99
votes
Accepted
Is the Riemann zeta function surjective?
The Riemann zeta function is surjective. First, $\zeta(1/z)$ is holomorphic in the punctured disk $0<|z|<1$. Looking at $z=(1/2+it)^{-1}$ with $t\to\infty$ reveals that $\zeta(1/z)$ has an ...
79
votes
Is the Riemann zeta function surjective?
$\zeta$ function has only one pole at $z=1$. It also has order $1$. If $\zeta$ omits $c\in C$ then $g:=1/(\zeta-c)$ is entire with one simple zero at $1$.
As it is of order $1$, it must be $g(z)=(z-1)...
78
votes
Accepted
A mysterious connection between primes and $\pi$
Here is a proof of the conjecture. We shall use Hecke's theorem that the angles of the lattice points $(x_p,y_p)$ are asymptotically equidistributed in $[\pi/4,\pi/2]$, cf. this MO post.
Let $t_p\in[\...
73
votes
"Long-standing conjectures in analysis ... often turn out to be false"
I don't know about analysis in general, but I think it's definitely fair to say "often" in functional analysis. My feeling is that we have a solid, thorough, elegant body of theory which usually leads ...
Community wiki
58
votes
Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros
It has significant implications on the error term of the PNT for arithmetic progressions.
PNT and Siegel-Walfisz theorem
Let $\psi(x;q,a)$ be the sum of $\Lambda(n)$ over $n\le x$ and $n\equiv a\pmod ...
56
votes
Accepted
Circle $x^2 + y^2 = n!$ doesn't hit any lattice points for any $n$ except for $0$, $1$, $2$ and $6$ or does it?
For $n\geq 7$, Erdős proved in 1932 that there is a prime $n/2<p\leq n$ of the form $p=4k+3$. From this he deduces (in the same paper) that $1!$, $2!$, $6!$ are the only factorials which can be ...
52
votes
Accepted
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
The complex-analytic proof of the prime number theorem can help inform the elementary one.
The von Mangoldt function $\Lambda$ is related to the Riemann zeta function $\zeta$ by the formula
$$ \sum_n ...
51
votes
Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros
There will be many important consequences of Zhang's result, if correct. One specific result is that it will reduce one of the last open problem from the era of Gauss and Euler to a finite amount of ...
49
votes
Accepted
Why is so much work done on numerical verification of the Riemann Hypothesis?
People are interested in computing the zeros of $\zeta(s)$ and related functions not only as numerical support for RH. Going beyond RH, there are conjectures about the vertical distribution of the ...
49
votes
Accepted
Closed formula for a certain infinite series
Apply Möbius summation, the formula for $\sum_{n>=1}\cos(2\pi n x)/n^2$ to obtain:
$$11/4-45\zeta(3)/\pi^3=1.00543...\;$$
48
votes
Accepted
Is this equivalent to RH - Riemann hypothesis?
Yes, this is equivalent to RH (but not in any significant way). Recall the completed Riemann $\xi$-function
$$
\xi(s) = s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s),
$$
which, by Hadamard's ...
48
votes
Accepted
Reasons behind assuming the existence of Siegel zeros can be used to prove something stronger than assuming GRH?
Roughly speaking, GRH asserts that the Möbius function $\mu$ is "orthogonal" to all Dirichlet characters $\chi$, in the sense that correlations such as $\sum_{n \leq x} \mu(n) \overline{\chi(n)}$ are ...
48
votes
"Long-standing conjectures in analysis ... often turn out to be false"
If RH is "analysis", then surely Littlewood's 1914 theorem that $\pi(x)$ (the prime counting function) and $\mathrm{li}(x)$ (the logarithmic integral) alternate in size infinitely often... despite all ...
Community wiki
48
votes
Accepted
About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
If one starts with the Weierstrass factorisation
$$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod_{k=1}^\infty \left(1 + \frac{z}{k}\right)^{-1} e^{z/k}$$
of the Gamma function, applied to $z = -x, -\...
46
votes
Closed formula for a certain infinite series
Let me add the details for Henri Cohen's nice answer, without claiming any originality. We have
$$\sum_{m,n\geq 1}\frac{\cos\left(\frac{m}n\right)}{m^2n^2}=\zeta(4)\sum_{(a,b)=1}\frac{\cos\left(\frac{...
43
votes
Accepted
Lagrange four squares theorem
The set $X$ doesn't have to be the set of non-negative integers. This was known already to Härtter and Zöllner in 1977, who constructed an $X$ of the form $\{ 0, 1, 2, \ldots \} \setminus S $ for an ...
43
votes
Expressing the Riemann Zeta function in terms of GCD and LCM
Let me denote your LHS by $f(n,s)$. For fixed even $n$ I shall show that $f(n,s)-1\sim\zeta(s+1)-1$ as $s\to\infty$, that is,
$$\lim_{s\to\infty}\frac{f(n,s)-1}{\zeta(s+1)-1}=1.$$
This result nicely ...
42
votes
Accepted
Prime square offsets: Why is +7 more frequent than -7?
Consider $n^2+7$ and $n^2-7$ modulo $3$. If these are to be prime they must be non-zero $\pmod 3$, and in the first case $n$ can be anything mod $3$, whereas in the second case $n$ must be $0 \pmod ...
41
votes
Heuristic argument for the Riemann Hypothesis
The Riemann hypothesis is true, if primes are random in certain ways.
41
votes
Accepted
How and when do I learn so much mathematics?
The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally.
First, there is no such thing as ...
Community wiki
40
votes
Intuition behind Harmonic Analysis in Analytic Number Theory
Harmonic analysis is the theory of representations of locally compact abelian groups. The integers, and the integers mod $n$, are such groups. For problems dealing with functions on these groups ...
39
votes
Expressing the Riemann Zeta function in terms of GCD and LCM
A variety of formulas of this type (in the sense of a relation between $\zeta(s)$ and a sum over gcd or lcm) has been derived by Titus Hilberdink and László Tóth in On the average value of the least ...
39
votes
Accepted
Parity of the multiplicative order of 2 modulo p
This problem was asked by Sierpiński in 1958 and answered by Hasse in the 1960s.
For each nonzero rational number $a$ (take $a \in \mathbf Z$ if you wish) and each prime $\ell$, let $S_{a,\ell}$ be ...
39
votes
Accepted
Iterated logarithms in analytic number theory
There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are:
...
38
votes
How and when do I learn so much mathematics?
It may seem like a mountain. But remember that a few years ago you knew absolutely nothing, and you have mastered a lot of material already! Three or four years is a lot of time, and almost certainly ...
Community wiki
37
votes
Motivated account of the prime number theorem and related topics
To a certain extent, I think that analytic number theory really is magical, and there's a limit to how natural and motivated it can be. Of the accounts I have seen, the one in Donald Newman's book ...
37
votes
Accepted
Is there a real nonintegral number $x >1$ such that $\lfloor x^n \rfloor$ is a square integer for all $n \in \mathbb{N}$?
There is no such number. Suppose $\alpha>1$ is a real number such that $\lfloor \alpha^n \rfloor$ is a square for all $n\in {\Bbb N}$. Put $\beta=\sqrt{\alpha}$.
Now for each $n$ we have
$$
m^2 +...
36
votes
Accepted
How many primes can there be in a short interval?
As observed by Hensley and Richards in
Douglas Hensley and Ian Richards, Primes in intervals, Acta Arith. 25 (1973-74), 375--391,
if the prime tuples conjecture is true, then $\limsup_{n \to \infty}...
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