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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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
2 votes
1 answer
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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
3 votes
0 answers
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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
138 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
4 votes
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139 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
185 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
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4 votes
2 answers
397 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
351 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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Interpretation of the rank of a system $h=(h_1,\dots,h_R)$ of forms when $R=1$

I am reading a paper Liu and Zhao - On forms in prime variables. The paper establishes the minimum number of variables required by a general system of forms of degree $d\ge2$ to have solutions in ...
Anish Ray's user avatar
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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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-6 votes
0 answers
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Average Goldbach and Riemann Hypothesis

This question is a follow up to About Goldbach's conjecture. To make it self contained, I paste below the required prerequisites of this former question of mine: "let's consider a composite ...
Sylvain JULIEN's user avatar
4 votes
0 answers
434 views

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
423 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
623 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
97 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
177 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
152 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
0 votes
1 answer
276 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
189 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
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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
39 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
298 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
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2 votes
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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
152 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
205 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
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3 votes
1 answer
284 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
3 votes
1 answer
265 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 ...
qifeng618's user avatar
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0 votes
1 answer
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Real part of the Riemann zeta function

Consider the real part of the Riemann zeta function on the critical line. Are there any results for the number of zeros of this real function in the interval [0,T]?
Autovetor's user avatar
2 votes
0 answers
144 views

Large prime divisors of values of a polynomial, in a given residue class

Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
Jakub Konieczny's user avatar
1 vote
0 answers
165 views

Upper bound on sum of Lambda(n) over short interval

I am looking for a bound of type $$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$ (or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
H A Helfgott's user avatar
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6 votes
1 answer
299 views

A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?

This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function: $$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
Agno's user avatar
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2 votes
0 answers
131 views

Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$

I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
LWW's user avatar
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6 votes
0 answers
266 views

Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
H A Helfgott's user avatar
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0 votes
0 answers
193 views

Does there exist an $L$-function for any subset of $\mathbb{N}$?

Consider the following prime sum: \begin{aligned} \sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}} \end{aligned} whose spikes appear at the Riemann $\zeta$ zeros as shown here. Taking these detected spikes (...
martin's user avatar
  • 1,893
1 vote
0 answers
261 views

On fifth powers forming a Sidon set

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct. Erdős conjectured ...
Sayan Dutta's user avatar
0 votes
1 answer
229 views

Is there any use of logarithmic derivatives of modular forms?

Does taking the logarithmic derivative of a modular form have any uses, such as identifying patterns in its coefficients or possible zeros of its corresponding L function?
Samay Varjangbhay's user avatar
27 votes
1 answer
2k views

Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
Pietro Majer's user avatar
5 votes
0 answers
125 views

Taking integer values of a sequence of Beurling primes

Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
Anon12345's user avatar
2 votes
1 answer
221 views

'$\times$' or '$\otimes$' when writing $L$-functions?

Recently, I came across the Langlands correspondence theorem, there is the following line: $$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$ where $\sigma$ and $\tau$ are ...
Misaka 16559's user avatar
18 votes
1 answer
1k views

Does summing divergent series using cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...
not all wrong's user avatar
2 votes
0 answers
179 views

Can all modular forms be written as Eta Quotients?

I have been going through a couple of introductory courses in modular forms and am quite curious whether all modular forms can be written as eta quotients of the Dedekind eta function?
Samay Varjangbhay's user avatar
4 votes
2 answers
578 views

Computing hypergeometric function at 1

I'm looking to compute $${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr 1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$ for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
JMK's user avatar
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1 vote
0 answers
103 views

Behavior of Dirichlet L-functions at the edge of the critical strip

Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
edward cornfoot's user avatar
1 vote
0 answers
156 views

Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
53Demonslayer's user avatar
8 votes
1 answer
627 views

Does this partial sum over primes spike at all zeta zeros?

Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \ p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines: Below ...
martin's user avatar
  • 1,893
0 votes
0 answers
69 views

Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?

What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$, every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
H A Helfgott's user avatar
  • 19.3k
2 votes
2 answers
279 views

$L^1$ norm for a product of cosines

Let $k$ be an integer and consider the function $$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t). $$ I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
Itachi's user avatar
  • 167
2 votes
2 answers
594 views

Can any Hurwitz zeta function be written as an Euler product?

I am attempting to understand the behavior of Hurwitz zeta functions and for what $a$ do they have an analytic continuation. Is it possible to write any Hurwitz zeta as an Euler product or are there ...
Samay Varjangbhay's user avatar

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