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.
0
votes
0answers
42 views
Is the de Bruijn-Newman constant non-positive?
Define
$$H_{t}(z)=\int_{0}^{\infty} e^{tu^2}\phi(u)\cos(zu) \mathrm{d}u$$
where $\phi$ is a certain exponentially decaying function. It is known that there exists some constant $\Lambda$ (the de ...
5
votes
2answers
194 views
asymptotic for li(x)-Ri(x)
Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$
where
$$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
0
votes
0answers
62 views
How to prove a Mersenne number is not pseudoprime to the base 3?
I start think in finding a small divisor $p$ of $M_n$ then I obtain that $p\mid (3^{M_{n}-1}-1)$ hence $ord_{p}(3)\mid M_{n}-1$
Now can I prove p is the same as $M_{n}$ so that $M_{n}$ is not ...
1
vote
0answers
69 views
Counting "simultaneous squares' over the Gaussian integers
Let $n$ be a square-free integer. Then for a given integer $m$, $m$ is a square modulo $n$ if and only if the sum
$$\displaystyle \sum_{d | n} \left(\frac{m}{d}\right) > 0.$$
In fact one can ...
3
votes
0answers
108 views
Modular root of $-1$
Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
0
votes
1answer
129 views
Parity and number of squares taken by polynomials in a range?
I have a polynomial $f(x)=a^2x^2+bx+c\in\mathbb Z[x]$ with $f(x)$ not a constant times a square and $abc\neq0$ and I want to know how many $x$ between $-a$ and $a$ the polynomial is a perfect square. ...
-4
votes
0answers
244 views
About a pattern on prime numbers [on hold]
I have read in https://www.sciencealert.com/mathematicians-discover-a-strange-pattern-hiding-in-prime-numbers
that says: "But this doesn't explain the magnitude of the bias the team found, or why ...
11
votes
1answer
148 views
Partial product of Euler factors
Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let
$$
\zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}},
$$
where $\mathrm{Re}(s)>1$.
Is there any $T$ such that $T$ ...
8
votes
1answer
813 views
Are there infinitely many primes of this form?
The semiprime $87 = 3*29$ has a curious property: it's the fact that both
$87^2 + 29^2 + 3^2 = 8419$
and
$87^2 - 29^2 - 3^2 = 6719$
are prime numbers.
This intrigued me and led me to wonder if ...
0
votes
0answers
65 views
Infinite difference length of integer subsets
Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...
2
votes
0answers
65 views
Twisted modular equation
Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$
are integral over $\mathbf Z[j]$. Under what conditions is ...
1
vote
1answer
203 views
Sum of log over friables
Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$\sum_{\substack{n \leq x \\ P(n)...
2
votes
0answers
90 views
Expressing modular functions of level 9 and 32 as rational functions
Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
3
votes
0answers
29 views
Analytic continuation of a Dirichlet series with several complex variables
For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define
\begin{equation*}
U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...
5
votes
1answer
471 views
An asymptotic formula for this sum
Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.
$$\sum_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$
where $a$ and $b$ are two coprime ...
7
votes
1answer
313 views
Goldbach's conjecture for the Liouville function
Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ?
Here $\lambda$ is the Liouville function.
3
votes
1answer
80 views
Number of $k$-free integers of bounded radical
Let $n \in \mathbb{N}$. Define the radical $R(n)$ of $n$ by
$$\displaystyle R(n) = \prod_{p | n} p.$$
In other words, $R(n)$ is the largest square-free number which divides $n$.
For an integer $k \...
-1
votes
0answers
120 views
Deuring-Heilbronn phenomenon: would a lower bound of $\lambda/\mu$ imply GRH?
Tonight I learnt the existence of the Deuring-Heilbron phenomenon which, if I understood correctly, implies that, if GRH fails for a single Dirichlet L-function, then all zeros up to some height of ...
4
votes
1answer
139 views
The number of solutions of the equation $ax_1x_2+by_1y_2=n$
The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive ...
10
votes
1answer
442 views
How many zeta zeros are needed to accurately calculate five digits for π(1000000), where π(x) is the prime counting function?
John Derbyshire in his book PRIME OBSESSION says on page 343:
"I’ll round off with a complete calculation of $\pi(1,000,000)$, the
number of primes up to one million, using Riemann’s formula -- ...
10
votes
1answer
636 views
Why are the coefficients of the modular equation so large?
The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions
$j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.
For example, we ...
34
votes
5answers
6k views
“Long-standing conjectures in analysis … often turn out to be false”
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
11
votes
2answers
296 views
Equidistribution of CM points in the principal genus
It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say ...
3
votes
1answer
212 views
On the Dirichlet series for $1/\zeta(s)$ at $\Re(s)=1/2$
Suppose that $1/2+it$ is not a zero of the Riemann zeta function $\zeta$, where $t \in \mathbb{R}$. Can $1/\zeta(1/2+it)$ be expressed as a Dirichlet series ?
-2
votes
0answers
56 views
Left-right discrepancy in a short interval containing k0 primes
Assuming Goldbach's conjecture, let $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ , $ k_{0}(n) : =\pi(n+r_{0}(n))-\pi(n-r_{0}(n)) $ , $ k_{0}^{+}(n) : =\pi(n+r_{0}(n))-\pi(n) $ and $ ...
1
vote
2answers
164 views
Number theory on Banach space $L^2(\mathbb R)$ meets linear independence?
Consider an orthonormal basis $(\varphi_k)$ of $L^2(\mathbb R)$ with Lebesgue measure.
I came along a nice number theoretic question in analysis:
Write $$f_k(x):=\int_{\left\lvert y \right\rvert \...
