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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votes
0answers
19 views
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.
2
votes
0answers
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+\...
1
vote
1answer
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(...
-4
votes
0answers
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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votes
0answers
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....
0
votes
0answers
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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0answers
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:
$$\cal{A} = (a_n) , n\le x$$,...
-1
votes
1answer
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 ...
0
votes
0answers
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(...
23
votes
1answer
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)^{...
4
votes
0answers
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 $\...
6
votes
2answers
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 $q_1,q_2>p^{1/4}$...
-2
votes
1answer
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 $\zeta$ ...
2
votes
3answers
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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votes
0answers
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.
...
6
votes
1answer
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 ...
-1
votes
1answer
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 | \...
1
vote
0answers
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 $ ...
2
votes
1answer
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
$$S(n) \ll \sum_{p \le n}1 \le \frac{n}{\log n}...
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votes
0answers
118 views
On a certain representation for the Riemann zeta function in Montgomery-Vaughan
On page 338 of Montgomery-Vaughan's ''Multiplicative Number Theory'', there is a somewhat nice representation for the Riemann zeta function. That is, let $0<\delta\leq 1/2$. Then one has
$$\zeta(1/...
-2
votes
0answers
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
$$\sum_{n \equiv \alpha \mod q} f(dn) (dn)^{-s}=\frac{F(s)}{dq} +O(|s| \...
2
votes
0answers
82 views
Coppersmith's method to quadrivariate degree $2$ polynomials that behave as bivariate?
We have a polynomial $f(x_1,x_2,x_3,x_4)\in\mathbb Z[x_1,x_2,x_3,x_4]$ where the only monomials are either from set $$\{x_1,x_1x_2,x_2,x_3,x_3x_4,x_4\}$$ and we seek solutions $(x_1,x_2,x_3,x_4)\in\...
4
votes
0answers
169 views
Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?
A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that
$$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ ...
1
vote
0answers
76 views
Probability distribution from equidistribution - II
Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ and denote $N(a,b)$ to be minimum $\ell_2$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
-1
votes
0answers
67 views
An asymptotics for $ r_{0}(n) $ through Euler totient
Assuming Goldbach's conjecture, let's denote by $ r_{0}(n) $ the smallest positive integer $r $ such that both $ n-r $ and $ n+r $ are prime, provided $ n $ is a large enough positive composite ...
2
votes
0answers
65 views
Generalization of regularly varying functions
A continuous function $L :\mathbb{R}_+ \to \mathbb{R}_+$ is called regularly varying (at infinity) if for all $a > 0$,
$$
\lim_{x\to \infty} \frac{L(ax)}{L(x)}= g(a)
$$
for some function $g(a)&...
0
votes
0answers
99 views
Reference request for bounds of $n$-th composite
Motivation
I will briefly elaborate here my motivations for asking the question. If you are not interested in it then please go to the questions.
Recently during trying to understand and prove the ...
2
votes
0answers
99 views
Chen primes and permutations
In 1973 the Chinese mathematician J.-R. Chen proved that there are infinitely many primes $p$ such that $p+2$ is a product of at most two primes. Nowadays such primes $p$ are called Chen primes.
For $...
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votes
0answers
62 views
Meyer's class number formulas
In my previous question I asked for a reference for $L$-series of quadratic orders in connection with a certain class number formula. It seems that this had been investigated by Curt Meyer. Is there ...
1
vote
0answers
78 views
Dirichlet series of Euler's totient function for Gaussian integers?
Define the Euler's totient function for Gaussian integers $f:\mathbb Z[i]_{\ne 0}\mapsto \mathbb Z_{>0}$:
$$f(z):=\sum_{\substack{q\in\mathbb Z[i]_{\ne 0}\\|q|\le|z|, |\gcd(q,z)|=1}}1,$$
and the ...
7
votes
0answers
151 views
Number fields ordered by discriminant
Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are ...
1
vote
0answers
120 views
Prime generating polynomials
Continuation to this previous question.
According to Lemke-Oliver, an irreducible polynomial $G$ of degree $g$ with positive leading coefficient and $\Gamma_G\neq0$ (with $\Gamma_G$ a certain factor ...
3
votes
2answers
314 views
Distance between primes that are quadratic residues modulo an other prime
Question: Is there an infinite sequence of primes $\{q_i\}_{i=1}^{\infty}$ that is not too sparse ( $q_n =O(poly(n))$ for a fixed polynomial) for which it is true that for every $k$ there is an $N(k)$ ...
9
votes
0answers
161 views
Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
4
votes
1answer
105 views
Almost-prime values attained by a product of quadratic polynomials
Let $F(x) = \prod_{i=1}^{k} (a_i x +b_i)$ be a product of $k$ linear polynomials, where $a_i,b_i$ are integers. Under very reasonable conditions, it is known that a constant $C_k$ exists with the ...
4
votes
1answer
189 views
On approximation of $\sum_{a,b=1}^n\gcd(a,b)$
Denote $g(n)=\sum_{a,b=1}^n\gcd(a,b)$, can we prove that
$$g(n)=\frac6{\pi^2}n^2\ln n+Cn^2+O(n\ln n)$$, where $C=-\frac12+\frac{6}{\pi^2}(-\frac12+\gamma-\ln(2\pi)+12\ln A),$ where $\gamma$ denotes ...
10
votes
1answer
192 views
Gauss sums for general number fields
There is a wide litterature for the classical Gauss sums. For $\chi$ a primitive Dirichlet character modulo $N$, it is given by
$$\tau(\chi) = \sum_{n \text{ mod } N} \chi(n) \exp(2i\pi n/N).$$
An ...
3
votes
0answers
81 views
$L$-functions for quadratic orders and Siegel's solution of the class number problem
Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and
$$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-...
4
votes
1answer
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 $\...
11
votes
1answer
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 ...
30
votes
0answers
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 $\mathbb A = (j_\ast j^{-1} \mathcal O_\mathbb Z)_p$ ...
9
votes
1answer
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 ...
-1
votes
1answer
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
$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$
3
votes
1answer
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
$$I_{a,b}=\int_{-\infty}^{\...
5
votes
0answers
192 views
A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2
Erdős asked1 whether the series
$$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges.
Here, $p_n$ denotes the n-th prime.
I can show that this series converges simultaneously with the series $\sum_{...
3
votes
2answers
524 views
Heuristics behind the Circle problem?
Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and ...
2
votes
0answers
38 views
Uniformity in Wirsing's Mean Value Theorems
In two important papers of Wirsing, namely "Das asymptotische Verhalten von Summen über multiplikative Funktionen" (1961) and its follow up (1967), several results on mean values of multiplicative ...
3
votes
0answers
157 views
Class fields without class field theory
Is there an English reference for the analytic construction of the Hilbert class field of an imaginary quadratic field without using class field theory? I am in particular interested in a proof of the ...
1
vote
1answer
207 views
$L_2$ bounds for $\zeta(1/2 + it)$ and a related integral
Denote by $\zeta$ the Riemann zeta function. I just learnt from this question $L_2$ bounds for tails of $\zeta(s)$ on a vertical line
that $\int_{T}^{\infty} \frac{\zeta(1/2 + it)}{1/4 + t^2}\mathrm{...
1
vote
0answers
98 views
On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$
I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality
$$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$
holds uniformly for $T\geq 2$, ...
