# 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.

2,203
questions

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91 views

### Watson's triple product for automorphic forms shifted by Maass rising operators

Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...

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78 views

### Origin of the term “singular integral” in the circle method

Ever since I learned about the circle method, I have implicitly held the following beliefs about the topic in the title:
The terms "singular integral" and "singular series" were ...

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79 views

### Compression bounds on the Copeland–Erdős constant

Motivation:
Given the set of prime numbers $\mathbb{P} \subset \mathbb{N}$, the Copeland–Erdős constant $\mathcal{C}$ is defined as [1]:
\begin{equation}
\mathcal{C} = \sum_{n=1}^\infty p_n \cdot 10^{-...

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67 views

### On a certain inverse Mellin transform involving $\zeta$

Let $c \in (1/2,3/2)$. In my research I have encountered the inverse Mellin transform of $\zeta(s-1)/s$:
$$f(x) = \frac{1}{2\pi i} \int_{(c)} \frac{\zeta(s-1)}{s}x^{-s} ds.$$
Does it have a closed ...

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**1**answer

552 views

### Possible contemporary improvement to bounded gaps between primes?

In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps ...

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35 views

### k specific prime factors guess and related prime guess [duplicate]

there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...

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110 views

### Is there an example of a non-principal Dirichlet character $\chi$ such that $\chi(F_n)\in \{0,1\}$ when evaluated at Fibonacci numbers $F_n$?

This is related to Sum of Fibonacci sequence evaluated at a Dirichlet character, but can be also be considered as a stand-alone.
I did an exhaustive search on non-principal (not necessarily primitive) ...

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81 views

### Sum of reciprocals of friable (i.e. smooth) numbers

Consider the `logarithmic' sum $$F(x,y) = \sum_{n > x, \, n \text{ is }y\text{-friable}} \frac{1}{n}.$$
What are it's asymptotics (for general $x,y\ge 1$).
I would expect that this sum was studied ...

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452 views

### Update on “A Mad day's work” by Cartier

In his paper "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry," Cartier discusses in the last sections 8 and 9 the role of ...

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66 views

### What are some bounds on $\displaystyle \sum_{n\leq N} \mu(n)e(\alpha n)$? [duplicate]

As usual, let $e(x):=2\pi i x$. I already know of a bound on $\sum_{n\leq N}\mu(n)$: by the prime number theorem, it is $\ll x\exp(-c(\log x)^{\frac{1}{2}})$ for some $c>0$.
However, what happens ...

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102 views

### On primes of specified length and bit pattern

Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...

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109 views

### On the connection between sums of prime numbers and distribution of prime numbers

As an amateur mathematician, I have always been fascinated by the magic of prime numbers, and their apparently random distribution. I was utterly amazed when I found the following connection between ...

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163 views

### What are some estimates for $\displaystyle \sum_{\substack{n, m \leq N\\ n, m \equiv a \bmod q\\n \not= m}}\mu(n)\mu(m)$?

Consider the sum $$\sum_{\substack{n,m\leq N\\ n, m \equiv a \bmod q\\n \not= m}}\mu(n)\mu(m).$$ What are some of the best known estimates regarding this sum? For the case where we allow $n=m$, I am ...

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656 views

### Modular forms with finitely many or very few non-zero Fourier coefficients

I have an elementary question on modular forms, but which I don't know how to solve.
a) Is there a congruence subgroup $\Gamma \leq \mathrm{SL}_2(\Bbb Z)$, an integer $k \in \Bbb Z$ and a non-...

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151 views

### Goldfeld resolution of the quadratic class number problem

Goldfeld proved the following result. Let $E$ be an elliptic curve (with conductor $N$) over $\mathbb{Q}$ whose Hasse-Weil L-function has a zero at $s = 1$ with multiplicity $g$ then for sufficiently ...

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106 views

### On integral relating logarithmic of absolute value of Zeta function:

Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?

