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.
2,906
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Heuristic argument for the prime number theorem?
Here is a bad heuristic argument for the prime number theorem. Let $n$ be a positive integer and assume that PNT holds up to $n$. Then $n$ itself is prime if and only if for each prime $p<n$ the ...
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vote
1
answer
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GRH and the Euler product
Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be ...
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3
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Yitang Zhang's 2007 preprint on Landau–Siegel zeros
The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint On the Landau-Siegel zeros conjecture? If this result is correct, ...
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2
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Counting integers with k large prime divisors
If $x \ge y \ge 1$ are real numbers and if $k$ is a positive integer, take $\Phi_k(x, y)$ to be the number of integers $\le x$ with exactly $k$ prime factors and no prime factor $\le y$. If $y$ is ...
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On number of shifted integer solutions to a linear system
Let $M$ be a random diagonally dominant non-singular $3\times 3$ integer matrix
$$\begin{bmatrix}
m_{11}&m_{12}&m_{13}\\
m_{21}&m_{22}&m_{23}\\
m_{31}&m_{32}&m_{33}
\end{...
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votes
1
answer
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The nontrivial zeros of the zeta function and the prime counting function
The truncated explicit formula has the shape
\begin{equation}\label{1}
\psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
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1
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Absolute convergence of Rankin–Selberg series
Let $\pi$ and $\pi'$ be two general automorphic representations on $\operatorname{GL}(n)$ and $\operatorname{GL}(n')$ over $\mathbb{Q}$.
I heard that the Rankin-Selberg $L$-function $L(s,\pi\times\pi')...
- 99.2k
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2
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Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?
(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...
- 88.9k
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1
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Voronoï summation for cusp forms with characters
In an attempt to solve an unrelated problem, I was led to the task of estimating/bounding from above sums of the form
$$\sum_{m=1}^\infty\lambda(m)e\left(-\frac{am}{q}\right)h(m)$$
where $\sum_{m=1}^\...
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Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
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Expected error term in the distribution of Friedlander-Iwaniec primes
In 1998 John Friedlander and Henryk Iwaniec famously proved the asymptotic formula
$$\displaystyle \mathop{\sum \sum}_{a^2 + b^4 \leq x}\Lambda(a^2 + b^4) = \frac{4x^{\frac{3}{4}}}{\pi} \int_0^1 (1 - ...
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2
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Explicit bounds on number of primes of given size
How many prime numbers of $b$ bits are there?
Beyond the prime number theorem, one can give explicit bounds on the number of primes below some integer $n$, or in a given interval. For instance, Rosser ...
- 99.2k
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Bounding number of solutions of a congruence
Let $d$ be a positive integer. Let $f(d,a)$ be the number of values of $x$ in $[1,d]$ such that $$x^{a}\equiv 1\pmod{d}.$$ I wanted to know if for some $0<\epsilon<1$, we can prove the following ...
- 1,032
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0
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Degree four polynomials with no real roots
Consider a degree four polynomial
$$
f = a_4x^4 + a_3x^3 + a_2x^2 + a_1x+ a_0 \in \mathbb{R}[x]
$$
with real coefficients. The discriminant $\Delta_f$ of $f$ is a homogeneous polynomials of degree six ...
- 8,852
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votes
1
answer
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Asymptotic for a sum involving GCD and Euler totient function
Let $\varphi$ denote the Euler's totient function. Is there any reference in literature for the value of sum $$\sum_{\substack{r\le x\\ d\mid r}}\gcd(\phi(d),r)$$ where $d$ is some fixed positive ...
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Asymptotics for a sum involving GCD and multiplicative order
Let $n$ be a positive integer and $\mathrm{ord}_{n}(a)$ be the least positive integer $d$ such that $n\mid a^{d}-1$. I wanted to know if for some choice of $y=y(x)$, one can obtain asymptotics for ...
- 1,032
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An approximation for the prime counting function
NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses.
SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, ...
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2
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Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers
Let $B_{n}$ for $n\ge0$ denote the Bernoulli number generated by
\begin{equation*}
\frac{z}{\textrm{e}^z-1}=\sum_{n=0}^\infty B_n\frac{z^n}{n!}=1-\frac{z}2+\sum_{n=1}^\infty B_{2n}\frac{z^{2n}}{(2n)!},...
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Ask for a proof of an identity involving the product of two Bernoulli numbers
It is well known that the Bernoulli numbers $B_{k}$ for $k\in\{0,1,2,\dotsc\}$ can be generated by
\begin{equation*}
\frac{z}{\textrm{e}^z-1}=\sum_{k=0}^\infty B_k\frac{z^k}{k!}=1-\frac{z}2+\sum_{k=1}^...
- 838
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0
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Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
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245
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A weighted count of Egyptian fraction representations
Previously asked and bountied at MSE:
Given a positive rational $q$, let
$$\mathsf{E}(q)=\left\{X\in[\mathbb{N}]^{\mathit{fin}}: q=\sum_{x\in X}{1\over x}\right\}$$
be the set of Egyptian fraction ...
- 19.3k
10
votes
0
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Are prime numbers among sums of prime numbers distributed as $\frac n{2\ln(n)}$?
Let $(s_n)_{n\in\mathbb N}$ be defined as follows:
For $n\in\mathbb N$, $s_n:=2+3+5+\cdots+p_n$ is the sum of the first $n$ prime numbers (e.g.: $s_1=2$, $s_2=5$, $s_3=10$, $s_4=17$, $\ldots$).
