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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Geometric mean of prime factors of all numbers up to n

Through numerical calculations I have discovered that for any natural number $n \geq 2$, the geometric mean of the prime factors of all natural numbers $\leq n$ can be approximated well by $1.6653 \...
Marcos Cramer's user avatar
4 votes
1 answer
168 views

Conditional convergence of Artin $L$-functions

Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v ...
Daniel Loughran's user avatar
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Analyzing a Dirichlet series with log-oscillating terms via Fourier methods

I am investigating the series $S(z)$ defined as follows: $$ S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)), $$ where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$. I want ...
swami's user avatar
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In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?

Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements: $\lambda$ being a random large prime such as $w^\lambda > 2\times m$ $1 < n < m−1$. m is ...
user2284570's user avatar
2 votes
1 answer
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Are the coefficients in the stationary phase approximation computed explicitly somewhere

In Stein's "Harmonic analysis" book, page 334, one can find the asymptotic expansion An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
Medo's user avatar
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12 votes
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Prime differences and zero multiplicity

Concerning gaps between consecutive primes, Paul Erdős conjectured that: $$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$ Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), ...
Felixson's user avatar
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Gaussian primes in translations of lattices in $\mathbb{Z}[i]$

I am considering undertaking some independent research in my summer break studying Gaussian primes in translations of lattices in $\mathbb{Z}[i]$, i.e. sets of the form $ \{ a+sx+tw:s,t \in \mathbb{Z} ...
Daniel Lang's user avatar
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Tuples of natural numbers with no mutual divisibility and large reciprocal sums

Standard apology in case this is something simple, as I'm not a number theorist. Let $E_1, \dots, E_n$ be disjoint finite sets of natural numbers, such that for any $a_1 \in E_1, \dots, a_n \in E_n$, ...
Sophie M's user avatar
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Analytic number theory and condensed mathematics

As of 2024, are there current or planned applications of condensed mathematics to analytic number theory? If so, what are suggested readings?
Jon23's user avatar
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How to check that a number probably/likely has a divisor having a specific bit length/in range?

Given a randomly generated $\alpha\in\Bbb N$ where $\alpha$ is large thus hard to factor (no small prime composites). How to check that a divisor $F\in\Bbb N$ with a specific bitlengh $n\in\Bbb N∧n<...
user2284570's user avatar
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A question on generalized bases

I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
Dumbest person on earth's user avatar
5 votes
1 answer
364 views

Lindelöf hypotheses for derivatives of zeta

The Lindelöf hypothesis says that if we have: $$\zeta(\sigma+iT)=\mathcal O(T^a)$$ Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
psubodiosa's user avatar
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Average of number of divisors of shifted exponential sequence

Let $a$ be a fixed positive integer greater than 1. We define the sequence $u_n=a^{n}-1$ for all positive integers $n$. Then are there any results in literature for asymptotic value of the sum $$\sum_{...
Hhhhhhhhhhh's user avatar
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Asymptotics of $\vartheta(x+y)-\vartheta(x)$, where $\vartheta$ is the Chebyshev function, when $y\in[x^\alpha, x]$ for some $\alpha\in(0,1)$

Introduction Consider $\vartheta$ to be the Chebyshev function, that is, $\vartheta(x)$ denotes, for $x\in\mathbb N$, the sum $\sum_{p\le x,\ p\text{ prime}} \ln p$. I am interested in asymptotics for ...
Maximilian Janisch's user avatar
2 votes
1 answer
136 views

Upper bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\ \gcd(k,n)=1} \sin\Big(t-\frac{k \pi}{n}\Big),\quad t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial upper ...
AgnostMystic's user avatar
4 votes
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159 views

Effective bound for odd numbers expressed as sums of three primes

I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...
Anton Rechenauer's user avatar
6 votes
0 answers
572 views

Generating functions in countable commutative monoids

Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
Tian Vlašić's user avatar
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Sum of reciprocals of primes dividing Mersenne numbers

Let $p$ be a prime. Then it has been proven by Erdős that $$\sum_{p\mid 2^{n}-1}\frac1{p} \ll \log \log \log n.$$ Let $c<1$ be a positive constant. Then can we prove that $$\sum_{p\mid 2^{n}-1}\...
Hhhhhhhhhhh's user avatar
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Techniques of showing the Order

