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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4
votes
1answer
55 views

Density of primes of the form $x^2+my^2$

I was playing with numbers and have the nice conjectures: Let $m$ be a fixed positive integer, and $\pi(N)$ denote the numbers of primes not exceeding $N$ and $\pi_m(N)$ denote the number of prime ...
4
votes
1answer
86 views

How to estimate $\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}$?

How to estimate $$\sum_{p\leqslant x}\sum_{q\leqslant x}\frac{1}{p+q}, \qquad\qquad(1)$$ where $p$, $q$ are prime numbers. We have the Mertens' formula $$ \sum_{p\leqslant x} \frac{1}{p} = \log\log ...
0
votes
1answer
28 views

Zeta function of the hypersurface of some homogeneous polynomial

Let $f(y)\in \Bbb Z_p[y_0,y_1,....,y_n]$ be a homogeneous polynomial. Let $N_s$ be the number of zeros of $f$ in $\Bbb P^n(F_{p^s})$. Here, $\Bbb P^n(F_{p^s})$ denotes the $n$-th projective space ...
1
vote
1answer
40 views

Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
0
votes
2answers
61 views

Using Dirichlet's theorem to show existence of number coprime to $n$

I have the following question: Let $n$ be a positive integer and $d$ be divisor of $n$. Use Dirichlet's theorem to show that there exists an integer $k$, where $1\le k\le d-1$ such that the number $m:=...
2
votes
1answer
40 views

Counting number of ideals in quadratic number field

Let $K$ be a quadratic number field and $R$ be its number ring, and if $a(n)$ denotes number of ideals of norm $n$, if $n$ is a prime number, then number of ideals of norm $n$ is $1+(d|n)$, where $d$ ...
53
votes
6answers
10k views

How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
3
votes
1answer
60 views

the exponent of convergence of $\frac{p_{n+1}}{p_n}$ to $1$

Let $p_k$ be the $k$-th prime. Then $\lim_{n\to\infty}\frac{p_{n+1}}{p_n}=1$ -- this is well known. I was looking for more specific information: What is the exponent of convergence of $\frac{p_{n+...
29
votes
1answer
836 views

Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = \sum_{n=...
0
votes
0answers
24 views

Convergence region of local and global zeta functions

Let $\chi = \prod_{v} \chi_{v}: \mathbb{A}^{\times}/F^{\times} \to \mathbb{C}^{\times}$ be a finite order Hecke character and let $\Phi = \prod_{v} \Phi_{v}$ be a Schwartz function on $\mathbb{A}$. ...
1
vote
0answers
37 views

Do the properties defining the Selberg class imply the distribution of real parts of non trivial zeros of an L-function is strongly unimodal?

Selberg defined what is now known as the Selberg class as a class of L-functions fulfilling for essential properties, which are analyticity, Euler product, functional equation and Ramanujan-Patersson ...
1
vote
1answer
73 views

The equation $\zeta(q)=0$ for $q$ a quaternion

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists ...
1
vote
0answers
23 views

Bound for sum of squares $r_{2k}(m)$ for $m \geq 1$

I happened to read in (Iwaniec-Kowalkski) Analytic Number Theory book that the Sum of Squares function satisfies the bound $r_{2k}(m) << m^{k-1+\epsilon}$ for $m \geq 1$. But $\epsilon$ is not ...
3
votes
1answer
58 views

Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
2
votes
3answers
1k views

How to prove $\sum_{d|n} {\tau}^3(d)=\left(\sum_{d|n}{\tau}(d)\right)^2$

For every positive integer $d$, we let $\tau\left(d\right)$ be the number of positive divisors of $d$. Prove that \begin{align} \sum_{d|n} \tau^3(d) = \left(\sum_{d|n} \tau (d)\right)^2 \end{...
7
votes
1answer
674 views

one to one mapping between the floor function and the Riemann prime counting function

We have the following 'transform' of a real valued, piecewise continuous function $f(x)$ : $$T[f(x)]=1+\sum_{n=1}^{\infty}\int_{\mathbb{R}^{n}_{+}}f\left(\frac{x}{\Lambda _{n}} \right )\frac{1}{n!}\...
10
votes
2answers
2k views

how to prove this extended prime number theorem?