Let $\...
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1
answer
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Is it possible to improve on Siegel's theorem for exceptional zeroes?
Let $\chi$ be a real nonprincipal character modulo $q$. Siegel's theorem on exceptional zeroes states that for any $\epsilon >0$ there exists a positive number $C(\epsilon)$ such that, for any real ...
- 4,618
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1
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Consecutive non-powerful integers
Pair of sequences $\ v_n\ $ and $\ U_n\ $ of integers start as in the following table:
[\begin{array}{rrrrrrrrrr}
n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \ldots \\
...
- 8,994
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The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution
I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
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Sums of $\Lambda(n)$ or $\mu(n)$ with hyperbolic-function weights: surprise!
I was leafing through Gradshteyn–Ryzhik in bed yesterday, as one does, and noticed on the last page that the Mellin transforms of several hyperbolic functions have a factor of $\zeta(s-1)$ or $\zeta(s)...
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votes
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Approximating a function by a convolution of given function?
Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...
- 88.9k
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Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential decay
Let $f:\mathbb{R}\to \mathbb{R}$ be a function in $L^2$ satisfying $|f(x)|\ll e^{-a_1 x}$, $a_1>0$, for $x\to \infty$. (Variant: assume as well that $|f(x)|\ll e^{a_2 x}$, $a_2>0$, for $x\to -\...
- 88.9k
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A question on the use of fractional derivatives in Riemann Hypothesis
We already know that Riemann-zeta function on the critical band is defined as follows:
$$(1-2^{1-\alpha})\zeta(\alpha) = \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha},\quad \Re(\alpha) \in ]0, 1[ $$
Is ...
- 61
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On Buchstab et al's "forgotten" sieve and the Goldbach conjecture for certain integers
There is a somewhat forgotten sieve-theoretic approach to the Goldbach conjecture, due to Buchstab et al, see e.g. pp.247-248 of R.D. James.
On p.247, James defines some function $F$ such that for any ...
- 99.2k
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Do primes of the form $4k+1$ ever lead the greatest prime factor race?
Analogous to Chebyshev's race between primes, I examined the race between primes in the greatest prime factors, GPF, of natural numbers. Similar to the regular prime race, in the GPF race, the ...
- 11.4k
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votes
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answers
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Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
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votes
0
answers
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Eigenfunction of $h\mapsto H(h')|_{[-1,1]}$?
Let $H$ be the Hilbert transform. Is there a continuous, even function $h:\mathbb{R}\to \mathbb{R}$ with support on $[-1,1]$ such that, for some $\lambda\in \mathbb{R}$,
$$H(h')(t) = \lambda h(t)$$
...
- 19.4k
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votes
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Fourier optimization problem related to the Prime Number Theorem
Let $\kappa>0$ be given. What is the function
$f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that
$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
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Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
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Counting powerful integers. Lower bounds
Remark: The upper bounds are perhaps still more interesting; I may address them in another post.
PROBLEM: Find simple (numerically efficient) lower bounds for the number of powerful integers (...
- 99.2k
3
votes
1
answer
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Best available bounds for $\pi(Y)-\pi(Y-X)$?
I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's Sieves in Number Theory, there is a theorem 4 which says that
If $Y\ge X \ge 2$, ...
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2
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Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
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Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
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6
answers
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What does Mellin inversion "really mean"?
Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
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A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
- 11.9k
3
votes
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answers
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Green-Tao's "Polylogarithmic bound for $r_4(N)$"
On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of $\psi\big(\frac{k}{N}\big)$ for all $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
- 5,632
2
votes
0
answers
171
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How to best approximate $1/\zeta(s)$ by a finite sum
I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum:
$$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) +
\epsilon(t)$$
with $\eta$ a function of compact ...
- 19.4k
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votes
0
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On the elementary proof of Dirichlet theorem on arithmetic progressions
In [Cassels, JWS, Rational quadratic forms, p. 333], the autor says: "In fact the elementary proof of Dirichlet's theorem [Selberg (1949)] makes essential use of the existence of genera".
In ...
9
votes
0
answers
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Best smoothing for the Prime Number Theorem?
There are plenty of proofs of the Prime Number Theorem with explicit error terms - it actually looks like a rather competitive field (see Remark 1.4 in https://arxiv.org/pdf/2204.02588.pdf). Several ...
- 19.4k
3
votes
1
answer
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Successive minima and the basis of lattice
I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...
- 60.4k
2
votes
1
answer
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Overall idea of estimating major arcs in Waring's problem
This is copied from math.SE after a kind comment's suggestion as I am sure people here are very well knowledged in this method :)
I am currently reading Vaughan's "The Hardy-Littlewood Method&...
- 1,305
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votes
0
answers
156
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Similar problems for the Dedekind psi function than those that are in the literature for the Euler's totient function: reference request or proposals
I know the linked articles from Wikipedia for the Euler's totient function and the Dedekind psi function. I know that are in the literature famous, interesting and difficult problems involving the ...
1
vote
0
answers
114
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Estimating a sum with a fractional part
Is it possible to pick up the value $\log^2 x<y<x$ to get this expressions simultaneously
$$\sum_{d\leq y}\left \{ \frac{x}{d} \right \}\leqslant \frac{y}{\log^{3} y}?$$
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Existence of a smooth function that approximates a characteristic function of an interval with certain property
Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the ...
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