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.
2,902
questions
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Measuring philoprimality/misoprimality
Given a natural integer $x$, let
$$\alpha(x)=(\log x)^2\sum_{p\in\mathcal P\setminus\{x\}}\frac{1}{(x-p)^2}$$
(with $\mathcal P$ denoting the set of prime-numbers)
measure its "philoprimality&...
5
votes
3
answers
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Consequences of Goldbach's Conjecture
In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement ...
3
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What is the smallest sequence $a_k$ with nondecreasing $\frac{\sigma(a_k)-H_{a_k}}{\exp(H_{a_k})\log(H_{a_k})}$?
This is inspired by the Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH , an answer and some comments there.
For $n\geqslant2$ denote
$$
L(n):=\frac{\sigma(n)-H_n}{\exp(...
21
votes
2
answers
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What are the consequences of an ineffective proof of the Riemann Hypothesis?
Suppose a proof came out (and was verified by credible peer review) of the following statement:
There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$
where $...
4
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2
answers
405
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Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH
In page 197, Equivalents of the Riemann Hypothesis Vol 1, the following statement caught my eye
There is an editorial comment in [102] that includes an observation by
the GCHQ Problem Solving Group. ...
5
votes
1
answer
330
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Coefficients of modular forms and the Sato-Tate distribution
Let $a(n)$ be the $n$th Fourier coefficient of a normalized Hecke eigenform $f(z)=\sum_{n=1}^{\infty}a(n)q^n$ of weight $k$ with respect to the full modular group, where $q=e^{i2\pi z}$.
A new paper [...
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0
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154
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Reference request for the following results
I am looking for references on the following results. In what follows $\pi(x)$ denotes the prime counting function.
Result 1. For all real $k>1$ there exists $x^k_0 \in \mathbb{R}$ such that for ...
4
votes
1
answer
230
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Short proof of the error bound in PNT assuming a zero-free strip?
I am looking for a short proof of the fact that $\zeta(z)\neq 0$ for $\Re z>a$ implies the prime number theorem with an error bound $O(x^{a+\varepsilon})$ for any $\varepsilon>0$, which would be ...
8
votes
1
answer
458
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Conjecture about the density of primes
Conjecture
For any sufficiently large integer $kn$ , the sequence representing
the number of primes in each block obtained by splitting $kn$ into $k$
equal blocks, is a strictly decreasing sequence, ...
29
votes
2
answers
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Closed formula for a certain infinite series
I came across this problem while doing some simplifications.
So, I like to ask
QUESTION. Is there a closed formula for the evaluation of this series?
$$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)}{...
4
votes
0
answers
79
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Joint mean values of arithmetic functions in sequences and families of sequences
This is a bit of a follow up question to this question I asked a couple days ago. The main content of that post can be phrased as asking for a nontrivial lower bound on the sum
$$
\sum_{n\leq x} \...
7
votes
1
answer
743
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$\mathit{NP}$-hard statements which are $\mathit{NP}$-complete under the Riemann Hypothesis
$\newcommand\NP{\mathit{NP}}\newcommand\SAT{\mathit{SAT}}\newcommand\CH{\mathit{CH}}\newcommand\PSPACE{\mathit{PSPACE}}$Are there $\NP$-hard problems which are $\NP$-complete under the Riemann ...
3
votes
1
answer
358
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How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?
Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
6
votes
0
answers
139
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Dickson's conjecture for Beatty sequences
A particular case of Dickson's Conjecture states that for $a_1,q_1,a_2,q_2$ with $(a_1,q_1)=(a_2,q_2)=1$, there are infinitely many $n$ for which $q_1 n + a_1$ and $q_2 n+a_2$ are both prime, provided ...
8
votes
1
answer
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A question on an equivalence of RH
In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that
$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$
where $\rho$ is ...
1
vote
0
answers
180
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Density of integers with prime factors at most $k$
I start with a quick example: Given a large random integer $N$, we are told to round it down to the closest power of $2$ and find what fraction of $N$ must be lopped off. Clearly the answer is between ...
