# 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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### 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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113 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 ...

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48 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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74 views

### Dynamics of extrema of the Landau Riemann $\xi(1/2+it)$ function

"Relations and positivity results for derivatives of the Riemann xi function" by Coffey characterizes the Riemann xi function and its relation to the Riemann zeta function, and there has been much ado ...

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### 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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100 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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172 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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122 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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### 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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132 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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90 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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92 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 ...

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74 views

### On Prime Numbers which can be Norms of an Integral Ideal of a Number Field

We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed ...

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96 views

### What is the collection of series that amount to $\gamma$ deduced by Ramanujan?

On p. 20 of an article by Borwein et al., it is stated that Ramanujan could generalize the following formula due to Glaisher $$\gamma = 2 - 2\log2 -2\sum_{n=3, \text{ odd}} \frac{\zeta(n)-1}{n(n+1)} $$...

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53 views

### $L^p$ bound on a Weyl sum

Consider the Weyl sum
$$S(x,t)=\sum_{n=1}^Ne(nx+nt^2).$$
We have the Strichartz estimate
$$\left\|S\right\|_{L^p(\mathbb{T}^2)}\lesssim N^{1-\frac{3}{p}},\ \text{for all }p>6.$$
We also have for $...

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65 views

### Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...

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97 views

### Euler product over subsets of primes

It is well known that
$$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$
Given an arbitrary prime $\,q\,$ is it true that
$$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$
Thanks.

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77 views

### Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...

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163 views

### The existence of rational points [closed]

Solving some problem parametrically, I got the following answer:
$$ \dfrac{5x}{4} + \sqrt{\dfrac{y^2}{4} - \dfrac{x^2}{16}} + \dfrac{1}{10} \sqrt{10x^2 + 9y^2} + \dfrac{1}{5} \sqrt{5x^2 + 16y^2} $$
...

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214 views

### Strengthening Sylvester's theorem

I am working on a problem in commutative ring theory, that deals with $p$-adic valuations. This leads to a number theoretical question that I want to explain in the following.
Let $n \in \mathbb{N}$ ...

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72 views

### Improved upper bound for second moment of reduced residues modulo $q$?

The question asked here is a follow up to this question, which was answered by user GH from MO. [EDIT: The question is edited for clarification after receiving a comment on the original posting.]
As ...

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59 views

### On norms of Boolean functions

Let $f: \mathbb{F}_2^n \rightarrow \{-1,1\}$ be a Boolean function, represented by a $N=2^n$ dimensional vector, $f \in \{-1,+1\}^N$.
Define the Fourier transform of $f$ to be $\hat{f}$, where $$\hat{...

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109 views

### Write $2n+1=p+q$ with $p$ prime and $q$ practical

Recall that a positive integer $n$ is callled practical if every $m=1,\ldots,n$ can be written as the sum of some distinct divisors of $n$. The only odd practical number is $1$.
In 1996 G. Melfi [J. ...

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119 views

### Studying the vast world of Number Theory [closed]

I'm a high school student, interested in mathematics, especially in number theory.
While preparing for the IMO test, and thinking about generalizations or the root of many olympiad problems led me to ...

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66 views

### Reference needed for proof of a Tauberian theorem

I was reading the following paper by Hubert Delange: http://www.numdam.org/article/ASENS_1956_3_73_1_15_0.pdf 1. In page 26, he provides a proof of Theorem b, the bulk of which relies on a result in ...

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213 views

### Another reference for higher order Fourier analysis

I am trying to read Tao's Higher order Fourier analysis but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (...

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### Sumset of $k$-smooth numbers

Take the set $T(k,n)=M_1(k,n)$ of all $k$ smooth numbers less than $n$.
What is the cardinality of $$\{1,\dots,n\}\cap M_2(k,n)$$ where every integer in $M_2(k,n)$ is the sum of two integers in $M_1(...

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49 views

### Erdos multiplication table problem avoiding integers with too many distinct prime factors

Consider multiplication operation $$f(x_1,x_2)=x_1x_2$$ where $x_i\in\{1,\dots, n_i\}\backslash T_i$ with $n_1, n_2\in\{1,\dots,\infty\} $ where $T_i$ is set of positive integers less than $n_i$ which ...

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### Not sure about meaning of a term in English in a French research paper

I am self studying a research paper which is in French and i am not a native french speaker so I used Google Translator and Deepl translator . But I am confused over meaning of a term and have no ...

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### Distribution of table entries in Erdos multiplication table problem

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
At $k =2$ with $n_1=n_2$ this is the standard Erdos ...

