# Questions tagged [analytic-number-theory]

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

3,702 questions
Filter by
Sorted by
Tagged with
1 vote
13 views

### 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 ...
• 1,269
87 views

1 vote
60 views

### 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 ...
1 vote
62 views

### 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 ...
• 1,498
35 views

### 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 ...
• 31.4k
1 vote
61 views

### 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 ...
59 views

### 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 ...
1 vote
33 views

### 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 $$\chi(n)\tau(\overline{\chi})=\sum_{h=1}^{q} \overline{\chi}(m) e_q(nh)$$ ...
39 views

• 81
1 vote
66 views

### 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 ...
• 1,463
42 views

### 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 $\zeta$ ...
• 277
48 views

### 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 ...
85 views

• 3,666
29 views

### 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 ...
• 65
156 views

### 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" ...
1 vote
45 views

### 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 $\sum_n a(n)$ ...
• 193k
21 views

### 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 ...
1 vote
43 views

### 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 ...
• 206
59 views

### 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 ...
• 387
41 views

• 759
1 vote
53 views

### 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 ...
115 views

### 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 ...
• 5,900
102 views

• 1,498
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 $\lambda(n) = (-1)^{\Omega(n)... 3 votes 1 answer 46 views ### Upper bound of prime counter function I am trying to prove a relatively simple bound for the problem below. I want someone to check if my solution is good :) For a$P=\{ p_1,p_2,...,p_k \}$a set of prime factors, we define as$N_P= \{ n\...
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(\...