In their paper The integral of Riemann Xi Function, Lagarias and Montague show that the integral $$\Xi_{\lambda}^{-1}(z)=2\int_0^{\infty}e^{\lambda u^2}\phi(u)\Big( \frac{\sin zu}{u}\Big)\;du$$ is ...
Tokita Ohma's user avatar
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1 answer
286 views

A basic conjecture/observation on the Riemann $\xi$-function

Based on computations I have made the following mini-conjecture: For any zeta zero $s_0$ with $|s_0|\geq|1-s_0|$ and for $0<\tau<1$ define $M_\tau=|s_0|(1+(1-\tau)^2)$. Let $\xi$ stand for the ...
Alexis's user avatar
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1 answer
670 views

Infinite dimensional lattice for integers and the Riemann hypothesis?

It is known that for each finite set of primes $p$ we have: $\log(p)$ are linear independent over the rational numbers. We have $\log(ab) = \log(a)+\log(b)$ and $\log(n) = \sum_{p |n}v_p(n) \log(p)$. ...
mathoverflowUser's user avatar
1 vote
1 answer
178 views

Classification of L functions and Dirichlet series by poles

I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions. Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
asdfasdgasdfgasfgasdf's user avatar
3 votes
2 answers
255 views

Equidistribution on $\mathrm{SU}_2$

Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant ...
Local's user avatar
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1 answer
166 views

Modified Gauss Sum when the characters have different period

Let $\chi$ be a Dirichlet character mod q, and \begin{eqnarray} t(\chi)=\sum_{n=1}^{q}\chi(n)e(\frac{n}{2q}). \end{eqnarray} Do we have a bound or formula for $t(\chi)$ similar to that of the usual ...
Jonathan Lam's user avatar
3 votes
1 answer
183 views

Generators of the ideal class group

Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following: Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
Rashad Ek's user avatar
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Error function of the second moment of the divisor function

It is easy to show that the second moment of the divisor function has asymptotics: $$\sum_{n\leq x} d_0(n)^2 = xP(\log(x))+E_2(x)$$ Where $P$ is some polynomial and that: $$E_2 = o(x)$$ Previously, ...
psubodiosa's user avatar
2 votes
0 answers
150 views

Electrostatic potential energy of point-charges at primes up to $x$

Given a positive real (or integral) number $x$ we consider the electrostatic potential energy of equal point charges at all primes up to $x$ given by $$E(x)=\sum_{p_1<p_2\leq x}\frac{1}{p_2-p_1}$$ ...
Roland Bacher's user avatar
5 votes
0 answers
167 views

Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$

It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$ for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
H A Helfgott's user avatar
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Density of a set of numbers whose prime factors are defined by congruences

Let $S$ be the set of positive integers not divisible by $3$ where if $p$ is a prime factor of $n \in S$ and $p \equiv 1\bmod 3$ then $p^2$ does not divide $n$, but if $p\equiv2 \bmod 3$ then $p^2$ ...
AsksQuestionsAboutMath's user avatar
3 votes
0 answers
212 views

Number of zeros of the zeta function along horizontal lines

Are there any known results about the number of zeros of the zeta function along horizontal lines of the complex plane? The Riemann hypothesis states that for any such line the number is at most 1, ...
Alexis's user avatar
  • 31
4 votes
2 answers
429 views

Reference request - Pillai-Selberg Theorem

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
alidixon222's user avatar
2 votes
1 answer
365 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
T. Amdeberhan's user avatar
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1 answer
125 views

A question about the setup of zero density estimates for Dirichlet $L$-functions

For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
Josh's user avatar
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4 votes
0 answers
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There are infinitely many prime which have arbitrary large gap in their digits in particular base expansion

Consider $m$ and $r$ is any fixed positive integer and $t$ is a variable $(t=0,1,2,3,...)$. Below, $[a]$ denotes the greatest integer function of $a$ (or floor function). Claim 1 : There exists a ...
Pruthviraj's user avatar
1 vote
1 answer
435 views