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**1**answer

1k views

### Optimality of the Riemann Hypothesis

The Riemann hypothesis is equivalent to the assertion that the prime counting function $\pi(x) := \sum_{p \le x} 1$ deviates from the logarithmic integral $Li(x) = \int_2^x \frac{dx}{\log x}$ in the ...

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59 views

### Ordinary primes for a weak form corresponding to a CM newform

Setup: Let $f$ be a harmonic Maass form of weight $2-k$ ($k \in \mathbb{N}$), level $N$, and character $\chi$. Letting $q := e^{2\pi i z}$ and considering the Fourier expansion of any harmonic Maass ...

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295 views

### Partial sums of the Möbius function on arithmetic progressions

There is a result of Soundararajan on the upper bound of the partial sums of the Möbius function assuming GRH here. Suger and Halupczok find an analogous bound for $\displaystyle \sum_{\substack{n\leq ...

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476 views

### Advice: What topics to study now in analytic number Theory( And if there are video lectures( Open Online course) / Course notes available on website)

I am a person living in a 3rd world country and completed my masters in mathematics in July 2020. Then I began to study some additional topics in Pure Mathematics as I was applying for Ph.D. abroad( ...

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**1**answer

137 views

### Complexity of a Diophantine equation having $\leq1$ solutions

We are provided a single Diophantine equation
$$f(x_1,\dots,x_n)=0$$
having degree $\geq2$ and having the promise it has $\leq1$ solutions in the set $\{0,\dots,m-1\}^n$ and $t$ is the number of terms ...

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184 views

### A question on assigning finite values to divergent sums involving expression of primes

We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...

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228 views

### What's the average order of the reduction of a section of an elliptic curve

Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n_p$ in the group $E(...

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152 views

### Does one have $\alpha_{n}\ll\sum_{p^m\leq n, m\geq 2}\Lambda(p^{m})\log n$?

This question is a follow-up to my question "About Goldbach's conjecture" (direct link: About Goldbach's conjecture) whose beginning I copy-paste below:
"Let's consider a composite ...

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169 views

### Maximum value of newform from Galois representation

One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the ...

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139 views

### What is a non-trivial upper bound on the $k$th prime below a given prime $p$?

Given a prime number $p_0$, by Bertrand's postulate we know that
\begin{gather}
p_1\ge\frac{p_0}{2}\\
p_2\ge\frac{p_1}{2}\ge\frac{p_0}{2^2}\\
\vdots\\
p_k\ge\frac{p_0}{2^k}
\end{gather}
where $p_1,p_2,...

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118 views

### Integers having the same digits as their prime factors

Say an integer is "digitally conservative" in base $b$ if the set of its digits coincides with the set of the digits of its prime factors, like $37127=137\times 271$ in base $10$, and denote ...

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210 views

### Lower bounds for class number of function fields with fixed $q$, growing $g$

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...

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677 views

### What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?

Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges?
Can one compute a few digits based on euristic considerations or plausible ...

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298 views

### Reference for behavior of Artin $L$-functions at $\Re(s) = 1$

Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters.
I assume this was done in ...

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104 views

### Obtain a series expansion of $a^2(q)a^2(q^4)$

Let $a(q)$ denote the Borwein function $$a(q)=\sum_{m,n=-\infty}^\infty q^{n^2+nm+m^2}.$$
In this research paper the author has obtained the series expansion for $a(q)a(q^4)$. I want the series ...

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299 views

### A generating function for non-trivial zeros of Riemann zeta function

Suppose $0^+_\zeta$ is the set of non-trivial zeros of the Riemann zeta function $\zeta(s)$ which lie on or to the right of the critical line and above the $x$-axis, i.e,
$$0^+_\zeta = \{s \in \mathbb{...

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81 views

### Smooth number pairs satisfying a congruence

Let $\mathcal P=\{p_1,\dots,p_{2t}\}$ be $2t$ primes between $2^\ell$ and $2^{\ell+1}$ and fix an exponent bound $a\in\mathbb Z_{\geq2}$.
Fix $N\in\mathbb N$ whose prime factors $p$ satisfy $p>2^{\...