0
votes
0answers
60 views
Summation formula for twisted L-function
Does any expert here know something about the summation formula of the Voronoi type for the sum $$\sum_{n\le X} a_{f}(n)\chi(n) e\left(\frac{an}{c}\right)?$$
Here $f$ is a newform of level $N$, $\chi$...
10
votes
1answer
348 views
Spacing of fractions with prime denominator
Let $X\ge 2$ be large. Let $$A = \left\{\frac{a}{q}: 1\le a < q\le X,\ (a, q) = 1\right\},$$ and let $$B = \left\{\frac{a}{p}: 1\le a < p\le X,\ (a, p) = 1,\text{ and $p$ is prime}\right\}.$$ ...
9
votes
0answers
197 views
The trace formula over function fields
There are many examples in number theory where an "arithmetic" problem (i.e. for number fields) has an easier analogue for function fields over finite fields. This is also true for questions ...
4
votes
1answer
348 views
Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?
Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled ...
3
votes
0answers
119 views
Cancellation in this exponential sum?
I would like to know whether it is possible to obtain cancellation in the sum
$$\sum_{p \leq X} e^{{2\pi iX}/{p}}$$
where $X$ is a real number that goes to $\infty$, and $p$ denotes a prime number.
1
vote
1answer
122 views
zeros of sums of complex exponential functions
Let $a_i,b_i$ be $n(\geq 2)$ non-zero real numbers. Assuming that $\sum_{i=1}^n a_ie^{\sqrt{-1}b_i x}=1$ has infinite real solutions for $x$, prove or disprove that $b_i(1\leq i\leq n)$ is linearly ...
0
votes
4answers
309 views
On the real part of the Riemann zeta function inside the critical strip
Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
1
vote
1answer
182 views
On a certain integral representation for Dirichlet L-functions
It is an ancient result of Jensen that
$$(s-1)\zeta(s)=\frac{\pi}{2} \int_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$
where $\zeta$ denotes the Riemann zeta function.
Is ...
4
votes
2answers
293 views
Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
1
vote
0answers
88 views
Fejer-Jackson-like inequality with divisor sum
A question was recently asked about a generalization of the Fejer-Jackson inequality $$\sum_{k=1}^n \frac{\sin kx}{k}\gt 0 \quad \forall\: n\in\mathbb{Z}^+\: \text{and}\: 0\lt x\lt\pi$$
to ...
0
votes
1answer
94 views
Is $P_{2n} \ge 2P_n$ and $\pi(2x) \le 2\pi(x)$ inconsistent to the Prime k-tuple conjecture?
I posed a conjecture as follows that is a special case of Second Hardy–Littlewood conjecture:
For $n, x \ge 2$ be two integers then:
$$P_{2n} \ge 2P_n$$
and
$$\pi(2x) \le 2\pi(x)...
25
votes
1answer
2k views
Lindelöf hypothesis claim
I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but ...
3
votes
0answers
78 views
Why is the smallest (fractional) absolute central moment of a Gaussian distribution almost at $\sqrt{3}/2$?
Let $X$ be a standard normal random variable. What $\alpha$ minimizes $E|X|^{\alpha}$?
Numerically, $\alpha$ turns out to be equal to $\sqrt{3}/2-\varepsilon$ where $\varepsilon$ is of the order $10^...
0
votes
1answer
110 views
Sergei numbers : even integers n being a prime gap at least n times
Let's introduce Sergei (for SElf-Referential Gaps Extensible to Infinity, and as a wink to a mathematician friend of mine of Russian descent whose given name is Serge and quite interested in number ...
-3
votes
1answer
163 views
Can this weakening of Polignac's conjecture be proven?
Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
12
votes
2answers
1k 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 $$x=...
-1
votes
1answer
68 views
Explicit second order bounds for the prime counting function
One of the most important theorems proved in the 19th century is the prime number theorem. Put $\pi(x)$ for the number of prime numbers $p$ satisfying $1 \leq p \leq x$. Then the prime number theorem ...
5
votes
1answer
481 views
Prove: If $P_n$ is $n$-$th$ prime number then $P_{n+m} \ge P_n+P_m$
Let $n > 1$ and $m > 0$ be two integers and $P_n$ be the $n^{th}$ prime.
Prove: $$P_{n+m} \ge P_n + P_m .$$
Can you give a hint, reference, comment, or proof?
3
votes
1answer
140 views
The Gauss Circle Problem asymptotic in dimension
The circle problem in $k$ dimensions: "For $n>0$, how many points $z\in \ \mathbb{Z}^k$ have $\|z\|^2\leq n$?"
For large $n$, the answer is $\approx n^{k/2}\cdot \operatorname{Vol}(B^k(0,1))+\...
5
votes
2answers
708 views
$\pi((n+1)^2)-\pi(n^2) \le \pi(n)$ for all $n \ge 370$?
There are some conjectures of the form: There always exist at least $X$ prime numbers between $A$ and $B$. Examples:
Bertrand's postulate: for every $n>1$ there is always at least one prime $p$ ...
4
votes
1answer
220 views
3-smooth number
Let $(u_n)_{n\in\mathbb N}$ be the sequence of $3$-smooth numbers (that is whole number that can be written as $2^a3^b$ with $a,b\in\mathbb N$) sorted by increasing order. I am looking for the ...
3
votes
1answer
174 views
Sum of Legendre symbol over primes
Given some $X, Y\ge 1$ and some $d\le Y$ not a perfect square, is it possible to bound
$$\sum_{p\le X}\left(\frac{d}{p}\right)?$$
As long as $Y$ is not too large compared to $X$, I would expect that ...
2
votes
0answers
102 views
Dirichlet series as rational zeta expressions
Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product ...
23
votes
0answers
564 views
How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?
For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have
$$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$
Note that this ...
