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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37 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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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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1answer
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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1answer
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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62 views

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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255 views

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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45 views

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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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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149 views

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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1answer
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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204 views

“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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66 views

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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110 views

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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1answer
90 views

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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1answer
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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47 views

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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1answer
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{...

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