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.
3,066 questions
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How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
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Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
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How dense are (very but not extremely) smooth numbers? Can they be found in most (not very) short intervals?
An integer is said to be $y$-smooth if it has no prime factors $>y$. Let $y$ be "medium sized", meaning $(\log x)^{1+\epsilon} < y < \exp((\log x)^{2/3})$ or so. (Why this range of ...
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$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
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Replacing the sequence in Chowla's conjecture and positiveness of the entropy
For any fixed integer $m>0$ and not-all-even $(a_1,\ldots,a_m)\in\mathbb N^m$, one version of Chowla's conjecture states that
$$
\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\mu(n+1)^{a_1}\...
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Smooth partitions of unity?
I read some paper of moment of Dirichlet L-function, and they often use approximate functional equation to continue, and some of them use the trick of smooth partition of unity, such as:
$$\sum_{k=-\...
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Hypergeometric sheaves on $\mathbb{A}^{1}_{E}$
Let $m, n$ be non-negative integers. Assume that $\boldsymbol{\chi} = \left( \chi_i \right)_{1 \leq i \leq m}$ and $\boldsymbol{\eta} = \left( \eta_j \right)_{1 \leq j \leq n}$ are two collections of ...
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Why is $\sum_{n=1}^\infty \frac{\sigma_a(pn)}{n^s}=(1+p^a-p^{a-s}) \zeta(s) \zeta(s-a)$ only when $p$ is a prime number?
I tried to find the summation for $a,b\in N$ and $s>a+1$
$$ \Omega_a(b,s)=\sum_{n=1}^\infty \frac{\sigma_a(bn)}{n^s}$$
where $\sigma_a(n)$ is sum of positive divisors function which defined by
$$ \...
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Uniform distribution of sequence mod 1
Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?
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On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$
I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin.
Consider the quantities defined here in pg. $617$
$$\tilde{F_n}:= \frac{1}{...
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Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?
For proving that the sequence
\begin{equation}\label{first-proof-decreas-seq}
\frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr|
\end{equation}
is decreasing in $k\...
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Sum over three squares
Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum
$$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
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Explicit expression for a function in number theory
In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of ...
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Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
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Average of gcd of sum of two $k$th powers
I am interested bounding the following quantity. Given fixed $k \in \mathbb{N}$, $a,b \in \mathbb{Z}$, $\sigma \in [0,1)$, and intervals $I_1,I_2 \subset \mathbb{Z}$ can we establish the bound
$$S = \...
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Chowla's theorem on class number of real quadratic field
Let $p\equiv1\bmod 4$ be a prime number and $h$
the class number of real quadratic field $\mathbb Q(\sqrt{p})$, $\epsilon=\frac{t+u\sqrt{p}}{2}$ its fundamental unit. In this paper https://www.pnas....
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The error term of ternary-Goldbach problem
Are there some results give the $X$ power saving of the error term in Ternary Goldbach problem, not just the $\log$ power saving?
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Prerequisites for Chen's theorem?
I am an undergraduate theoretical physics student, and I am trying to understand Chen's theorem. But when I tried to read Chen Jingrun's 1973 paper (https://www.sciengine.com/Math%20A0/doi/10.1360/...
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Sieve Method works for variant question?
There are multiple results on the sieve method, and I wanted to ask about the following variant
(to know if it is trivial by one of the current versions of the sieve method, or seems a challenging ...
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Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
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Product of two Dirichlet characters with non-coprime moduli
If I have Dirichlet characters $\chi ,\chi '$ modulo $q,q'$ is there anything useful I can say about the individual characters or moduli if I'm given that $$\chi \chi '=\text { id}_{[q,q']}$$
where ...
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Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
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Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
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Optimal implied constant for the classic divisor function bound
One of the most fundamental and useful asymptotic results in number theory is the divisor bound, namely
$$\displaystyle \tau(n) = \# \{x,y \in \mathbb{Z}_{\geq 1} : xy = n\} = O\left(n^{\frac{1}{\log \...
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Equidistribution of Frobenius Classes
Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy ...
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Are there any positive integers $n$, $k$ such that $n > 2, k > 6$, and all prime factors of $n^k - 1$ are less than $n$?
I noticed that the prime factorization of $68^6 - 1$ is $3^2 \cdot 7^2 \cdot 13 \cdot 19^2 \cdot 23 \cdot 31 \cdot 67$, which makes all of its prime factors less than 68. This made me wonder the ...
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On the local factor of Rankin-Selberg L-functions
I have a puzzle on the local factors of Rankin-Selberg $L$-functions. Consider two newforms on $\text{GL}_2$. Let $f$ be a newform of square-free level $N$, and $g$ a newform of trivial level. As ...
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Finiteness and bounds for elliptic curves realizing a given galois representation
Let $\rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \text{GL}_2(\mathbb{Z}_p)$ be a continuous, irreducible Galois representation. Consider the set $\mathcal{L}_\rho$ of all elliptic curves $...
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Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
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Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
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From $\Lambda_k$ and $\Lambda$ to $\mu$ (or $\lambda$)
Let $\{a_n\}_{n=1}^\infty$, $a_n \in \mathbb{C}$, $|a_n|\leq 1$. Let $\Lambda_k = \mu \ast \log^k$; in particular, $\Lambda_1$ equals the von Mangoldt function $\Lambda$. Suppose that we have ...
