# 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.

1,537 questions

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194 views

### upper bound of consecutive integers which are not coprime with $n!$

Is there any research on getting upper bound of the maximal possible number of consecutive positive integers which are less than $n!$ and NOT coprime with $n!$?
Easy to see that lower bound $\ge n$, ...

**5**

votes

**1**answer

166 views

### Functional equation for general number fields

When it comes to general number fields beyond $\mathbb{Q}$, the litterature is not so abundant in analytic number theory. For instance over $\mathbb{Q}$, for primitve Dirichlet characters modulo $q$, ...

**2**

votes

**1**answer

159 views

### A truncated divisor sum

I am interested in an upper bound for
$$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...

**0**

votes

**0**answers

70 views

### Number of $b$-separated Sidon sets with pairwise difference set intersection bounds

Given two integers integers $0<b<p$ and a real $\alpha\in(0,1)$ call a set of $m$ integers $a_1<\dots<a_m$ in the interval $(p^\alpha,p-p^\alpha)$ to be $b$-separated Sidon if:
$a_i-a_j\...

**8**

votes

**1**answer

209 views

### Conjecture about an Exponential Sum

Let $X \subset \mathbb{N}$ and say that $X$ is super-equidistributed if
for all $\alpha \in \mathbb{R} \setminus \mathbb{Z}$ there exists $C(\alpha) > 0$ such that for all $N$
$$
\left| \sum_{x \in ...

**0**

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**0**answers

106 views

### Degree bounds and coefficient size in elimination theory?

Suppose we have polynomials is of form
$$h_1(x_1,\dots,x_{dn})-c_1\in\mathbb Z[x_1,\dots,x_{dn}]$$
$$\vdots$$
$$h_{nd}(x_1,\dots,x_{dn})-c_{nd}\in\mathbb Z[x_1,\dots,x_{dn}]$$
where $h_1(x_1,\dots,x_{...

**2**

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**0**answers

66 views

### L-functions of tempered automorphic representations

Let $G$ be a reductive group over the adele ring $\mathbb A_F$ of a number field $F$. Let also $r$ be a complex finite dimensional representation of the $L$-group $r:{^LG}\to GL(V)$.
It is generally ...

**3**

votes

**1**answer

152 views

+50

### How often does the Mertens function vanish?

It is well known that the Mertens function
$$M(x)=\sum _{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a short proof.
Are there known results about how often the Mertens function ...

**3**

votes

**0**answers

117 views

### Maximum number of integral roots in degree $d$ polynomial?

Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that
Each coefficient is bound in absolute value by $B$
Degree of each variable in any monomial is bound by $d$
Total degree is $d'$
$f(x_1,\...

**8**

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**1**answer

231 views

### Question about Friedlander, Iwaniec: “The polynomial $X^2+Y^4$ captures its primes”

I have a question about the argumentation at the beginning of section 15 in this paper. The goal is to estimate the sum
$$V(\beta) = 2 \sum_{(z_1,z_2)=1} \beta_{z_1} \overline{\beta_{z_2}} \sum_{\...

**4**

votes

**2**answers

603 views

### Chebyshev's bias-conjecture and the Riemann Hypothesis

Chebyshev's bias-conjecture that says "there are more primes of the form 4k + 3 than of the form 4k + 1" and the Riemann Hypothesis are equivalent? That means, one implies the other (if and only if)?

**4**

votes

**1**answer

243 views

### Proper Way To Compute An Upper Bound

I regard to the proof of Lemma 10 in "A remark on a conjecture of Chowla" by M. R. Murty, A. Vatwani, J. Ramanujan Math. Soc., 33, No. 2, 2018, 111-123,
the authors used the average value $(\log x)^...

**9**

votes

**0**answers

188 views

### How many partition values are expected to be prime?

Let $p(n)$ be the partition function. Let $P(N)$ count how many $1\leq n\leq N$ are such that $p(n)$ is prime.
Are there any heuristics for how $P(N)$ should behave?
A crude guess at how this ...

**-1**

votes

**0**answers

48 views

### Probability distribution from standard domain (multiple pairs single prime) - V

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_2(a,b,n)$ 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(...

**1**

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**0**answers

61 views

### Probability distribution from standard domain (two primes) - IV

Pick a random pair $(a,b)\in\mathbb Z_n^2\setminus\{0,0\}$. Denote $N_2(a,b,n)$ 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**

vote

**1**answer

257 views

### How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?

Let $\zeta$ be the Riemann zeta function.
My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?...

