# 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.

1,986
questions

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### Mean square estimate for the Kloosterman sums

For $m,n\in \mathbb{N}$, denote the Kloosterman sum
$$S(m,n;c)=\sum_{a\bmod c}e\left( \frac{ma+n \overline{a}}{c}\right),$$where $\overline{a}$
denotes the multiplicative inverse of $a\bmod c$.
Does ...

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60 views

### Integer partitions into restricted parts

Given a linear diophantine equation $$x_1+\dots+x_n=m\leq nn'$$ how many solutions does it have with each $x_i\in[0,n']\cap\mathbb Z$? Looking for asymptotics that parametrizes well with both $n$ and $...

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140 views

### A generalization of strong primes

In this post we denote the sequence of prime numbers as $p_k$ for integers $k\geq 1$. I don't know if the following definition is in the literature.
Definition. We define the $\theta$-strong primes, ...

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67 views

### General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...

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133 views

### Polynomials of minimum degree that interpolate primes in intervals

Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of ...

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143 views

### When did the main conjecture in Vinogradov's mean value theorem first appear in literature?

Recently I was asked about the history of Vinogradov's mean value theorem that I was hoping someone here could clarify. Let me first start with some terminology. Let $J_{s, k}(X)$ be the number of $2s$...

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164 views

### On the error bound for the Prime Number Theorem for arithmetic progressions

Let $\chi$ be a Dirichlet character, $L(s,\chi)$ be the corresponding L-functions and $\Theta_{\chi}$ be the supremum of the real parts of the zeros of $L(s, \chi)$. Define $\pi(x; a, q)$ to be the ...

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193 views

### How to determine the coefficient of the main term of $S_{k}(x)$?

Let $k\geqslant 2$ be an integer, suppose that $p_1,p_2,\dotsc,p_k$ are primes not exceeding $x$. Write
$$ S_{k}(x) = \sum_{p_1 \leqslant x} \dotsb \sum_{p_k \leqslant x} \frac{1}{p_1+\dotsb +p_k}. $$...

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80 views

### Average value of a fractional part of a function

Let $f(x): \mathbb{R} \to \mathbb{R}_{\geq 0}$ be a smooth function. I am interested in estimating
sums of the form
$$
\sum_{ A < n \leq B } \{ f(n)\}
$$
where $\{ c \}$ denotes the fractional part ...

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261 views

### Should I expect functions to have analytic continuations?

I spend lots of time working with Dirichlet series with bounded coefficients, and I often need to find whether or not they have analytic continuations to the full complex plane. When proving that some ...

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33 views

### On $\sum_{\substack{1\leq d\leq x\\(d,r)=1}}\frac{\mu(d)}{d}W(\frac{x}{d})$, with $\mu(n)$ the Möbius function and $W(x)$ the Lambert $W$ function

I wondered if it is possible to posed a similar question than Question 2 by Olivier Ramaré from [1] (page 231), although the computational evidence that I have for my conjecture is very small.
...

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202 views

### Is the parity of $\omega(n)$ equally distributed?

I recently learned that the prime omega function $\Omega(n)=\Omega\left(p_1^{\alpha_1}p_2^{\alpha_2}...p_k^{\alpha_k}\right)=\alpha_1+\alpha_2...+\alpha_k$ is very well studied. In particular, we know ...

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109 views

### Analytic continuation over boundaries

In D.J Newman's paper
A simple analytic proof of the prime number theorem
there is the following theorem:
Suppose $|a_n|<1$ and form the Dirichlet series $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...

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648 views

### What is the significance of Friedlander-Iwaniec and related theorems?

On p.177 of Number Theory Revealed: A Masterclass by Andrew Granville, the author states that "One can ask for prime values of polynomials in two or more variables." (though he later ...

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93 views

### On $(\prod_{\substack{1\leq s\leq X\\s\text{ semiprime}}}s)(\sum_{\substack{1\leq s\leq X\\s\text{ semiprime}}}\frac{1}{s})$ as $X\to\infty$

Few weeks ago an user from Mathematics Stack Exchange answered my question On an inequality that involves products and sums related to the sequence of semiprimes (asked May 26). It seems that for ...

