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Questions tagged [analytic-number-theory]

On the 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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Möbius square root function: existence of multiplicative and bounded function

With $\mu$ being the Möbius function, there exist infinite possibilities of square roots. For example, for each $n$ such that $\mu(n)\neq 0$ there is a choice: if $\mu(n)=-1$, we can choose to define $...
Virgile Dine's user avatar
2 votes
1 answer
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Closed form expression for this zeta-like series involving GCD and LCM

I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$: $$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
Alexandre's user avatar
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5 votes
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172 views

"Genus theory" for elliptic curve $L$-functions

Let $d$ be a positive integer, so that $-d$ is a fundamental discriminant. This means that $d$ is square-free at odd primes and $\nu_2(d) \in \{0, 2, 3\}$. Further, if $\nu_2(d) > 0$ then $-d 2^{-\...
Stanley Yao Xiao's user avatar
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1 answer
222 views

Trying to understand last part of the proof of normalized prime gap

We know that $$\liminf_{n\to\infty}{\frac{p_{n+1}-p_n}{\log p_n}}=0.$$ I'm trying to figure out the proof and I have read a lot of documents, I asked a question here. Still I can't see what's going on....
Arda Yonet's user avatar
2 votes
1 answer
110 views

Asymptotic behavior in a modular color-cycling problem

Consider the following problem: We have $k$ rooms, each equipped with a light that cycles through three colors – red, green, and blue – in a cyclic order. Initially, all lights are set to red. Each of ...
PianothShaveck's user avatar
1 vote
1 answer
168 views

Error term of Fourier series for fractional part

In a paper of Heath-Bown The Pjateckiĭ-Šapiro prime number theorem, Journal of Number Theory 16 Issue 2 (1983) pp 242–266, MR698168, Zbl 0513.10042. on page 245, there is a Fourier expension of ...
Jia's user avatar
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2 answers
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Understanding the proof of Goldston–Pintz–Yíldírím's theorem

I hope this question fits the mission of this site. In "Primes in Tuples I" theorem 2 says, $$\liminf_{n\to\infty}{\frac{p_{n+1}-p_n}{\log p_n}}=0.$$ After a sieving progress you get $$h>\...
Arda Yonet's user avatar
5 votes
0 answers
273 views

$\sum_n a_n/n$, $\sum_n a_n/n^\rho$, $\sum_n a_n$… Tauberian theorems?

In analytic number theory, it is common to prove that $$\sum_{n\leq N} \frac{a_n}{n} = o(\log N)\tag{$\star$}\label{476699_star}$$ for some sequence $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{C}$. (It is ...
H A Helfgott's user avatar
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On lacunary series connected with prime number theory

Consider the following lacunary sum with parameter $x$: $$S(x)=\sum_{n=5}^{\infty}\sin^2\left(\frac{x\Gamma(n)}{n}\right).$$ As we can see for $x=\frac{\pi}{2}$ the sum becomes$$\sum_p\cos^2\left(\...
TPC's user avatar
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approximate value of Dedekind eta function [closed]

approximate value of $\eta(\frac{i}{\sqrt{24N}})$ , Where $N$ is a large positive integer ?
user534741's user avatar
1 vote
1 answer
160 views

Are there infinitely many primes $p$ such that $p +2$ has at most two distinct prime factors?

using lower bound sieve, one can show that there are infinitely many prime $p$ such that $p+2$ has at most four distinct prime factors [Theorem 10.2.1, 1]. Has there been any improvement of the above ...
Nicky's user avatar
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Explicit bounds on gaps between zeros of $\zeta^\prime(s)$

In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
Stopple's user avatar
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What are necessary and/or sufficient conditions for a Dirichlet series to admit analytic continuation?

Let $A = \{a(n)\}_{n \geq 1}$ be a sequence of complex numbers. By normalizing, we may as well assume that $|a(n)| \leq 1$ for all $n \geq 1$. Under this assumption, the Dirichlet series $\...
Stanley Yao Xiao's user avatar
2 votes
1 answer
173 views

Best known results on lattice point counting inside an ellipse

Consider a positive definite and primitive integral binary quadratic form $\displaystyle f(x,y) = ax^2 + bxy + cy^2$ which is reduced in the Gaussian sense: that is, we have the inequalities $\...
Stanley Yao Xiao's user avatar
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0 answers
88 views

Accentuating the appearance of convergence of the Möbius function Dirichlet series on the line $\sigma = \frac{2}{3}$ in the critical strip

Set the constant $c$ to: $$c = -\frac{3}{4}$$ which is in the interval: $$-1 < c < 0$$ and let the matrix $A$ be: $$A(n,k)=[k|n] - [n=k](1+c)$$ Then form the matrix power series: $$M=\sum _{n \...
Mats Granvik's user avatar
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4 votes
0 answers
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Bounding an expression equivalent to Mertens function

Cross-posted from MathStackExchange, where the question is bountied but has not received any comment or answer) Some months ago, I derived the following formula for the Merten's function $M(n)$ using ...
Juan Moreno's user avatar
3 votes
0 answers
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On a functional equation of Mahler?

Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
Prelude's user avatar
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A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound

In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$ I ...
user avatar
3 votes
2 answers
511 views

Exotic series for some mathematical constants from String Theory

Following these MO posts from Timothy Chow and Henri Cohen and their comments, based on Saha & Sinha's String Theory article, I have found three series for $\pi$, Apéry's constant $\zeta(3)$ and ...
Jorge Zuniga's user avatar
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4 votes
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Behavior of $m(x)\sqrt{x}$ where $m(x)=\sum_{n\leq x}\frac{\mu(n)}{n}$

Let $M(x) = \sum_{n\leq x} \mu(n)$ and $m(x) = \sum_{n\leq x} \frac{\mu(n)}{n}$, where $\mu(n)$ is the Möbius function. We know that (it is not the best known bounds): $$\limsup_{x \to \infty} M(x)x^{-...
 Babar's user avatar
  • 611
2 votes
1 answer
125 views

Reference for Mellin inversion; Meijer G-function

We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)\,ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory". I would like a similar formula ...
tomos's user avatar
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0 votes
0 answers
96 views

Series with fractional part and inverses

For $(d,d')=1$ let $$d\overline d\equiv 1(d')\hspace {10mm}d'\overline {d'}\equiv 1(d).$$ For a multiplicative function $g$ (maybe even just take $g$ the identity or $g(n)=1/n$) is there anything at ...
tomos's user avatar
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2 votes
0 answers
136 views

Elementary functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't

Background The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
Max Lonysa Muller's user avatar
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268 views

Do plots $(5)$ and $(6)$ go to infinity not at the same rate but at similar rates?

The following has been proven by joriki and GH from MO (see here): assuming that $n>1$, then the von Mangoldt function $$ \Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\...
Mats Granvik's user avatar
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6 votes
1 answer
247 views

Convergence and meromorphic continuation of a Dirichlet series under RH

Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series $$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$ converges ...
 Babar's user avatar
  • 611
0 votes
0 answers
60 views

On an oscillatory property of the Riemann Xi-function

In their paper "The Integral of the Riemann Xi-Function", Lagarias and Montague mention Wintner's 1947 proof that $$ \Xi^{(-1)}(t) > 0 \quad \text{when} \quad t > 0. $$ This result ...
Tokita Ohma's user avatar
6 votes
1 answer
642 views

Generalizations of Hamburger's Theorem

(Despite the name, the theorem in question is not a joke nor is it a statement about a delicious food). An old theorem of Hans Hamburger from 1921, as stated in Marvin Knopp's paper "On Dirichlet ...
Stanley Yao Xiao's user avatar
3 votes
1 answer
274 views

Rate of convergence of the Riemann zeta function and the Euler product formula

We consider the Euler product formula $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_p \frac{1}{1-p^{-s}}$$ I have two questions about this equality: 1)Does the rate of convergence of each side ...
Ali Taghavi's user avatar
0 votes
0 answers
193 views

Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$

Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...
Yep's user avatar
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1 vote
0 answers
65 views

General argument strategy for counting functions in AP's

For concreteness I'll refer to the divisor function but just pick your favourite multiplicative $f:\mathbb N\rightarrow \mathbb C$. If I want to estimate $$\sum _{n\leq x}d(n)$$ then I can write it as ...
tomos's user avatar
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8 votes
2 answers
393 views

Can exist a positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ are not prime for all $n \ge 1$?

Using my computer, I found that the most of positive integer number $x$ such that $a_1=x$ and $a_n=2a_{n-1}+1$ is prime number after a few iterations. But exist some positive integer numbers, my ...
Đào Thanh Oai's user avatar
8 votes
2 answers
178 views

Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators $$ L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
yoyo's user avatar
  • 609
3 votes
0 answers
1k views

Formula for $\pi$ involving exponents of Mersenne primes

Can someone provide a proof for the following claim? $$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
Pedja's user avatar
  • 2,661
7 votes
2 answers
389 views

Systematic way to compute $\sum_{n=1}^\infty P(n) / Q(n)$ for polynomials $P$ and $Q$

This may be well known so feel free to downvote. When $P = 1$ and $Q(x) = x^k$, this is of course the Riemann zeta function. But what about other cases? For instance is it always possible to express $\...
John Jiang's user avatar
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2 votes
0 answers
102 views

Division based recurrence with negative coefficients, e.g. $F(n)= -F(\lfloor n/2\rfloor) - F(\lfloor n/3\rfloor)$

A famous problem of Erdos dealt with the division-based recurrence $a_n = a_{\lfloor n/2\rfloor}+a_{\lfloor n/3\rfloor}+a_{\lfloor n/6\rfloor}$ with $a_0=1$ (and was about the limit $\lim_{n\to\infty} ...
D.R.'s user avatar
  • 831
11 votes
0 answers
436 views

Can we rule out the possibility that $\sqrt[3]{2}$ is small modulo every prime?

