# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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### Drichlet theorem on primes in ap with a prime index

Someone know a result like this:
Given $a,b\in\mathbb{N}$, with $a>b>0$ and $(a,b)=1$,then there exist at least one prime $p$ such that $ap+b$ is a prime number too.

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63 views

### Number of lattice points on spheres with center not at the origin

Let $k\ge1$. It is known that the number of lattice points on the $k$-sphere $S^k(0)$ (center at the origin, radius $R$), namely the size of $\mathbb{Z}^{k+1}\cap S^k(0)$, is bounded by $R^{k-1+\...

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148 views

### Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$

Let $\zeta$ be the zeta function of Riemann. Is the bound for
$$I_{T}=\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$$
known ?
It seems to me that $I_{T} \ll T\log T$ since $\log|\zeta(...

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250 views

### Would this be a significant/publishable result on the Riemann zeta function? [on hold]

I recently proved that the Riemann zeta function $\zeta(s)$ has only finitely many zeros on any line $\Re(s)>1/2$. I've shown my work to a couple of professors (non-number theorists) and they both ...

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73 views

### How to understand the sign periodicity of any conditionally convergent series? [on hold]

Given any conditionally convergent series $\sum_{n\geq1} a_n$. I wonder if there is a "standard way/method" to "investigate/estimate" the sign periodicity of $a_n$ by some explicit functions $f(x)$ i....

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74 views

### Upper bound of $\zeta$-function on critical strip

How can I determine any upper bound for $|\frac{\zeta^4(s)}{\zeta(2s)}+d\cdot\zeta^2(s)|$ on the critical strip $s=\frac12+it$ for an integer $d$?

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141 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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112 views

### summation of Euler totient function

Let $\phi(n)$ be the Euler totient function and let $2\leq k\in\mathbb{N}$. For $m\in\mathbb{N}$, are there any known results, upper bounds (tighter than just removing the coprimality) or ...

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70 views

### How many pairs of integer numbers with bounded product?

Let $r\in (0,1)$ and denote by $A_r$ the set
$A_r=\left\{ a,b\in\mathbb{N} ~:~ a,b\leq N, a\cdot b\leq rN^2 \right\}$. Is it possible to find a good estimation for $|A_r|$?
It is known that $|A_r|= r(...

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689 views

### Intuitive reason why the $j$-invariant is a cube?

Let $\tau$ be a CM point of discriminant $D$. Assume that $D$ is not divisible by $3$. Then $j(\tau)$ is an algebraic integer of degree equal to the class number $h(D)$. Let $ \gamma_2(\tau)=j(\tau)^{...

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98 views

### Abelian variety over Q with many roots of unity

Given an abelian variety $A$ over the rational integers $\mathbb{Q}$, and a prime $p$, we know that $\mathbb{Q}(\zeta_p)$ is contained in $\mathbb{Q}(A[p])$, the $p$-division field of $A$, and where $\...

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259 views

### Reference Request for a result on divisors of $p-1$

I have seen this result in several places without an English reference:
There exist infinitely many primes $p$ such that $p-1=2q_1q_2$ where $q_1$ and $q_2$ are prime numbers with $q_1,q_2>p^{1/4}$...

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219 views

### A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that
$$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$.
where $\zeta$ ...

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203 views

### Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...

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119 views

### On segments of the series $\sum_p\frac1{p-1}$

Here I ask a question concerning segments of the divergent series
$$\sum_p\frac1{p-1}=\sum_{k=1}^\infty\frac1{p_k-1},\tag{$*$}$$
where $p$ runs over all the primes, and $p_k$ denotes the $k$-th prime.
...

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**1**answer

436 views

### Are the ideles literally a picard group?

I understand that in the number field / function field analogy, the ideles $\mathbb I_K$ of a number field $K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this ...

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**1**answer

83 views

### Discrepancy in non-homogeneous arithmetic progressions

I have a doubt, Roth's discrepancy theorem [1] says that there is a subset of arithmetic progressions $A \in [n]$ where any function $f:N\rightarrow \left \{ -1,1 \right \} $ implies
$ \left | \...

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87 views

### Sum of a prime and a number one can write as a sum of 2 squares in exactly $ k $ ways

Following this interesting question : Sieve bound for the sum of two squares I define $ P_{k}(n)=1 $ if and only if $ n $ is the sum of an odd prime and an integer that can be written in exactly $ ...

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98 views

### Sieve bound for the sum of two squares

Let $$S(n) = \sum_{p \le n} b(n-p),$$ where
$b(a)=1$ is $a$ is a sum of two squares of positive integers and $b(a)=0$ otherwise.
Trivially by PNT we have
$$S(n) \ll \sum_{p \le n}1 \le \frac{n}{\log n}...