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Deeper meaning behind the occurrence of the factor $\frac{\log q}{i}$ in Deninger's results
In two papers Deninger proved the following:
If $q=p^{n}$ and $p$ is a finite prime of $\mathbb{Z}$, $B=\mathbb{C}[\mathbb{C}]$ is generated by symbols of the form $e^{\alpha}$, $\alpha\in\mathbb{C}$,...
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Other, easier, approaches to proving: for $k$ coprime integers summing to zero, there is no bound on $\min\{\Omega(a_1),\ldots,\Omega(a_k)\}$
I put this on Stackexchange, and the question was closed without them specifying why (they just said "no context", even though I did mention the context). Never mind - I have deleted the ...
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Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
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Must bounded sequences be well-distributed to most *composite* moduli?
Let $\{a_n\}_{n=1}^N$, $|a_n|\leq 1$. Let $Q=\sqrt{N}$. Then $a_n$ is well-distributed modulo most prime $p\leq Q$, in the following sense:
$$\sum_{p\leq Q} \frac{1}{p} \left(\frac{1}{N/p} \sum_{\...
- 20.2k
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Counting matrices of bounded norm in SL_n(Z)
I'm looking for the asymptotic order of growth of the number of points in algebraic groups, such as $\mathrm{SL}_n(\mathbb{Z})$, of height/norm at most $X$, i.e. all entries are at most $X$ in ...
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Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
user536681
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Sum of reciprocals of rough numbers
Let $x$ and $y$ be given real numbers. We may suppose that $2\leqslant x \leqslant y$ and that $u:= \log(y)/\log(x)$ remains bounded in a compact set away from $1$ as $x,y\to\infty$. An integer $n$ is ...
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What rational zeta series with non-integer arguments appear in mathematics?
Background
Rational zeta series are series of the form $$\sum_{n=2}^{\infty} q_{n} \zeta(n + p, m), \label{1} \tag{1} $$ where $\zeta(x,m)$ is the Hurwitz zeta function and $q_{n}, \ p \in \mathbb{Q} \...
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Finding $\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma$ from the generating function of $\zeta(•)$
In equation (130) of this page, the identity $$\lim_{n \to \infty} \sum_{k=2}^{n-2} \zeta(k) \zeta(n-k) x^{k-1} = x^{-1} - \psi_{0}(-x) - \gamma \label{1} \tag{1} $$ is stated. Here, $\zeta(\cdot)$ is ...
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Cancellation in correlations of the Möbius function over function fields
Let $p$ be an odd prime and $q$ a power of $p$. For a polynomial $f \in \mathbb{F}_q[T]$, let $\mu(f)$ be the Möbius function of $f$. For a positive integer $d$, let $M_d$ be the set of monic ...
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Connection between a special integral transform and averages of L-functions
Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ and $\mathcal{H}$ be the upper half-plane. For $A>1$, define the truncated Eisenstein series $E_A(z,s)$ as $$E_A(z,s) = \begin{cases} E(z,s), & \...
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Using Lang–Trotter to get bounds on averages of Fourier coefficients
Let $E$ be an elliptic curve over $\mathbf{Q}$ and let $(a_n)$ be the sequence of Fourier coefficients for the weight two newform attached to $E$. The coefficients $a_p$ are the Frobenius traces given ...
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Integrating on $\mathbb{R}$ by summing on $\mathbb{Q}^+$
Does the following integration method hold for regular enough functions $F:\mathbb{R}\to\mathbb{R}$?
\begin{align}
&\zeta(2)\sum_{\frac{a}{b}\in\mathbb{Q}_n} \frac{F(\log \frac{a}{b})}{\sqrt{abn}...
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Ratios Conjecture for L-functions Associated to Hecke Characters of Number Fields
Let $K$ be a number field and consider a family of L-functions $\{L(s,\chi_i)\}_{i=1}^N$ associated to Hecke characters of $K$. Assuming the Generalized Riemann Hypothesis (GRH) for this family, we're ...
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How does the geometry of the set R(P) govern the 'smoothness' of sieve bounds?
Consider the problem of estimating sums over primes, where the sequence being summed satisfies specific Type I and Type II estimates controlled by parameters $\gamma$, $\theta$, and $\nu$. Let $R(P)$ ...
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Sieve theory obstruction: prime-sparse and nearly full-differenced sets?
Let $D(A) = {|a-b| : a, b \in A}$ denote the difference set of $A \subseteq \mathbb{Z}$. A set $A \subseteq (x/2, x]$ is almost full-differenced if $|D(A)| \geq \frac{x}{2} - \log x$. Let $C_x$ denote ...
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1
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Jacobi two square's theorem last step to conclusion
The Jacobi two-square theorem is: "The number of representations of n as a sum of two squares is four times the difference between the number of divisors of n congruent to 1 modulo 4 and the ...
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A consequence of Firoozbakht's conjecture?
This is a question out of curiosity, while looking at the Firoozbakht's conjecture. It might not be research related, but as usual, I am not really sure if a question ever is research related or not, ...
- 3,064
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The twin prime problem and the Jurkat-Richert Theorem
Where does the Jurkat-Richert Theorem for linear sieves fail when applied to the twin prime problem?
I'm reading the last two chapters of Additive Number Theory The Classical Bases. The Jurkat-...
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