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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.

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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} \...
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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 ...
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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
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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 ...
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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. ...
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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
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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 ...
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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
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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
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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
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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
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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
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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 \...
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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
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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
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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
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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 ...
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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
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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
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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 ?
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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
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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 \...
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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
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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
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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
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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
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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.
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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 ...
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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 $\frac{1}{2}<\Re(s)\leq 1$ ?
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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
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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 $0\leq a<b \leq 1$?
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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 ...
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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
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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 ...
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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^...
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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 ...
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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
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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=...
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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
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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
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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
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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
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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 ...