Conjectured upper bound on the maximum value of the absolute value of the Möbius function in the poset of multiplicative partitions under refinement

PRELIMINARIES: Consider the poset $(\mathcal{P}_n, \leq_r)$ of the (unordered) multiplicative partitions of $n$ partially ordered under refinement (for all $\lambda, \lambda’ \in \mathcal{P}_n$, we ...
Tian Vlašić's user avatar
6 votes
2 answers
641 views

On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$

I am interested in determining the behaviour of the the series/function $$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$ near $s=0$. It is clear that $f(0)$ is undefined....
Tian Vlašić's user avatar
3 votes
0 answers
104 views

Is there any notion of Poincaré series for Hermitian modular forms?

I have been studying modular forms and their generalisations for a year or so. It is a very interesting fact that the space of cusp forms $S_k$ is generated by the Poincaré series of exponential type (...
Ancient Antagonist's user avatar
1 vote
0 answers
182 views

Asymptotic behaviour of a sum involving Möbius function

(This is a cross-posted simplification of this question posted in MSE which did not have a complete answer.) I am trying to get the asymptotic behaviour when $n$ grows to infinity of a partial sum of ...
Juan Moreno's user avatar
1 vote
1 answer
155 views

prime distribution lower bound: Does the inequality hold for $s=19/10$?

Define a transformation of the prime counting function, $\pi(k)$, by $$J: (0,1) \longrightarrow (0,1) $$ where $$J(x)= \lim_{r \to \infty} \frac{\int_2^{rx} \pi(k) \, dk }{\int_2^r \pi(k) \, dk}$$ ...
53Demonslayer's user avatar
1 vote
1 answer
289 views

Is there any way to estimate this functions: $f(n)=\sum_{d|n}d\varphi(d)$ and $g(n)=\sum_{d|n}\frac{\varphi(d)}{d}$?

Let that $n$ be a natural number and $\varphi(n)$ be the Euler totient function. Is there any formula or estimation for computing functions $f,g$ such that: $$ f(n)=\sum_{d\mid n}d\varphi(d) $$ and $$ ...
Jamal Farokhi's user avatar
6 votes
1 answer
190 views

A strictly increasing, analytic function that goes through key points of the iterated logarithm?

Is it possible to create a function $f(x)$ that: is strictly increasing (at least for $x>0$) is real analytic goes through all the points where the iterated logarithm would increment value? i.e. [...
user5399200's user avatar
0 votes
0 answers
58 views

What is the lattice point distribution over binary quadratic forms?

Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$. For simplicity, we keep things only on quadrant I of the ...
ReverseFlowControl's user avatar
1 vote
0 answers
40 views

What is the form of the incomplete Eisenstein series on PGL_2(C)?

Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
Misaka 16559's user avatar
2 votes
0 answers
190 views

Zeros of the semiprimes

Let $P$ be the prime zeta function $$ P (s) = \sum_{p\, \in\text{ primes}} \frac 1 {p^s} = \frac {1} {2^s} + \frac {1} {3^s} + \frac 1 {5^s} + \frac 1 {7^s} + \frac 1 {11^s} + \cdots $$ and define the ...
martin's user avatar
  • 1,893
13 votes
0 answers
305 views

Upper bound on prime powers in interval

I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses: (a) the ...
H A Helfgott's user avatar
  • 19.3k
2 votes
0 answers
68 views

The logarithmic derivative of a twisted L-function?

Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have $$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$ (I ...
Misaka 16559's user avatar
1 vote
0 answers
158 views

Integers with $k$ prime factors, in terms of the Möbius function

A $k$-free integer is an integer $n$ such that there is no $k$th power dividing $n$. It is well known (see Murty's Problems in Analytic Number Theory q1.18 for instance) that \begin{equation}\sum_{d^k|...
alidixon222's user avatar
5 votes
1 answer
212 views

A limit related to quasi-periodic function

Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that $$ \frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2} $$ ...
Sean's user avatar
  • 313
3 votes
1 answer
286 views

Does this condition characterise intervals, among subsets of the real line?

For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$: $\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
Pietro Majer's user avatar
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3 votes
2 answers
302 views

Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?

It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by \begin{equation*} \frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
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