A Generalized Prime Number Theorem? Conjecture Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest ...
1
vote
0answers
14 views

application of distribution of primes in arithmetic progressions

I try to understand an application of distribution of primes in arithmetic progressions Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$ So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23 ...
3
votes
0answers
65 views

On the proof that $\phi(n)/n$ has a limit law

In this question, $\mathbb{N}$ denotes the set of positive integers. Also, $\overline{\mathrm{d}}$, and $\mathrm{d}$ means upper natural density, and natural densitiy respectively. (They are the ...
2
votes
1answer
38 views

How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem: Let $f$ be Dirichlet series with real, positive coefficients $(a_n>0)$. If $f$ is holomorphic on $\Re(z)\ge1$, but has one singularity at $z=1$, then $\lim_\limits{...
2
votes
1answer
37 views

$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$

I wish to prove the following equality $$\sum_{p \le x, p \equiv 3 \bmod 10} \frac{1}{p} = \frac{1}{4} \log\log(x)+A+O(\frac{1}{\log x})$$ For some constant A Own work: Let $$A(x) = \sum_ {p \le x} ...
2
votes
2answers
43 views

$F(x) = L(1, \chi ) \log x + O(1)$

I wish to prove $$F(x) = L(1, \chi ) \log x + O(1)$$ when $A(n) = \sum_{d|n} \chi (d)$ and $F(x) = \sum_{n \leq x} \frac{A(n)}{n}$ I started of course by substituting $A(n)$ in $F(x)$, which becomes ...
0
votes
0answers
22 views

Effectiveness of Landau's Prime Ideal Theorem

Is Landau's Prime Ideal Theorem effective? It seems clear that bounds of the strongest kind known on the error term cannot be made fully effective, due to the possibility of Siegel zeroes. At the same ...
0
votes
1answer
961 views

Euler's summation formula proof

The following proof is from Apostol's book: Questions: On the first line of the proof, he uses '{}' just as brackets or do they have other meaning like $[x]$ being the floor function? right before ...
0
votes
0answers
55 views

“on average” in the Bombieri-Vinogradov theorem

TLDR: I don't understand the bit in bold, i.e. where in the formula is the average q? Thanks. The Bombieri-Vinogradov theorem states the following: For any $A > 0$ there exists a $B = B(A)$ such ...
1
vote
1answer
66 views

Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$

Guy Robin proved that $$\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation}$$ is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). The paper where ...
4
votes
0answers
268 views

Liouville function and PNT

The Big Omega function is defined as the number on non-distinct prime factors of an integer. I.e. $\Omega (2^a3^b...p^z)=a+b+...+z$, and the Liouville function is defined as $\lambda(x)=(-1)^{\Omega(x)...
0
votes
0answers
35 views

Bound on log integral

I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+...
0
votes
0answers
55 views

Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
2
votes
0answers
33 views

Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
5
votes
1answer
60 views

Lower Bound on the Sum of Reciprocal of LCM

While reading online, I encountered this post which the author claims that \begin{align} S(N, 1):=\sum_{1\le i, j \le N} \frac{1}{\text{lcm}(i, j)} \geq 3H_N-2 \end{align} and $S(N, 1) \geq H_N^2$ ...
0
votes
1answer
21 views

To show a function is bounded by a function when x is large

I want to show $\sum_{p^a\leq x}\log p = O(\sqrt{x}\log^2 x)$,where sum runs over $a\geq2$. I only know that $\sum_{\sqrt{x}<p \leq x}\log p \leq 2x\log x$. I tried using above property but I am ...
2
votes
1answer
56 views

Inferences about sign of a function from abscissa point

Suppose we have a function $g(x)$ and an integral $$F(s)=\int_1^\infty \frac{g(x)}{x^{s+1}}dx$$ and $F(s)$ converges for $s>\beta$ and diverges for $s = s<\beta.$ Assume also that $\beta$ is ...
1
vote
1answer
84 views

Partial Euler product

The Riemann Zeta function defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$ For $\Re(s)>1$ is convergent and admits the Euler product representation $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ For ...
2
votes
0answers
33 views

Summatory function of Moebius and Euler's totient function over $y$-smooth numbers

Let $y \geq 1$. We say that a positive integer $n$ is $y$-smooth if $n$ has no prime factors larger than $y$. Let $x \geq y$. Let $\mu$ and $\varphi$ be the Moebius and Euler's totient function ...
1
vote
2answers
149 views

Is there any possibilities that the following partial sum of the Dirichlet eta function can be zero?

