**0**

votes

**1**answer

169 views

### Prime quadratic non-residue

NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime.
I have following ...

**0**

votes

**0**answers

126 views

### $L(x)$ growing faster than $\text{Mertens}(x)$ $\implies$ RH $\ $?

$M(x)$ is the Mertens function, $L(x)$ is the summatory Liouville function,
$f(x) {\, \scriptstyle\gg \!\!\ll} \ g(x)$ means that $f(x) = \mathcal{O}(\sup_{y < x}|g(y)|)\ $ and $g(x) = \mathcal{O}...

**1**

vote

**2**answers

629 views

### Statements going against the grain of Riemann Hypothesis (R.H.) [closed]

Let $M(N) := \sum_{n=1}^N \mu(n)$
It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H.
A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...

**7**

votes

**1**answer

191 views

### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...

**2**

votes

**0**answers

210 views

### On a sequence of L-functions having same zeros in critical strip and GRH

I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...

**9**

votes

**0**answers

364 views

### Riemann hypothesis in Zilber's field

Question. What is known about the situation (truth or falsity) of Riemann hypothesis in the Zilber's field?

**-1**

votes

**1**answer

1k views

### Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]

Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-...

**0**

votes

**2**answers

327 views

### Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...

**-7**

votes

**1**answer

1k views

### Non-standard numbers and exponential form of Zeta function [closed]

Basic idea
For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...

**1**

vote

**0**answers

191 views

### Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemann hypothesis they used?

Does the proof of Conjectures B and D of Hardy and Littlewood have any implication on the generalized Riemannn hypothesis they used?
In their paper,
Some problems of 'Partitio numerorum'; III - On ...

**5**

votes

**0**answers

192 views

### Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$.
The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...

**6**

votes

**0**answers

139 views

### How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum _{n=1}^{\...

**2**

votes

**0**answers

151 views

### Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?

This question expands on this one and seems to have a stronger result.
Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...

**2**

votes

**0**answers

68 views

### Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

With $s \in \mathbb{C}, a \in \mathbb{R}$,
numerical evidence strongly suggests that the complex zeros in the critical strip of:
$$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$...

**4**

votes

**0**answers

363 views

### Is there a hidden symmetry in the prime numbers distribution?

Under Goldbach's conjecture, let's consider once again the map $r_0\colon n\mapsto r_{0}(n)$ such that $r_{0}(n)$ is the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime.
Let'...

**9**

votes

**0**answers

326 views

### Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...

**2**

votes

**1**answer

282 views

### Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x \...

**4**

votes

**0**answers

213 views

### Possible weakening of Robins criterion for RH

I hope 2015 doesn't start with gross nonsense question for me.
Robins equivalence for RH is
$$ \frac{\sigma(n)}{n \log\log n} < e^{\gamma}\qquad(1)$$
for $n \ge 5041$. We have
$ \limsup \frac{\...

**8**

votes

**1**answer

683 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**1**

vote

**0**answers

147 views

### Except for a finite few outside the strip, do all complex zeros of $\zeta(a+s)\pm \zeta(a+1-s)$ reside on the critical line for all $a\lt 0$?

Assume $a \in \mathbb{R}$ and $s \in \mathbb{C}$.
Numerical evidence suggests that all complex zeros, except for a finite few outside the strip, of:
$$\zeta(a+s)\pm \zeta(a+1-s)$$
lie on the line $\...

**9**

votes

**0**answers

246 views

### Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...

**1**

vote

**0**answers

94 views

### Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms:
$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$
Assume $z=i$:
$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$
with $\eta(...

**1**

vote

**0**answers

101 views

### Are all complex zeros of $Li_s(z)\, \pm \, Li_{1-s}(z)$ on the critical line or outside the critical strip for $z \le -1$?

This question loosely builds on this one, however is a bit simpler and I found the results to be more robust.
It seems that all zeros in the critical strip $0 \lt \Re(s) < 1$ of:
$$Li_s(z)\, \pm \...

**6**

votes

**0**answers

591 views

### Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of:
$$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$
are on the ...

**1**

vote

**2**answers

141 views

### Asymptotics and error terms for an arithmetic function built upon $\omega$ and $\Omega$ functions

For any real number $x$, let's define $Om_{k}(x)$ as the number of positive integers $m$ below $x$ such that $\Omega(m)-\omega(m)=k$, where $\omega(n)$ is the number of distinct primes dividing $n$, ...