Consider a prime $p$ such that the polynomial $X^3-2$ splits into linear factors over $\mathbb{F}_p$: $X^3-2 = (X-\alpha_p)(X-\beta_p)(X-\gamma_p)$. It seems reasonable to expect that (identifying $\...
Jakub Konieczny's user avatar
4 votes
1 answer
150 views

Subconvexity bounds for Dedekind zeta functions of cyclotomic fields

I am looking for hybrid subconvexity bounds on $|\zeta_K(1/2+it)|$ where $K=\mathbb{Q}(\mu_n)$ is the cyclotomic field with $n$th roots of unity. Basically, I am looking for $$|\zeta_K(1/2+it)| \ll |\...
Breakfastisready's user avatar
1 vote
0 answers
199 views

Other kind of supercongruences for rational Ramanujan-like series

We can write rational Ramanujan-like series with rational parameters in the following form: $$ \sum_{n=0}^{\infty} \left( \prod_{i=0}^{2m} \frac{(s_i)_{n}}{(1)_{n}} \right) z_0^{n} \sum_{k=0}^m a_k n^...
Jesús Guillera's user avatar
2 votes
0 answers
107 views

Record for determining complete list of imaginary quadratic fields with small class number

In 2003, Mark Watkins (Class numbers of imaginary quadratic fields) determined all imaginary quadratic fields having class number at most 100. Has this list been improved? That is, what is the largest ...
Stanley Yao Xiao's user avatar
2 votes
0 answers
191 views

The exponential sum over primes on average

In https://academic.oup.com/blms/article-abstract/20/2/121/266256?redirectedFrom=fulltext Vaughan shows the following bounds for the $L^1$-mean of the exponential sum over primes $$\sqrt x\ll \int _0^...
tomos's user avatar
  • 1,381
14 votes
1 answer
1k views

The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$

Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
Roberto Trocchi's user avatar
2 votes
0 answers
146 views

Reference for accelerated sum to compute the Meissel-Mertens constant

The Meissel-Mertens constant $$ B_1 = \lim_{n \to \infty} \left(\sum_{p \leq n} \frac{1}{p} - \log\log n\right) $$ has the series representation $$ \begin{equation} \tag{1} B_1 = \gamma + \sum_{n=2}^{...
Greg Hurst's user avatar
1 vote
1 answer
221 views

Density of numbers with a large prime factor in specified arithmetic progression

I am looking for an answer to the following question. Fix some coprime integers $a$ and $b$ and let $S_x$ be the set of positive integers $n<x$ such that there exists a prime factor $p$ of $n$ with ...
AsksQuestionsAboutMath's user avatar
4 votes
1 answer
202 views

Is "self-dual" equivalent to "dihedral" for Maass forms on $\mathrm{GL}(2)$?

Suppose $f$ is a Maass cusp form on $\Gamma_0(D)$. The associated symmetric square $L$-function $L(\mathrm{sym}^2 f, s)$ has a pole at $s = 1$ if and only if $f = \overline{f}$ (if $f$ is self-dual). ...
davidlowryduda's user avatar
11 votes
1 answer
637 views

Primes such that a given number has very small order

The following came up in (a previous version of) this answer. Question. Let $a > 1$ be a positive integer, and $f \in \mathbf Z[x]$ a polynomial with positive leading term. Does there always exist ...
R. van Dobben de Bruyn's user avatar
-1 votes
1 answer
122 views

On $\Re\frac{\zeta'}{\zeta}(s) \geq -A \log(|t|+4)$ for some $A>0$

On page 438 of "Multiplicative Number Theory I" by Montgomery-Vaughan one finds the following statement and its verification : Assuming RH, there exists an absolute constant $A>0$ such ...
12321's user avatar
  • 59
0 votes
1 answer
211 views

Sign of $\Re\frac{\xi'(s)}{\xi(s)}$ locally around a zeta zero

In the article ”On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$–function” the authors check the positivity of $\Re \frac{\xi'}{\xi}(s)$ for $\frac{1}{2}<\...
12321's user avatar
  • 59
1 vote
0 answers
133 views

Automorphy of the twisted representation

The Artin reciprocity says that if $$ \chi: \operatorname{Gal}(K/\mathbb Q) \to \mathbb C $$ is a 1-dimensional representation of a finite Galois extension $K/ \mathbb Q$, then it corresponds to a ...
LWW's user avatar
  • 663
0 votes
1 answer
191 views

Estimates for the first coefficient of the Taylor expansion of $\zeta$ around a zero

Let's suppose that $s_0=\frac{1}{2}-\Delta+it$ with $0<\Delta<\frac{1}{2}$ is a simple zeta zero (i.e a zero not on the critical line). Then $1-\overline{s_0}$ is also a zero. If we take the ...
12321's user avatar
  • 59
2 votes
1 answer
202 views

Exponential sums involving smooth truncated divisor functions

Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as $...
user152169's user avatar

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