies following properties: $f'' \in \mathrm{C}[a,b]$, $\exists A,B\geq 1$ such that $f''(x)\geq 1/A$ and $|f'(x)|<D$ for all $x\in [a,b]$. Prove that $$\displaystyle \...
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Lower bound for divisor counting function
Let,
$$\tau(n)=\sum_{d|n}1$$
Be the divisors counting function.
Then is it true that,
There exists infinitely many $n$ satisfying,
$$\tau(n)>\left(\ln(n)\right)^{a}$$
Where $a\in[1,\infty)$?
My ...
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Evaluation of nontrivial zeros of $\zeta$ in explicit formulae
Im sure this question is completely trivial, but I just want to check my understanding:
For the various explicit formulae in Analytic Number Theory involving sums over the nontrivial zeros of $\zeta$ ...
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Is there a way to prove that $\left|\sum_{n \leqq x} \chi(n) e(\alpha n)\right| \ll_\epsilon \frac{x}{\log q}$ without using Burgess inequality? .
I am working in my undergrad thesis and I have came across the inequality $\left|\sum_{n \leqq x} \chi(n) e(\alpha n)\right| \ll_\epsilon \frac{x}{\log q}$, where $\chi$ is a dirichlet character ...
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What is the limiting mean value of the product of the exponents in the prime factorization of numbers?
Let $n = p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m}$ be the prime factorization of $n$ and let $f(n) = a_1 a_2 \cdots a_m$. Using a heuristic argument I am able to show that the mean value of $\lim_{n \to \...
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Meromorphic continuation of $\zeta(s)$ to ${\rm Re}(s)>0$.
Riemann zeta function is one of the most mysterious functions that we encounter in mathematics.We require a meromorphic continuation of this function to ${\rm Re}(s)>0$ in order to prove the prime ...
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modified riemann zeta function $\zeta ^*(s)$?
I remember there being a function $\zeta ^*(s)$ where
$$\zeta ^*(s)=\zeta (s), \ s\neq 1$$
$$\zeta ^*(1)=\gamma$$
but now I can't seem to find any record of it, does a function like this exist or am I ...
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What’s the best bound on the Dirichlet coefficients of $\zeta(s-1)^2/\zeta(s)$
We have $\frac{\zeta(s-1)^2}{\zeta(s)} = \sum\limits_{n\ge 1} \frac{a_n}{n^s}$, where $a_n = \sum\limits_{d|n} \mu(d) \sigma_0(\frac{n}{d}) \frac{n}{d} = \sum\limits_{d|n} \phi(d) \frac{n}{d}$. Here $\...
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What it means when automorphic function has " nebentypus" $\psi$ [duplicate]
I am reading a research paper and I am not able to understand what could be the definition of " nebentypus" here : " We study automorphic functions $f$ on $\Gamma= \Gamma_0(N)$ of ...
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Stronger result than bertrand's postulate
It is well known that there is a prime number between $n$ and $2n$ for all $n$. I decided to go deeper: is there a lower bound on the number of primes between $n$ and $2n$ for "large enough" ...
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Why does $a(p^k)\mapsto\frac1ka(p^k)$ send $\sum_n a(n)$ to $\ln\left(\sum_n a(n)\right)$
In a recent Numberphile video on the Chebychev bias, Grant Sanderson (of 3blue1brown fame) uses what I guess would be the following theorem:
Given a multiplicative function $a$ where $\sum_n a(n)$ ...
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Uniform convergence of $\sum\delta_n(s)$ using $|\delta_n(s)|\leq |s|/n^{\sigma+1}$ [duplicate]
In Complex Analysis by Stein and Shakarchi, on page 173 the authors state that
$\left| \delta_n( s) \right|\leq |s|/n^{\sigma+1}$
implies that the series $\sum\delta_n(s)$ converges uniformly on any ...
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Analytic continuation of Bessel series
What is the analytic continuation of $$f(x)=\sum_{n=1}^\infty\frac{2K_1(2\sqrt{n^x})}{\sqrt{n^x}}$$
where $K_1$ is a modified Bessel function of the second kind.
This converges for real $x>0.$
I ...
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Estimate for $\sum_{n > x} \frac{\mu(n)}{n^2}$
I am dealing with the term $\sum_{n \le x} \frac{\mu(n)}{n^2}$ (where $x \to \infty$). Here, $\mu$ denotes the Möbius function. Writing $D_\mu (s) = \sum_{n \in \mathbb{N}} \frac{\mu(n)}{n^s}$ for the ...
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Another Mellin transform question
Take $1\leq H\leq X$. Let $f(u)$ be a smooth function on $(X,2X)$ and satisfy there $f^{(j)}(u)\ll _jH^j/X^j$ for all $j$. Let $$\mathfrak g(s)=\frac {\Gamma (s/2)^3}{\Gamma ((1-s)/2)^3}\approx t^{3(...
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A property of simultaneous eigenforms
While reading the following theorem for Apostol's modular functions and dirichlet series in number theory, I have a question:
(Theorem 6.14, page 130)
Assume that k is even and $k\geq 4$. If the space ...
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What is the value of $L'(1,\chi)$ where $\chi$ is the non-principal Dirichlet character modulo 4?