**3**

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**0**answers

53 views

### Flow of zeros in the shifted exponential generating function?

Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...

**0**

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**0**answers

72 views

### Probabilistic Howgrave-Graham Bounds with Coppersmith Technique?

Howgrave-Graham condition says roots of $$f(x_1,\dots,x_n)\equiv0\bmod R$$ at an $R\in\mathbb N$ are roots of $$f(x_1,\dots,x_n)=0$$ over $\mathbb Z$ if $$\|f(x_1X_1,\dots,x_nX_n)\|<\frac{R}{\sqrt{\...

**0**

votes

**1**answer

262 views

### Reason Coppersmith fails here?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$.
$P$ has a binary expansion and so does $Q$. We can set the binary $0/1$ variables to be $x_1$ through $x_{\lceil\log P\rceil}$ and ...

**-2**

votes

**1**answer

216 views

### Negative Dirichlet Pigeonhole Principle

From Dirichlet Pigeonhole Principle if $p$ is a prime and if $a,b\in\mathbb Z$ are in $(0,p/2)$ then there is a $t\in(0,p)\cap\mathbb Z$ such that $\|(x,y)\|_\infty<\lceil\sqrt p\rceil$ holds where ...

**6**

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**1**answer

172 views

### Equidistribution of $\{p_n^2α\}$

Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha\}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...

**1**

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**1**answer

164 views

### Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that
$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...

**0**

votes

**0**answers

27 views

### Probability density from standard domain (Typical Box principle and Chinese Remainder Theorem) - III

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_2(a,b,n)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(...

**1**

vote

**1**answer

97 views

### Factoring with partial information on gaps

If $N=PQ$ is a semi-prime with $P=N^{\frac12 +\delta}$ and $Q=N^{\frac12-\delta}$ then if we know $\delta\in(0,\frac12)$ to a reasonable precision we can factor $N$ quickly. What precision (number of ...

**3**

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**1**answer

114 views

### Divergence of a series related to Schinzel's hypothesis H

The Series
Consider the series identity
$$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$
$$R(n) = \left\...

**5**

votes

**0**answers

141 views

### Gaussian square-free moat

Is there a sequence $\{z_n\}_{n=1}^\infty$ of distinct square-free
Gaussian integers with $$\sup_{n \geq 1} |z_{n+1} - z_n| < \infty ?$$
For the analogous problem with Gaussian primes instead, ...

**5**

votes

**0**answers

145 views

### Divisor bound for $r_2$ off the origin

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...

**-1**

votes

**1**answer

197 views

### On a certain representation of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...

**3**

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**0**answers

88 views

### Friedlander-Iwaniec Flipping moduli

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $X^2+Y^4$ captures its primes" by Friedlander and Iwaniec.
At page 997, just below equation (12.7) we start estimating the ...

**2**

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**0**answers

106 views

### Representation as $n=p^2+q^2-r^2$

What is known about the number of representations of a positive integer $n$ as
$$
\rho(n) = \# \{ (p,q,r): n=p^2+q^2-r^2\},
$$
where all the variables are primes?
What about the average number of ...

**0**

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**0**answers

127 views

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

Someone know a result like this:
Given $a,b\in\mathbb{N}$, with $a>b>0$, $(a,b)=1$ and a $\not \equiv b \pmod 2$ ,then there exist at least one prime $p$ such that $ap+b$ is a prime number too.

**3**

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**1**answer

143 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**

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**1**answer

167 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(...

**0**

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**0**answers

82 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 $d$?

**-1**

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**1**answer

123 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**

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**0**answers

72 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

**1**answer

720 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**

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**0**answers

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**

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**1**answer

266 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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votes

**1**answer

230 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

**3**answers

209 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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**0**answers

128 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.
...

**0**

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**0**answers

105 views

### Does Coppersmith technique suffice to factor?

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time?
Technically I am ...

**6**

votes

**1**answer

451 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

**1**answer

89 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**

votes

**0**answers

97 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

**1**answer

108 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}...

**0**

votes

**0**answers

122 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

**0**answers

195 views

### Algebraically independent vectors in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$
$$\|v_1'\|_2\...

**2**

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**0**answers

86 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\...