**-4**

votes

**1**answer

444 views

### Is SOC known to imply the Grand Riemann Hypothesis? [closed]

I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...

**25**

votes

**1**answer

940 views

### How strong is this conjecture? $(Z/nZ)^*$ is generated by “small” elements

Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My main ...

**3**

votes

**1**answer

176 views

### A Hadamard product of the zeros of the Riemann integral. Does it put any constraints on where the $\rho$'s can reside in the critical strip?

I have deleted a previous, now obsolete question on the same topic.
Take the well-known Riemann integral:
$$\displaystyle \pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\...

**2**

votes

**1**answer

278 views

### Is there anything known about the complex zeros of this integral related to $\zeta(s)$?

The right-hand side of the well known equation:
$$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}} + {x}^{\frac{1-s}{2}}\right)\,\...

**0**

votes

**1**answer

247 views

### A convergence issue [Edited]

Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors in a Hilbert space $$l^2_{k^{-2}}:=\{z=\{z(k)\}_{k=1}^\infty:\sum\limits_{k=1}^\infty z(k)^2k^{-2}<\infty\}.$$ It is known that for some $x\in l^...

**41**

votes

**1**answer

4k views

### Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset logarithmic ...

**23**

votes

**1**answer

1k views

### How good is “almost all” when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...

**5**

votes

**2**answers

421 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 ...

**1**

vote

**2**answers

434 views

### The implicit constant in GRH

One particularity of the Generalized Riemann Hypothesis seems to deserve some clarification. In particular, what is included in the commonly accepted version of the conjecture?
GRH states that
$$\...

**8**

votes

**1**answer

551 views

### Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...

**4**

votes

**1**answer

623 views

### definition of accretive operator

A relation T with domain and range in a Hilbert space is said to be accretive if the
transformation $ (T − \lambda)/(T + \bar \lambda\ ) $
with domain and range in the Hilbert space is contractive for ...

**8**

votes

**0**answers

2k views

### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...

**1**

vote

**0**answers

164 views

### Is RH equivalent to the following estimate?

This question is a follow up from About Goldbach's conjecture and comes from what I read about the Farey series related criterion for RH.
Let $r_{k}(n)$ be the $k+1$-th potential typical primality ...

**5**

votes

**2**answers

638 views

### On extended Riemann Hypothesis and coefficients of Selberg Class L-functions

There is the conjecture that Selberg Class L-functions satisfy RH.
So that an L-function needs to have its coefficient multiplicatives (plus other conditions: functional equation,...) in order to ...

**3**

votes

**1**answer

345 views

### On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...

**15**

votes

**3**answers

1k views

### Does this infinite sum provide a new analytic continuation for $\zeta(s)$?

It is well known that the infinite sum:
$$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$
only converges for $\Re(s)>1$.
The Dirichlet 'alternating' sum:
$$\displaystyle \zeta(s) = \...

**2**

votes

**1**answer

502 views

### Are the zeros of the sum/difference of these integrals all on the critical line?

The answers given to the question whether all zeros in the critical strip of $\zeta(s)\pm\zeta(1-s)$ lie on the critical line, suggest that this can indeed be proven, however only for those zeros ...

**5**

votes

**1**answer

720 views

### Sufficient condition for Riemann Hypothesis?

Is there an L-function ($L_s=\sum_{n =1}^{\infty} \frac{a_n}{n^s}$) having a functional equation coming from a relation of the type [1]:
$\sum_{n =1}^{\infty} a_n \; e^{-2\pi nx}= \frac{A}{x^k} \sum_{...

**7**

votes

**1**answer

1k views

### Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...

**47**

votes

**1**answer

4k views

### What if the Riemann Hypothesis were false?

There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false?

**7**

votes

**1**answer

1k views

### A reformulation of the Riemann Hypothesis

I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.
Let's define $R(x,...

**49**

votes

**5**answers

5k views

### Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line:
$$ \zeta(\frac{1}{2} + i k_j) = 0 $$
Let $f$ be the Fourier transform of the sum ...

**3**

votes

**1**answer

430 views

### Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize
if this is too much -- but I am still not understanding well why this conjecture is believed to be true.
...

**4**

votes

**1**answer

302 views

### Estimate on the prime-counting function $\psi(x)$.

There is an elementary statement that I believe I have read somewhere, but I can't remember where. I'd like to know if the statement is correct (in which case it is surely standard) and if so, where I ...

**10**

votes

**2**answers

799 views

### Effective Chebotarev without Artin's conjecture

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's $L$-function and ...