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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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....
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0
answers
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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 (...
1
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0
answers
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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 ...
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answer
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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
$$J(x)= \lim_{r \to \infty} \frac{\int_2^{rx} \pi(k) \, dk }{\int_2^r \pi(k) \, dk}$$
...
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vote
1
answer
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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
$$
...
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votes
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answer
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A strictly increasing, analytic function that goes through key points of the iterated logarithm?
Is it possible to create a function $f(x)$ that:
is strictly increasing (at least for $x>0$)
is real analytic
goes through all the points where the iterated logarithm would increment value?
i.e. [...
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0
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What is the lattice point distribution over binary quadratic forms?
Let $f(x,y)=x^2+ny^2$ be the binary quadratic form of interest and consider the lattice points $S=\{ (x,y,f(x,y)) \in \mathbb{N}^3 \}$.
For simplicity, we keep things only on quadrant I of the ...
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What is the form of the incomplete Eisenstein series on PGL_2(C)?
Let $F$ be an imaginary quadratic number field. Let $G = \mathrm{PGL}_2(\mathbb{C}) $ and $\varGamma = \mathrm{PSL}_2(\mathcal{O}_F)$. We have the Iwasawa decomposition $G = NAK$ where $K = \mathrm{...
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Zeros of the semiprimes
Let $P$ be the prime zeta function
$$
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
$$
and define the ...
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Upper bound on prime powers in interval
I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses:
(a) the ...
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The logarithmic derivative of a twisted L-function?
Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have
$$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$
(I ...
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Integers with $k$ prime factors, in terms of the Möbius function
A $k$-free integer is an integer $n$ such that there is no $k$th power dividing $n$. It is well known (see Murty's Problems in Analytic Number Theory q1.18 for instance) that \begin{equation}\sum_{d^k|...
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A limit related to quasi-periodic function
Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that
$$
\frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2}
$$
...
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votes
1
answer
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Does this condition characterise intervals, among subsets of the real line?
For a real number, $c\in \left]0,1\right[$, consider the following property $\mathbf(\mathbf P_c\mathbf)$ of subsets $A$ of $\mathbb R$:
$\mathbf(\mathbf P_c\mathbf)$ For every bounded set $B\subset \...
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Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?
It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
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Real part of the Riemann zeta function
Consider the real part of the Riemann zeta function on the critical line. Are there any results for the number of zeros of this real function in the interval [0,T]?
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answers
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Large prime divisors of values of a polynomial, in a given residue class
Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
- 1,762
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vote
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Upper bound on sum of Lambda(n) over short interval
I am looking for a bound of type
$$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$
(or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
- 19.4k
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A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?
This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function:
$$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
- 4,169
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votes
0
answers
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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 ...
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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 ...
- 19.4k
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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 (...
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0
answers
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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 ...
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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?
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answer
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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 ...
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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 $N_P$ ...
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answer
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'$\times$' 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 ...
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votes
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answer
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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 $\...
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0
answers
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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?
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2
answers
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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 ...
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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 ...
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0
answers
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Understanding the effect of PDE solution on critical strip?
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 ...
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votes
1
answer
652
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 ...
- 1,893
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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 ...
- 19.4k
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votes
2
answers
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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 ...
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2
votes
2
answers
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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 ...
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votes
2
answers
643
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
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votes
1
answer
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Generating function over primes in an arithmetic progression
Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function
$$
\sum_{p\equiv a\pmod{m}}a(p)q^p
$$
over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
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votes
1
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Smallest prime factor of numbers
The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
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views
Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions
Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that
$p$ does not divide the conductor of $d$,
$p$ splits ...
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votes
1
answer
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On a summation in "Artin's conjecture for primitive roots" by Heath-Brown
This is a question on the paper: D. R. Heath-Brown. Artin's conjecture for primitive roots. Quarterly J. Math. 37 (1): 27–38, 1986.
At the beginning of the proof of his main theorem on page 35, Heath-...
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votes
0
answers
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views
Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,888
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votes
1
answer
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Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$
Euler proved
$$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$
where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
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votes
1
answer
315
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
- 4,638
1
vote
0
answers
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views
Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves
In this article in Compositio (2011), Munshi proves a mean value result for
$$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$
where here $f$ is a primitive holomorphic form of level $q$ with ...
- 1,191
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votes
1
answer
709
views
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in ...
- 2,462
1
vote
0
answers
100
views
Relationship between two types of partition functions
Referring to this unanswered question on MS, I'm posting the same question here:
For $s\in \mathbb{C},\Re(s)>1 $, consider:
$$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}...
- 271
2
votes
1
answer
192
views
Series with the smallest number whose square is divisible by $n$
I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
0
votes
1
answer
209
views
Implications for large sums of roots of unity
I have some coefficients $(a_n)_{n \leq N} \subset \mathbb{R}$ such that $a_n \geq 0$ and their average value is one, i.e. $\frac{1}{N} \sum_{n \leq N} a_n = 1$. Suppose that
$$ \Bigl| \sum_{n \leq N} ...
3
votes
1
answer
210
views
Non-vanishing of archimedean integral representations
Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
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