If we accept-- $s_o$ -- as one of the non-trivial zeros of the Riemann zeta function by $0 <Re(s_o)<1$ and $Re(s_o)$ is the real part of a complex variable, we know: $$\eta(s_o ) = \...
0
votes
1answer
68 views

Show that $\sum_{pq\leq x}\frac{1}{pq}$ = $(\ln \ln x)^2+O(\ln \ln x)$

I know that $\sum_{pq\leq x}\frac{1}{pq}$=$\sum_{p\leq x}\frac{1}{p}\sum_{q\leq\frac{x}{p}} \frac{1}{q}$=$\sum_{p\leq x}\frac{1}{p}(\ln\ln(\frac{x}{p})+A+O(\frac{1}{\ln (\frac{x}{p})}))$. However, I'm ...
0
votes
0answers
21 views

Problem trying to show the following:$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$

So I have to show the following: $$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$$ But the problem is in finding a suitable bound for: $$\sum_{n\leq x} (\omega(n))^2$$ I have tried the ...
0
votes
1answer
13 views

If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $\exp_m(ab)=\exp_m(a)\exp_m(b).$

If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $$\exp_m(ab)=\exp_m(a)\exp_m(b).$$ The notation $\exp_m(a)$ is denote the smallest positive integer $n$ such that $a^n\equiv 1\pmod m$. ...
2
votes
1answer
69 views

Limit of a function, given the recurrence relation

Let $f(n)$ be a function defined for $n\ge 2$ and $n\in N$ which follows the recurrence(for $n\ge 3$) $$\displaystyle f(n)=f(n-1) +\frac {4\cdot (-1)^{(n-1)} \cdot \left(\sum_{d \vert (n-1)} (\chi (d))...
2
votes
0answers
53 views

Asymptotic for the gamma function on vertical lines

On page 135 of Joerg Bruedern's "Einfuehrung in die analytische Zahlentheorie" he claims that Stirling's formula implies for fixed $\sigma <0$, any $t\geq 1$, and some constant $C$ (I assume ...
4
votes
1answer
109 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 $s$ ...
0
votes
1answer
33 views

Bounding sum (log factor)

I want to prove that $$ \sum_{\substack{1\leq n\leq T \\ n\neq m}}n^{-\frac{1}{2}}\left|\log \frac{m}{n}\right|^{-1}\ll T^{\frac{1}{2}}\log T $$ for any $1\leq m \leq T$. Do you have any hint how I ...
0
votes
0answers
48 views

What information do the moments of the Riemann-Zeta function give us

I have seen an explicit formula for what a moment of the Riemann-Zeta function is but I am unsure what information this give us? If we are looking at the zero's of the function then this can be ...
0
votes
0answers
17 views

Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
1
vote
0answers
32 views

The sum $\sum_{r=1}^{p-1} r(r|p)$ when $p$ is an odd prime of the form $4k+3$, $k\geq 1$.

In the book Apostol Analytic Number Theory, $(r|p)$ denotes the Legendre Symbol. The exercise tell us to prove when $p\equiv 1\pmod 4$, $$\sum_{r=1}^{p-1}r(r|p)=0.$$ I can solve this quickly, but I ...
3
votes
0answers
43 views

Solving $\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du=0$ (an extension to the Ramanujan-Soldner constant)

For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$ Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be ...
1
vote
1answer
23 views

Let $P=\{1,2,\cdots,p-1\}$, $P=S\cup T$, prove that $S$ is quadratic residues and $T$ is quadratic nonresidues.

Let $p$ be an odd prime. Assume that the set $\{1,2,\cdots,p-1\}$ can be expressed as the union of two nonempty subsets $S$ and $T$. $S\neq T$, such that the product (mod $p$) of any two elements in ...
0
votes
0answers
11 views

Problem understanding summation of Big-O notation i.e $F(x)=\sum_{n\leq x}O(\frac{x}{n})$

I know that $O(f(x))+O(g(x))=O(g(x))$ if $f(x)=O(g(x))$. But I cant seem to find a bound for F(x) since n changes when x increases. Any ideas. Im assuming that $F(x)=O(\sum_{n\leq x}\frac{x}{n})$ ...
1
vote
2answers
1k views

A proof of $\sum{\mu(n)/n}=0$

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function. Many thanks !