4
votes
0
answers
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$\DeclareMathOperator\sym{sym}$Does $L(s, \sym^2 f \times \sym^2 g)$ have a pole at $s=1$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\sym{sym}$I encountered a question on the poles of $\GL_3\times \GL_3$ $L$-function, which needs the knowledge of the experts ...
5
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A modern reference for the Piltz divisor problem
apparently, the Dirichlet hyperbola method is no longer up-to-date, and instead Voronoi's identity is used in order to establish good bounds on the Dirichlet divisor problem.
The same applies to the ...
1
vote
1
answer
288
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How to relate Rankin triple L-function to its Dirichlet series
I have a very tricky question which may look naive to many experts here.
Let $f$ be a newform of level prime $P$, and $g,h$ two newforms of level 1, respectively. These three forms $f,g,h$ are all of ...
5
votes
1
answer
401
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Limit on a certain double sum
While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context
$$\sum_{n,m\geq1}\frac1{...
3
votes
1
answer
230
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Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$
This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$.
Can one derive from Robin's criterion for RH an ...
8
votes
0
answers
337
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circle method estimate integral over minor arcs
I have given the following lemma:
Let $\varepsilon>0$ and let $a, q, z$ be such that
$$
1 \leqslant a \leqslant q \leqslant B^{2}, \quad \operatorname{gcd}(a, q)=1, \quad|z| \leqslant \frac{1}{q^{2}...
2
votes
0
answers
550
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Weak Hardy-Littlewood-Goldbach conjecture
Assuming the weak Hardy-Littlewood-Goldbach conjecture as stated in this paper,
does the density $d(\delta,\varepsilon)$ of integers $m$ below $n$ such that
$$
\left\vert\frac{G(m)}{{\frak{S}}(m)m}-\...
0
votes
0
answers
90
views
A way to bound $\sum_{1 \leq n \leq X} \min ( \| \alpha n \|^{-1} , X/n)$?
Let $\alpha$ be a real number and $|| \cdot ||$ be the distance to
the nearest integer.
I want to find a non-trivial upper bound for
$$
\sum_{1 \leq n \leq X} \min ( || \alpha n ||^{-1} , X/n),
$$
...
5
votes
2
answers
353
views
A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument
There is a basis question which puzzles me for a while. The question is the following:
Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, ...
3
votes
1
answer
198
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Estimating $\sum_{x_i < X} \prod_i \phi(x_i)/ \mathrm{lcm}(x_i)^a$
I would like to estimate from above the following sum
$$
\sum_{1 \leq x_1 < X} .. . \sum_{1 \leq x_n < X} \frac{\prod_{1 \leq i \leq n } \phi(x_i)}{\mathrm{lcm}(x_1, .., x_n)^a}.
$$
$\phi$ is ...
1
vote
0
answers
90
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Any known relations to this doubly exponential constant?
Two years back I was working with lacunary series. In my explorations I had derived that the following series is periodic with period 1:
$$ f(x)= \sum_{n=0}^{\infty} \left[ e^{2^{x+n}} \right] + x - \...
5
votes
0
answers
124
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On Ford's "The distribution of integers with a divisor in a given interval"
Let $H(x,y,z)$ denote the number of positive integers of size at most $x$ which have a divisor in the range $(y,z]$. In his famous "The distribution of integers with a divisor in a given interval&...
2
votes
1
answer
454
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The number of ideals in a number ring below a given norm
I'm doing some number theory, and my deficient knowledge of algebraic number theory shows itself; I have to ask for help.
Suppose $K$ is a number field, and we want to count the ideals in $\mathcal ...
2
votes
1
answer
211
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Prime factor distribution over $\mathbb{N}$
I wonder if there is something like a general "prime component distribution pattern" of "the general natural number" $n$?
Using the following notation for the prime factorization $...
1
vote
0
answers
170
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Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions
We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...
4
votes
2
answers
436
views
Hardy-Littlewood circle method for non-diagonal quadratic forms
In short, the question is for any references describing how to use the Hardy-Littlewood circle method to find an asymptotic for the number of solutions to $F(x_1, ..., x_s) = k$ for $(x_1, ..., x_s) \...