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128 views

### Generalized Erdős multiplication table problem

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...

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163 views

### Efficient boxing for a mean value in the Bombieri Iwaniec method

One of the nice applications of decoupling is Bourgain’s record towards Lindelöf:
https://arxiv.org/pdf/1408.5794.pdf
Wooley has developed some techniques known as efficient congruencing which allow ...

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153 views

### On the Absolute Value of the Riemann Zeta Function on the Critical Line [closed]

Is there any sort of (closed form preferably, though if not, it's fine) function for $|\zeta(\frac12+it)|$ where $\zeta$ is the Riemann zeta function? Anything is welcome, so I can take it from there. ...

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### Possible rearrangments of double products containing sine function : [closed]

I know that the following question is not a fit (at all ) for this site , So , apologies ; but it interests me in very unusual way ; so I'm asking here . If not appropriate to post here tell me I'll ...

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246 views

### Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language

I thought of utilizing this lockdown period to study research papers in number theory by myself.
I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ...

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63 views

### Construction of weight function to satisfy condition on given functional

(Sorry for similar and trivial looking question ; But could have potential application in prime number theory )
Consider the following function :
$$F(z) = \omega(z)\frac{\sin^2\left(\frac{c\Gamma^2(...

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### Properties of Dirichlet series

I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me.
With $c\in [0,1]$,
$$f(n) = \pm 1,...

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108 views

### The equivalent proposition of Legendre's conjecture [closed]

Legendre's conjecture, proposed by Adrien-Marie Legendre, state that there is a prime number between $n^2$ and $(n+1)^2$ for every positive integer $n$.
My conjecture: Let $n$ be a positive integer, ...

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316 views

### What is known about the functional square root of the Riemann zeta function?

Let us consider the Riemann zeta function $\zeta(s)$, where $s$ can take on values on the domain $\mathbb{R}_{>1}$:
$$\zeta(s) := \sum_{n=1}^{\infty} \frac{1}{n^{s}} .$$
I wonder what is known ...

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557 views

### What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...

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### “On the distribution of reduced residues” by Montgomery and Vaughan – missing careful argument wanted

In their paper, On the distribution of reduced residues, Montgomery and Vaughan state early on that
With a more careful argument from (2) it is easily seen that
$$\tag{*}
qhP - qhPQ + O(qhP^2) \leq ...

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### On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$

The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...

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### What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]

My opinion is ;
We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question.
i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain
$F(S)=\sum_{n=...

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### On existence of conjecture relating prime zeta function:

There is an article on Wikipedia about prime zeta function (PZF):
https://en.m.wikipedia.org/wiki/Prime_zeta_function
In that article , there is table of fairly accurate values of PZF for different ...

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76 views

### On the asymptotics of the Chebyshev psi function

Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that
$$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...

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860 views

### A naive question about the prime number theorem

Let $\psi(x)=\sum_{n\leq x} \Lambda(n)$, where $\Lambda(n)$ is the von Mangoldt function.
Then as Chebyshev showed, the following equality holds
$$\sum_{n\leq x} \psi(x/n)=x\log(x)-x+O(\log(x)).$$
My ...

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152 views

### Understanding a deduction in research paper of Sprang, Fischler and Zudilin (“Many Odd zeta values are irrational”)

I am a master's student interested in number theory. I am reading a research paper in Analytic Number theory which is "Many Odd zeta values are irrational" by Stephane Fischler, Johannes Sprang and ...

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226 views

### Equivalence between Ramanujan and Selberg conjectures

At first the Ramanujan conjecture for automorphic forms and the Selberg conjecture appear to be understood as totally independent. However, they are now known to be tighyly connected once viewed in ...

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### Asymptotics of Hurwitz zeta function

Can someone help me please with a reference about the asymptotics (or just upper bounds) of Hurwitz $\zeta(s,z)$ as $|t|\rightarrow \infty$ with $Re(z)>0,$ $s=\sigma+i t$ and $\sigma<0$? I have ...

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110 views

### inequality for sum $\sum_{j_{i}=1,i=1,\cdots,k}^{n}\gcd(j_{1},j_{2},\cdots,j_{k})$

if $n>k>1$ be postive integer,show that
$$S_{k}(n)=\dfrac{1}{n^k}\sum_{j_{1}=1}^{n}\sum_{j_{2}=1}^{n}\cdots\sum_{j_{k}=1}^{n}\gcd(j_{1},j_{2},\cdots,j_{k})\le\dfrac{\zeta(k-1)}{\zeta(k)} \tag{...