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36 views

### prime decomposition in even dihedral extensions

Let $L/K$ be a finite extension of number fields of degree $n$ with $n$ an even integer such that the normal closure of $L$ has the Galois group isomorphic to $D_n$, the dihedral group of order $2n$. ...

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341 views

### Twisted functional equations in Goldfeld's book

I am confused about the contents of D. Goldfeld's book "Automorphic forms and L-functions for the group $\mathrm{GL}(n,\mathbb{R})$". In the process of deriving the converse theorem on $\...

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### Goldbach conjecture and other problems in additive combinatorics

The field is also known as additive number theory. I am interested in sums $z=x + y$ where $x \in S, y\in T$, and both $S, T$ are infinite sets of positive integers. For instance:
$S = T$ is the set ...

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140 views

### On a conjecture about the arithmetic function that counts the number of twin primes

This is cross-posted from the question that I've asked with same title on Mathematics Stack Exchange two months ago, which has remained answer.
Given a positive real number $x$ we will write positive ...

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158 views

### Asymptotics on a double sum over primes

I am attemping to find asymptotics of
$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \...

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137 views

### Binomial transform of Dirichlet series

Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence:
$$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$
And let $\left\{a_{n}\right\}...

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56 views

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### On variations of a claim due to Kaneko in terms of Lehmer means

This post is cross posted from Mathematics Stack Exchange, due that there was a mistake from my part (see the excellent partial answer and my thread of edits of my question on MSE) this post on ...

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35 views

### Montgomery-Hooley refinement of Barban-Davenport-Halberstam

Where can I find a (correct/precise) statement of the Montgomery/Hooley refinement of the Barban-Davenport-Halberstam Theorem without having to subscribe to a journal or buy a book?

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260 views

### Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

I was working with some Dirichlet series and I realized that I have never seen any general conditions under which
\begin{equation}
\sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...

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54 views

### Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line,
$$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$
where $\chi$ is a ...

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22 views

### Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers

I don't know if my question can be answered easily from your reasonings and knowledeges of the theory of odd perfect numbers. I wondered about it yesterday. I've crossposted this post here from the ...

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129 views

### Fourier series and fractional calculus

We already know that riemann-zeta function on the critical band
$$\zeta(\alpha) \propto \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha}, \Re(\alpha) \in ]0, 1[ $$
Is it possible to say that
$$ \zeta(\...

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242 views

### Is every integer $\ge 312$ the sum of two integers with triangular divisors?

We say that a natural number $n$ has triangular divisors if it has at least one triplet of divisors $n = d_1d_2d_3$, $1 \le d_1 \le d_2 \le d_3$, such that $d_1,d_2$ and $d_3$ form the sides of a ...

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### L. Gegenbauer's proof of Infinitude of Primes [closed]

I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that
L. Gegenbauer proved Infinitude of Primes by ...

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43 views

### Expected order of magnitude of character sums under GRH

Let $\chi$ be a nonprincipal character with modulus $q$. Under GRH, what is the expected order of magnitude of $\sum_{n \le x} \chi(n)$, where I think of $x$ and $q$ as growing, but $x$ is smaller ...

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141 views

### On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers

For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...

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117 views

### On $\mathsf{LCM}$ of a set of integers

For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
How ...

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142 views

### What is known about products of zeta values?

A couple of years ago, I asked this MSE question on the evaluation of the product of even zeta values: $$ \prod_{n=1}^\infty \zeta(2n) \approx 1.82 \quad .$$ While it can be shown that the product ...

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31 views

### Minimum size integer accommodating some divisors within some prescribed gaps

Assume we pick $t$ uniformly random integers $l_1$ to $l_t$ independently from $1$ to $2^v$.
Assume $k_1$ through $k_t$ are similarly picked from $1$ to $2^r$.
What is the minimum size of non-...