16
votes
1
answer
815
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Fourier Series with Mobius coefficients
I assume this question has been considered before, but I can't find an literature on it. Let $\mu(n)$ denote the usual Mobius function and define:
$F(x) : = \sum_{n=1}^{\infty} \frac{\mu(n)}{n}e(nx)$
...
2
votes
2
answers
500
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Polynomial whose values divide $n!$
Let $P(n)$ be an irreducible polynomial of degree $2$ over the positive integers. Do there exist infinitely many positive integers $n$ such that $P(n)$ divides $n!$?
Edit: motivation by examples:
A) ...
3
votes
1
answer
274
views
Fully explicit Linnik's Theorem
Linnik's Theorem states that there exist absolute constants $c$ and $L$ such that for every $m \in \mathbb{N}$ and every $a$ coprime to $m$, there is a prime $p$ with $p \equiv a \pmod{m}$ and $p < ...
1
vote
0
answers
100
views
Lower bound on a Truncated Divisor Sum
Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$.
I am interested in estimating, the following sum
$$
A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a
$$
...
3
votes
0
answers
163
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Generalizing an estimate of Jutila
I'm working on a problem right now in which I need an upper bound for an exponential sum of the form
$$
\tag{1}
\sum_{N < n \leq 2N} \tau_3(n) e(f(n)),
$$
where $\tau_3(n) = \sum_{d_1d_2d_3=n} 1$ ...
9
votes
1
answer
517
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Innovations in number theory leading to breakthroughs in statistical mechanics
Might there be a good reference on the interaction of number theory with statistical physics? I am particularly interested in innovations in number theory that have led to breakthroughs in statistical ...
6
votes
0
answers
306
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Hoheisel's paper
Does anyone know a digital link to Hoheisel's paper: "Primzahlprobleme in der Analysis"?
It appeared in the 30's, published by the Berlin Academy. There seems to be no digital version.
3
votes
1
answer
222
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On the arithmetic of powers of subseries of the exponential series
Let $p$ be a prime number and $q=p-1$. I’m trying to prove that the nonzero coefficients $a_{qk}$ ($k\ge1$) of the power series
$$ \sum_{k\ge1} a_{qk} z^{qk} := \left( \sum_{k\ge0} \frac{z^{qk+1}}{(qk+...
6
votes
1
answer
597
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The history and original paper of the Rosser–Iwaniec sieve
I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated ...
4
votes
0
answers
168
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Is there a relation or formula between the correlations of the nontrivial zeros of the Riemann zeta function and the correlations between high points
Consider an interval of length $(\log T)^{\theta}$ for some fixed $\theta > −1$, around a point $1/2 + i y$ on the critical line where $y\in[T,2T]$ and $T$ is large. How do the correlations between ...
11
votes
2
answers
709
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Improving the error term in a classic sieving problem
I'm new to asking questions on MathOverflow, so forgive me if this question is not the kind of thing to be asked here.
Let $q$ be a positive integer and let $N$ be an integer with $1 \leq N \leq q$. ...
3
votes
0
answers
211
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On partial sums of the Ramanujan sums
Let $n$ be a positive integer and $c_{m}(n)$ denote the $m^{th}$ Ramanujan sum at $n$. What is the best known estimate for $\sum_{m=1}^{N} c_{m}(n)$?
3
votes
1
answer
411
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The number of elements in {1,2,...,a}.{1,2,...,b}, where $ab=n^2$
Let $A_{a,b}$=$\{pq:p\leq a,q\leq b\}$, where $ab=n^2$ and $n^2$ is fixed.
How large is $A_{a,b}$? Does $A_{a,b}$ attain its lower value when $a=b=n$?
The case when $a=b=n$ is settled by Ford, and a ...
-1
votes
1
answer
1k
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Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
-4
votes
1
answer
222
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A generalization Bertrand's postulate [closed]
Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question ...
6
votes
2
answers
699
views
Does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
-3
votes
1
answer
267
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Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?
This question is a follow-up to About Goldbach's conjecture and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ ...
4
votes
1
answer
311
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Infinite, finitely generated linear group has indices of subgroups divisible by infinitely many primes
This question was previously asked at Math.SE, but didn't receive much attention.
Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, say $G \leq \operatorname{GL}_n(K)$ ...