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118 views

### On a certain representation for the Riemann zeta function in Montgomery-Vaughan

On page 338 of Montgomery-Vaughan's ''Multiplicative Number Theory'', there is a somewhat nice representation for the Riemann zeta function. That is, let $0<\delta\leq 1/2$. Then one has
$$\zeta(1/...

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120 views

### Question on page 432 of Iwaniec-Kowalski'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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82 views

### Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?

We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...

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169 views

### Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?

A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that
$$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ ...

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76 views

### Probability distribution from equidistribution - II

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...

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67 views

### An asymptotics for $ r_{0}(n) $ through Euler totient

Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) $ the smallest positive integer $r $ such that both $ n-r $ and $ n+r $ are prime, provided $ n $ is a large enough positive composite ...

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65 views

### Generalization of regularly varying functions

A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,
$$
\lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a)
$$
for some function $g(a)&...

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99 views

### Reference request for bounds of $n$-th composite

Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...

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99 views

### Chen primes and permutations

In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...

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62 views

### Meyer's class number formulas

In my previous question I asked for a reference for $L$-series of quadratic orders in connection with a certain class number formula. It seems that this had been investigated by Curt Meyer. Is there ...

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78 views

### Dirichlet series of Euler's totient function for Gaussian integers?

Define the Euler's totient function for Gaussian integers $f:\mathbb Z[i]_{\ne 0}\mapsto \mathbb Z_{>0}$:
$$f(z):=\sum_{\substack{q\in\mathbb Z[i]_{\ne 0}\\|q|\le|z|, |\gcd(q,z)|=1}}1,$$
and the ...

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151 views

### Number fields ordered by discriminant

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...

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120 views

### Prime generating polynomials

Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...

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314 views

### Distance between primes that are quadratic residues modulo an other prime

Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...

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161 views

### Semi-primes represented by quadratic polynomials

According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...

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105 views

### Almost-prime values attained by a product of quadratic polynomials

Let $F(x) = \prod_{i=1}^{k} (a_i x +b_i)$ be a product of $k$ linear polynomials, where $a_i,b_i$ are integers. Under very reasonable conditions, it is known that a constant $C_k$ exists with the ...

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189 views

### On approximation of $\sum_{a,b=1}^n\gcd(a,b)$

Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...

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192 views

### Gauss sums for general number fields

There is a wide litterature for the classical Gauss sums. For $\chi$ a primitive Dirichlet character modulo $N$, it is given by
$$\tau(\chi) = \sum_{n \text{ mod } N} \chi(n) \exp(2i\pi n/N).$$
An ...

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81 views

### $L$-functions for quadratic orders and Siegel's solution of the class number problem

Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and
$$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-...

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174 views

### Landau's theorem using nth roots

This question was asked earlier at MSE .
Let $\omega$(n) denote the number of distinct primes dividing $n$. The Mobius function is defined as $\mu(n) = (-1)^{\omega(n)}$ if $n$ is squarefree and $\...

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276 views

### Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...

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610 views

### Is there a rigid analytic geometry proof of the functional equation for the Riemann zeta function?

The adèles $\mathbb A$ arise naturally when considering the Berkovich space $\mathcal M(\mathbb Z)$ of the integers. Namely, they are the stalk $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...

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283 views

### Is $j(\tau)^{1/3}$ the hauptmodul for the congruence subgroup generated by $\tau\rightarrow\tau+3, \tau\rightarrow-1/\tau$?

The 3rd root of the modular invariant $j$ is
$$ j(\tau)^{1/3}=q^{-1/3}(1+ 248q+ 4124q^2+ 34752q^3+\cdots),$$
where $q=e^{2\pi i \tau}$.
I was wondering if $j(\tau)^{1/3}$ the hauptmodul for the ...

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217 views

### Enquiry on an equality involving the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. Does there exist a $t\geq 0$ such that
$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$

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135 views

### Closed form for an integral involving the Riemann zeta function at the critical line

After seeing this question $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral i became curious if/how the approach in the answer by reuns can be applied to evaluate
$$I_{a,b}=\int_{-\infty}^{\...

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192 views

### A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2

Erdős asked1 whether the series
$$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges.
Here, $p_n$ denotes the n-th prime.
I can show that this series converges simultaneously with the series $\sum_{...

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524 views

### Heuristics behind the Circle problem?

Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and ...

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38 views

### Uniformity in Wirsing's Mean Value Theorems

In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...

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157 views

### Class fields without class field theory

Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...

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207 views

### $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral

Denote by $\zeta$ the Riemann zeta function. I just learnt from this question $L_2$ bounds for tails of $\zeta(s)$ on a vertical line
that $\int_{T}^{\infty} \frac{\zeta(1/2 + it)}{1/4 + t^2}\mathrm{...

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98 views

### On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$

I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality
$$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$
holds uniformly for $T\geq 2$, ...