I was trying to compute the following sum:
$$\sum_{n\le x}{\frac{r_2(n)}{n}}$$
where $r_2(n)=\vert\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\vert$. Using Abel's summation formula with $a_n=r_2(n)$, $\varphi(t)...
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Distribution of elements in a Reduced Residue System
Let $R_n = \{ a\in 1\ldots n, \gcd(a,n)=1\}$ denote the reduced residue system modulo n, with its elements ordered from smallest to largest. By definition, each element has an inverse mod n, so we can ...
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Product of $n$ primes greater than $m$ [closed]
What are some good bounds for the logarithm of the product of the first $n$ primes greater than $m$?
I also want to address a confusion I have:
The logarithm of the product of the first $n$ primes is ...
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Show that $\frac{f'(z)}{f(z)}$ has a pole if and only if $f(z)$ has a zero or a pole.
In our analytic number theory course, our instructor told us that $\zeta(s)$ has a zero or a pole at $z_0$ if and only if $\frac{\zeta'(s)}{\zeta(s)}$ has a pole at $z_0$. In fact, this is true for ...
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Growth rate of Mellin transform
For $x>0$ and $0<\sigma <1/6$ consider $$\int _{\sigma \pm i\infty }\underbrace {\frac {\Gamma ^3(s/2)}{\Gamma ^3((1-s)/2)}}_{=:G}\frac {ds}{x^s}$$
which is absolutely convergent since $G\...
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Asymptotic equivalency of two infinite products based on prime numbers
I am trying to figure out if the infinite product $$\omega=\frac{5\sqrt{3}}{12}\prod\limits_{\substack{p\equiv 1\pmod3 \\
p\ge 13}}\left(\frac{p-2}{p-1}\right)\prod\limits_{\substack{p\equiv 2\pmod3 \\...
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Are there any results in the literature involving square-free factorizations of the integers?
For positive integers $n\in\mathbb{N}^*$, the radical of $n$ is defined as $\text{rad}(n):=\prod_{p\vert n}p$ where the product extends over the prime divisors of $n$. It can also be thought of as the ...
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How can every real number be expressed as an infinite sum of rationally weighted Hurwitz zeta values?
On the wiki on rational zeta series, it is stated that - given a real number $x$ - it can be represented as follows:
$$ x = \sum_{n=2}^{\infty} q_{n} \zeta(n,m). \tag{1} $$ Here, $q_{n}$ is a sequence ...
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Why am I getting $\pi(n)=\operatorname{li}(n)+O\left(\log^2 n\right)$
I was trying to see if I could prove the Prime Number Theorem using Legendre's formula. I did it the following way:
We hae
$$\log n!=n(\log n-1)+O(\log n)=\sum_{p\leq n}\frac{n-s_p(n)}{p-1}\log p=(n-O(...
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Is there a version of Dirichlet's theorem for a system of power congruences?
I know that Dirichlet's theorem says that there are infinitely many primes $p$ such that $p\equiv a$ (mod $n$) when gcd$(a,n)=1$. I'm wondering more generally about a system of power congruences
$x^{...
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Asymptotic mean value of certain GCDs
In their paper entitled "On the Number of Cycles of $p$-adic Dynamical Systems," the authors Khrennikov and Nilsson prove that for any $n\in \mathbb{N}$,
$\lim_{M\rightarrow \infty}\frac{1}{\...
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$\Gamma(1)\backslash \Bbb H$ quotient meaning
Let $\Bbb H$ be the upper-half complex plane and $\Gamma(1)=SL_2(\mathbb{Z})$ (acting on $\Bbb H$ by Möbius tranformations) . My modular forms notes frequently refer to the quotient $\Gamma(1) \...
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Size O(1/N) error term of sum of logs
I am trying to derive an asymptotic expression for $N\to \infty$ for the classic sum:
$$ \sum_{n\leq N} \log n = \log (N!)$$
But with the error of the size $O(1/N)$.
I am using the standard ...
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Asymptotics of $\sum_{p\leq n} \frac{1}{p(p-1)}$
Consider the sum $\sum\limits_{p\leq n} \frac{1}{p(p-1)}$, where $p$ are the primes.
This sum certainly converges to some value $<1$ (the sum over all integers is $1$).
Is there a closed form of ...
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Euler product involving divisor function
Take $k,N\in \mathbb N$ and $s\in \mathbb C$ with real part $\sigma \in [1-\delta ,1]$ for some small fixed $\delta $. In its simplest form my question is how do I sum $$\sum _{l\geq 0}\frac {d_k(p^{...
- 1,498
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Turán proof that constant sign of Liouville function implies RH
In Mat.-Fys. Medd. XXIV (1948) Paul Turán gives what he says is a proof of the statement that if the summatory $L(x) = \sum_{n\leq x} \lambda(n)$ of the Liouville function $\lambda(n) = (-1)^{\Omega(n)...
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Upper bound of prime counter function
I am trying to prove a relatively simple bound for the problem below. I want someone to check if my solution is good :)
For a $P=\{ p_1,p_2,...,p_k \}$ a set of prime factors, we define as
$N_P= \{ n\...
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Zero-free region for the zeta function
I am studying de la Vallee Poussin's proof of the zero-free region of the Riemann zeta function. Assuming the following bound
$$\Re \frac{\Gamma'}{\Gamma}(s/2) \leq \log \vert s/2\vert + \min \Big(\...
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