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263 views

### A question about Schwartz-type functions used in analytic number theory

In analytic number theory we like to weigh our counting functions with a smooth function $f$, so that we may apply Poisson's summation formula and take advantage of Fourier transforms. Typically the ...

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387 views

### Understanding Sylvester' s $1871$ paper of primes in arithmetic progression of the forms $4n+3$ and $6n+5$

The following is the proof of infinitude of primes in arithmetic progression of the form $4n+3$ and $ 6n+5$ done by Sylvester in $1871$ in his paper "On the theorem that an arithmetical progression ...

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273 views

### Proving certain inequality related to Primes

I was reading the following paper. But I can't understand why the last line concerning $\frac{2}{\pi}$ is true. The proof is a work of Sylvester.
I would be happy if someone helps me in understanding ...

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73 views

### Odd perfect numbers having as prime factors exclusively Mersenne primes and Fermat primes

I don't know if the following question is in the literature, please add a commment if it is in the literature. I add my thoughts and motivation below in last paragraph, it is discursive and ...

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247 views

### Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...

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47 views

### A question related to Hilbert modular form

Is it possible to construct a Hilbert modular form from a Hecke character of quadratic extension of a totally real field $F$?
Basically I would like to know if a generalization of the theorem 12.5 in ...

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124 views

### Sum of four squares from sum of $k\geq5$ squares - exact version $\mathsf{II}$

Every natural number is sum of $4$ squares and every natural number bigger than $169$ is sum of five squares.
Suppose $n>169$ and $x'x=n$ at a vector $x\in\mathbb Z^5$ then can we find two matrices ...

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51 views

### Sum of four squares from sum of $k\geq5$ squares - scaled version $\mathsf I$

Every natural number is sum of $4$ squares and every natural number bigger than $169$ is sum of five squares.
Suppose $n>169$ and $x'x=n$ at a vector $x\in\mathbb Z^5$ then can we find a matrix $M\...

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43 views

### Diophantine bound for homogeneous system under norm conditions for solutions and system

If $b_{ij}\in[0,2^t-1]\cap\mathbb Z$ holds at every $i\in\{1,\dots,k'\}$ and $j\in\{1,\dots,k\}$ where $1\leq k'\leq k$ holds then there are $x_1,\dots,x_k\in\mathbb Z:\sum_{j=1}^kx_jb_{1i}=0\wedge\...

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111 views

### Integral over an exponential sum with squares

How should I estimate the following integral
$$I = \int_0^1 \left( \sum_{n=0}^{p-1} e(n^2t) \right)^2 dt $$
where $p$ is a prime?
Here is the method I followed:
\begin{align*}
I & = \int_0^1 \...

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178 views

### Deformations of the Riemann zeta function

Consider the Dirichlet series (for fixed $0 < a \leq 1$):
$$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$
which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...

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**1**answer

146 views

### Asymptotic density of sums of consecutive primes

Call a positive integer respectable if it is a sum of consecutive prime numbers. For example, every prime numbers is respectable. So are $3+5=8$, $2+3+5=10$, $5+7=12$, $3+5+7=15$, $2+3+5+7=17$, $7+11=...

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118 views

### Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...

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139 views

### Arithmetic progressions, given a prime

I have recently become interested in reading a little more on certain directions regarding primes in arithmetic progressions (AP). I would appreciate specific paper references (with the journal and ...

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**1**answer

99 views

### Looking for a paper by Landau and one by Watson

For the purposes of a project, I've been looking for the following two papers referred to in Serre's "Divisibilité de certaines fonctions arithmétiques":
Landau (E.), - Über die Eitenlung der ...

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**2**answers

98 views

### A variant of Turán–Kubilius inequality

Let $\omega(n)$ the number of distinct prime factors of $n$ (counted without multiplicity). A famous consequence of Turán–Kubilius inequality is
$$
\sum_{n\leq x}(\omega(n)-\log\log x)^2=O(x\log \log ...