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.
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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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Weights in the proof of Chen's theorem in Nathanson's "Additive Number Theory The Classical Bases"
I was reading Professor Nathanson's graduate texts in mathematics 164: "Additive Number Theory The Classical Bases" (http://www.alefenu.com/libri/nathansonbases.pdf), and I was wondering if ...
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Problem in Uchida's Theorem
I m studying relative class number of CM fields and i found a very interesting theorem from Kόji Uchida's article: class numbers of imaginary abelian fields (1970) but one argument of the ...
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Why do we have fewer distinct Gauss sums over a field of characteristic $2$?
Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
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Brauer–Siegel's Theorem and application
$\newcommand\underto[1]{\xrightarrow[#1]{}}$Brauer–Siegel's Theorem says that if we consider an infinite sequence of number fields $K_i/\mathbb{Q}$ such that $$\frac{[K_i:\mathbb{Q}]}{\log{|D_i|}}\...
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Average bounds on Rankin-Selberg coefficients for modular forms
Let $f$ be a cuspidal Hecke newform of weight $k$ and level $N$, and denote by $a_f(n)$ its $n$-th Fourier coefficient. The newform $f$ is normalized so that $a_f(1) = 1$. As a consequence of Rankin-...
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Is there an asymptotic expansion for the reciprocal of the classical Euler beta function?
The classical Euler beta function can be defined by
$$
B(p,q)=\int_0^1t^{p-1}(1-t)^{q-1}\operatorname{d\!}t
$$
for $\Re(p),\Re(q)>0$.
The beta function and the classical Euler gamma function $\...
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Circle Method applications only for Quadratic & Cubic Form
So I have studied multiple applications of Circle Method.
1- Initially I read about the Classical Hardy Littlewood circle method, where they use generating functions, and determine hte coefficent ...
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Validity of a Tauberian theorem for Dirichlet series
I encountered this statement about Dirichlet series but couldn't find a similar result in Korevaar's "Tauberian Theory". Is this statement valid?
Statement:
Let $f(s) = \sum_{n=1}^{\infty} \...
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$\{ x/p\} $ on average
This is a vague question: Lemma 2.2 of Friedlander and Lagarias' "On the distribution in short intervals of integers having no large prime factor" says that $$\sum _{p\leq w}\left (\{ x/p\} -...
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Asymptotics for sums of two sets of positive integers
Assume that $A$ and $B$ are subsets of $\mathbb N$, with counting functions verifying $A(x)\gg x^\alpha$ and $B(x)\gg x^\beta$, with $\alpha+\beta<1$. Let $C=A+B$ and $C(x)$ its counting function.
...
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Locating volume 2 of certain conference proceedings in analytic number theory
Does anyone know where one might locate "Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Volume 2"? There exists Volume 1 here: https://link.springer.com/...
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Brun-Titchmarsh for sum over square divisors
Let $f(n)$ be a nonnegative arithmetic function satisfying
$f(p^l) \leq A_1^l$ for all primes $p$, integers $l\geq 1$, and some constant $A_1$;
$f(n) \leq A_2 n^\varepsilon$ for all $\varepsilon > ...
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How many Fourier coefficients vanish?
Let $G$ be a compact abelian connected metric group with Haar measure $\mu$ and let $f\colon G\to S^1$(=unit circle in $\mathbb{C}$) be a continuous function (not necessarily a group homomorphism) ...
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Stirling's formula and a Gamma function relation
I am trying to understand a paper by by A. Booker on poles of Artin $L$-functions where in one of the lemmas he uses the following identity, derived using Stirling's formula:
$$
\frac{\Gamma(s/2)^2}{2^...
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A strictly increasing, analytic function that goes through key points of the iterated logarithm?
Is it possible to create a function $f(x)$ that:
is strictly increasing (at least for $x>0$)
is real analytic
goes through all the points where the iterated logarithm would increment value?
i.e. [...
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Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
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Asymptotic behavior of partial sums of Dirichlet series
Consider the Dirichlet series:
$$\sum_{n \geq 1} \frac{a_n}{n^s} = \frac{\zeta(s+1/3)}{\zeta(s)}$$
where $\zeta(s)$ is the Riemann zeta function.
Question: Assuming the Riemann Hypothesis (RH), how ...
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Do plots $(5)$ and $(6)$ go to infinity not at the same rate but at similar rates?
The following has been proven by joriki and GH from MO (see here): assuming that $n>1$, then the von Mangoldt function
$$
\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\...
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How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
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Uniform distribution of sequence mod 1
Is it known whether "for most $r$" the sequence $$r \cdot 2^k \bmod 1, \qquad k \in \mathbb N $$ is uniformly disributed in $[0,1]$?
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Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
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How dense are (very but not extremely) smooth numbers? Can they be found in most (not very) short intervals?
An integer is said to be $y$-smooth if it has no prime factors $>y$. Let $y$ be "medium sized", meaning $(\log x)^{1+\epsilon} < y < \exp((\log x)^{2/3})$ or so. (Why this range of ...
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Replacing the sequence in Chowla's conjecture and positiveness of the entropy
For any fixed integer $m>0$ and not-all-even $(a_1,\ldots,a_m)\in\mathbb N^m$, one version of Chowla's conjecture states that
$$
\lim_{x\rightarrow\infty}\frac{1}{x}\sum_{n\leq x}\mu(n+1)^{a_1}\...
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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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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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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
$$S = \...
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Is the function $(z-1)(2^{z}-1)\zeta(z)$ logarithmically concave and convex in $z\in(0,\infty)$?
For proving that the sequence
\begin{equation}\label{first-proof-decreas-seq}
\frac{1}{(2k-1)(k+1)} \frac{2^{2k+2}-1}{2^{2k}-1} \biggl|\frac{B_{2k+2}}{B_{2k}}\biggr|
\end{equation}
is decreasing in $k\...
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Explicit expression for a function in number theory
In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of ...
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On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$
I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin.
Consider the quantities defined here in pg. $617$
$$\tilde{F_n}:= \frac{1}{...
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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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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)|...
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Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
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Semi-primes represented by quadratic polynomials
According to Lemke-Oliver, irreducible quadratic polynomials $G$ with positive leading coefficient and $\rho(2)<2$, (where $\rho(m)$ denotes the number of incongruent solutions to the congruence $G(...
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Could the complex zeros of Riemann zeta function be of the form $ s=0.5+ik$ with $k$ a positive integer? [closed]
I have checked in Andrew Odlyzko, Tables of zeros of the Riemann zeta function, to know if there is an example of zeros of Riemann zeta function with integer imaginary parts, but I don't see that. I ...
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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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Exponential sum vs. exponential integral via Poisson summation
When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...
CommunityBot
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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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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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Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
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Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
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Optimal implied constant for the classic divisor function bound
One of the most fundamental and useful asymptotic results in number theory is the divisor bound, namely
$$\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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Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
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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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