Skip to main content

All Questions

Filter by
Sorted by
Tagged with
43 votes
3 answers
3k views

Is this integral representation of $\zeta(2n+1)$ known?

Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
Andrew Knapp's user avatar
40 votes
5 answers
8k views

Is $\zeta(3)/\pi^3$ rational?

Apery proved in his paper from 1979 that $\zeta(3)$ is irrational, and we know that for all integers $n$, $\zeta(2n)=\alpha \pi^{2n}$ for some $\alpha\in \mathbb{Q}$. Given these facts, it seems ...
Thomas Bloom's user avatar
  • 7,013
30 votes
2 answers
4k views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
user135845's user avatar
29 votes
2 answers
4k views

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)}{...
T. Amdeberhan's user avatar
29 votes
4 answers
5k views

What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
Gene S. Kopp's user avatar
  • 2,200
29 votes
5 answers
5k views

Partial sums of multiplicative functions

It is well known that some statements about partial sums of multiplicative functions are extremely hard. For example, the Riemann hypothesis is equivalent to the assertion that $|\mu(1)+\mu(2)+\dots+\...
gowers's user avatar
  • 29k
26 votes
0 answers
567 views

Elliptic analogue of primes of the form $x^2 + 1$

I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
Marty's user avatar
  • 13.3k
24 votes
2 answers
1k views

Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element $$ S = \prod_{k=1}^{\frac{N-1}2} k^k $$ ...
Franz Lemmermeyer's user avatar
22 votes
3 answers
2k views

Hecke equidistribution

For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore $$ a+bi=p^{1/2}e^{i\varphi} $$ where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
M.B's user avatar
  • 2,508
21 votes
2 answers
1k views

Most squares in the first half-interval

It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
Andrea Ferretti's user avatar
21 votes
1 answer
771 views

Covering a set with geometric progressions

Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
shadow10's user avatar
  • 1,090
17 votes
2 answers
3k views

Consequences of the Birch and Swinnerton-Dyer Conjecture?

Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following What are the consequences of the Birch and ...
17 votes
2 answers
2k views

Is every odd positive integer of the form $P_{n+m}-P_n-P_m$?

I am looking for a comment, reference, remark, or proof of three conjectures as follows: Conjecture 1: Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P_{n+...
Đào Thanh Oai's user avatar
16 votes
1 answer
4k views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ \sum_{\substack{p<n\\\text{...
Daniel Soltész's user avatar
16 votes
1 answer
1k views

On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...
Fan Zheng's user avatar
  • 5,169
16 votes
1 answer
705 views

Connection between Bernoulli numbers and Riemann-Siegel theta function?

I have come across a strange approximation for the Riemann-Siegel theta function involving the Bernoulli numbers - namely that $$\frac{1}{2} \log \left| B_{2 n}\right|\approx \vartheta (2n)\ ,\quad n ...
martin's user avatar
  • 1,903
15 votes
5 answers
2k views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
Stopple's user avatar
  • 11.1k
15 votes
2 answers
1k views

Evaluating the integral $\int_{1}^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$

I am trying to find a formula for the following integral for non-negative integer $k$: $$\int_1^{\infty}\frac{\{u\}}{u^{2}}\left(\log u\right)^{k}du.$$ My first thought was to use the formula $$\...
Eric Naslund's user avatar
  • 11.4k
15 votes
1 answer
969 views

Counting lattice points inside a three-dimensional ellipsoid

I want to answer the following simple question: Given a three-dimensional ellipsoid defined by $Q(x, y, z) \leq Z$ for a positive definite quadratic form $Q$, how many lattice points in $\mathbb{Z}^...
Frank Thorne's user avatar
  • 7,347
14 votes
4 answers
3k views

Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals $$4(d_1(n)-d_3(n)),$$ where the function $d_i$ counts the number ...
Dr. Pi's user avatar
  • 3,062
14 votes
1 answer
497 views

Geometric Mean of $L(1,\chi)$ for quadratic Dirichlet characters

Let $S = \{D_1, D_2, D_3, \ldots \}$ be the set of all prime discriminants (or positive prime discriminants) of quadratic number fields. For such a discriminant let $\chi_j(n) = (\frac{D_j}n)$ be ...
Franz Lemmermeyer's user avatar
14 votes
1 answer
285 views

Lower bounds for class number of function fields with fixed $q$, growing $g$

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
David E Speyer's user avatar
13 votes
2 answers
1k views

Question about a lesser-known "class number formula" of Gauss

My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($D>0$): $$h(D) = \frac{...
user2554's user avatar
  • 2,099
13 votes
2 answers
644 views

Reference for a conjecture on the first primes congruent to 1 modulo other primes

Given a prime $p$, define $f(p)$ to be the smallest prime congruent to $1$ modulo $p$. For example, $f(7)=29$. It has been conjectured that $f(p)<p^2$ always: by Schinzel in his "Hypothesis H" ...
Greg Martin's user avatar
  • 12.8k
13 votes
2 answers
726 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
Myshkin's user avatar
  • 17.6k
13 votes
0 answers
328 views

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 ...
H A Helfgott's user avatar
  • 20.2k
12 votes
1 answer
526 views

Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes. Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then ...
Daniel Loughran's user avatar
12 votes
1 answer
1k views

Do Shintani zeta functions satisfy a functional equation?

Probably my questions are known or evident to the experts but I'm a bit puzzled. First of all there seem to be two kinds of zeta functions that go under the name of Shintani zeta functions. First, ...
user5831's user avatar
  • 2,029
11 votes
1 answer
2k views

Has this number-theoretic constant been studied?