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120 views

### $\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $

Consider,$$I=\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $$
where $\zeta$ is the Riemann Zeta function.
Since , Hardy (...

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217 views

### Why is the Barban-Halberstam-Davenport theorem important?

I have a slightly open-ended question about the Barban-Halberstam-Davenport theorem and hope that it is not off-topic. The theorem itself states that for any $A>0$ and $Q$ lying the range $x\log^{-...

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68 views

### Is there only one holomorphically induced parametrization $\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$?

A function $f:\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ is called a parametrization if for any $n\in \mathbb{Z}_{\geq 0}$ there exists $k\in \mathbb{Z}_{\geq 0}$ such that $n=f^{\circ k}(0)$.
A ...

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42 views

### Finding generator versus finding order for multiplicative groups modulo composites

Let $N$ be a composite and let $\lambda(N)$ be Carmichael Lambda of $N$. It is known finding $\lambda(N)$ reveals information about factors.
Is there a polynomial time (under suitable assumptions) or ...

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**1**answer

346 views

### Splitting of small primes in number fields generated by the torsion of elliptic curves

Suppose $E/\mathbb Q$ is a non CM elliptic curve and we look at the number field $K_d$ generated by the $d$-torsion of $E$. What is known about the (complete) splitting of small primes in $K_d$?
More ...

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235 views

### Proof of Mertens' formula

$$\sum_{p\leq x}\frac{1}{p} = \log(\log x) + a+ O\left(\frac{1}{\log x}\right)$$
$$ a=\int_{2}^{\infty}R(u)\frac{du}{u(\log u)^2} +1- \log(\log 2)$$
$$ e^{-1/p}(1-1/p)^{-1}= 1+O(1/p^2)$$
Below is ...

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315 views

### On infinite sum containing logarithmic derivative of Zeta function and Möbius function:

Consider the following function:
$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$
Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...

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29 views

### Asymptotics of a sum involving multiplicative partitions of an integer $n$ into $k$ possibly non-distinct parts $≥2$

Let $x\in\left(0,1\right)$. For each integer $n\geq2$, let $\Omega\left(n\right)$ denote the number of prime factors of $n$, counted according to multiplicities; thus $\Omega\left(2\right)=1$, $\Omega\...

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106 views

### Applications of Jensen's Formula to entire functions of finite order

I am trying to understand a frequently omitted technical detail in applications of Jensen's Formula to bound the number of zeros of entire functions of finite order.
We say that an entire function $f(...

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849 views

### Correlations of $\phi(n)/n$

We know that the average of $\phi(n)/n$ is approximated by a constant. Here $\phi $ is the Euler quotient function. One can furthermore show asymptotics with a secondary main term, at least
for the ...

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**1**answer

554 views

### Where should I learn about the p-adic L-functions of elliptic curves?

Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ...

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93 views

### Degree of compositum of all number fields under given discriminant

For an integer $n\geq 1$ define $f(n)$ to be the degree of the compositum of all number fields with discriminant at most $n$. What bounds are known on $f(n)$?

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861 views

### On modified Euler product

I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it.
Consider the modified Euler product as ...

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304 views

### Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?

Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...

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158 views

### Richaud-Degert type quadratic extensions

A Richaud-Degert type real quadratic field is a number field of the form $K = \mathbb{Q}(\sqrt{d})$ where $d = {(an)}^2 + ka > 0$ for positive integers $a, n$ and $k \in \{ \pm 1, \pm 2, \pm 4 \}$, ...

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**1**answer

97 views

### Sum of inverse squares of numbers divisible only by primes in the kernel of a quadratic character

Let $\chi$ be a primitive quadratic Dirichlet character of d modulus $m$, and consider the product
$$\prod_{\substack{p \text{ prime} \\ \chi(p) = 1}} (1-p^{-2})^{-1}.$$
What can we say about the ...

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56 views

### Stabilizers of points in the upper half-plane

Suppose that $\Gamma$ is a group acting discontinuously on $\mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 1\}$. In order to keep things simple, suppose that $\Gamma \subseteq \...