Unless I made a mistake, the expected value of the largest exponent in the prime factorization of random positive integer (defined in the appropriate way) is $$\eta := \sum_{n=1}^\infty \Big(1-\zeta(n)...
mathworker21's user avatar
  • 1,355
11 votes
4 answers
707 views

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 ...
H A Helfgott's user avatar
  • 20.2k
11 votes
6 answers
2k views

The Wiener-Ikehara approach to the PNT

Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the Tauberian theorem of Wiener and Ikehara or the other way around? In any case, do you know who ...
José Hdz. Stgo.'s user avatar
11 votes
3 answers
745 views

Counting points on lattices

I expect that the following is a standard problem from analytic number theory, but I don't know where exactly to look for an answer. Let f: ℤr→ H be a surjective homomorphism into a ...
Tzanko Matev's user avatar
11 votes
1 answer
328 views

Critical points of Dirichlet L functions

Let $L(s,\chi)$ denote a Dirichlet $L$-function for a real-valued non-principal character $\chi$. This has limiting value $L(\infty,\chi) = 1$ and we are interested in how this limit is approached ...
user2052's user avatar
  • 1,411
11 votes
2 answers
2k views

Good books on Dirichlet's class number formula

I refrained from asking the technical questions; maybe everyone didn't like my attitude. At least, help me finding good books. Can anyone suggest a good book that gives a complete reference to "...
user avatar
11 votes
1 answer
1k views

The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} a\...
Eric Naslund's user avatar
  • 11.4k
11 votes
1 answer
498 views

Siegel--Walfisz for number fields

For a number field $K$, we write $\Delta_K$ for its absolute discriminant. I was hoping for a Siegel--Walfisz type theorem of the following type: Let $A > 0$. Then for every $X > 0$, every ...
P. Koymans's user avatar
10 votes
1 answer
1k views

A generalisation of theorem of Landau on sum of two squares?

Let $r(B)$ be the number of integers $1 \leq n \leq B$ such that $n = x^2 + y^2$ for some $x, y \in \mathbb{Z}.$ Then it is a known theorem of Landau that $$ r(B) \sim C \frac{B}{\sqrt{\log B}} $$ ...
Johnny T.'s user avatar
  • 3,625
10 votes
2 answers
870 views

Bound on gcd of two integers

Well this is a problem I was fiddling with. I came up with it but it probably is not original. Suppose $a\in \mathbb{N}$ is not a perfect square. Then show that : $$\text{gcd}(n,\lfloor{n\sqrt{a}}...
shadow10's user avatar
  • 1,090
10 votes
3 answers
1k views

Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
user36212's user avatar
  • 1,687
10 votes
1 answer
314 views

Coefficient bounds on cusp forms, half-integer weight

Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \...
user105068's user avatar
10 votes
2 answers
849 views

Schur's proof of Hilbert's inequality: streamlining?

TL;DR: Is there a way to make Schur's (elegant) proof of Hilbert's inequality feel like less of a trick/miracle? Longer version: Let me go quickly over Schur's proof to show what I mean. Actually, let ...
H A Helfgott's user avatar
  • 20.2k
9 votes
1 answer
1k views

Sums of two squares in arithmetic progressions

Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$\sum _{n\leq x\atop {n\equiv a(q)}}r(n)$$ and in particular is there an asymptotic ...
caws's user avatar
  • 143
9 votes
1 answer
388 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

(I posted this question on Math SE but it has had no answer for a year now so I would like to ask if anyone here can provide one.) Thinking about the prime number theorem, I wondered whether it is ...
user21820's user avatar
  • 2,912
9 votes
2 answers
286 views

What is the density of integers of the form $a^2+nb^2$?

Landau proved that the mean density of integers of the form $a^2+b^2$ up to $x$ is $K\frac{x}{\sqrt{\log x}} (1+o(1))$, where $K$ is an explicit constant. One proof is based on the fact that a prime $...
Lior Bary-Soroker's user avatar
9 votes
2 answers
839 views

Reference and best bounds of $\sum_{n\leq x}\frac{\mu(n)}{n}$

Could someone please provide information about the best possible known bounds of the sum $$A(x)=\sum_{n\leq x}\frac{\mu(n)}{n}?$$ Unconditionally, $A(x)=O(e^{-c\sqrt{\log x}})$ is known to me. Does ...
Subhajit Jana's user avatar
9 votes
2 answers
547 views

Primes between $x$ and $x+x^\theta$

Iwaniec [1] proved that $$ \pi(x+x^\theta)-\pi(x) < \frac{(2+\varepsilon)x^\theta}{\eta(\theta)\log x},\ x>x_0(\varepsilon,\theta). $$ with $$ \eta(\theta)=\frac{15\theta-2}{9}. $$ (Actually, he ...
Charles's user avatar
  • 9,114
9 votes
3 answers
658 views

Vinogradov-Korobov for Dirichlet L-functions?

Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
H A Helfgott's user avatar
  • 20.2k
9 votes
1 answer
574 views

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 ...
Aidan Rocke's user avatar
  • 3,871
9 votes
2 answers
647 views

On bounds for idoneal integer

What is the best known lower bound and upper bound known for such a number if it exists and have there been any attempts (computational including) to eliminate the existence of such a number in known ...
user avatar
9 votes
2 answers
2k views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
mike's user avatar
  • 603

1
2 3 4 5 6