**0**

votes

**0**answers

60 views

### Why are these two gamma functions equal? [on hold]

$\gamma_{1}(s)=\pi^{\frac{1}{2}-s}\frac{\Gamma(\frac{s}{2})}{\Gamma(\frac{1}{2}(1-s))}$
$\gamma_{2}(s)=(\frac{b}{2\pi})^{\frac{1}{2}-a} e^{-2i\theta(b)}$
,where $s=a+bi$
I calculated ...

**21**

votes

**4**answers

2k views

### Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...

**0**

votes

**1**answer

791 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: ...

**0**

votes

**2**answers

274 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

666 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

171 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 ...

**-4**

votes

**1**answer

213 views

### Calculating Riemann Non-Trivial Zeros [closed]

I recently started studying about the Riemann Hypothesis. I just want to understand if there is any condition that the imaginary part i.e. t in (0.5 + it) should always be greater than 1? For Example, ...

**5**

votes

**0**answers

152 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 ...

**67**

votes

**5**answers

24k views

### Consequences of the Riemann hypothesis

I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice ...

**6**

votes

**0**answers

133 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 ...

**2**

votes

**0**answers

133 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

61 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 ...

**9**

votes

**0**answers

287 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 ...

**4**

votes

**0**answers

332 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.
...

**2**

votes

**1**answer

269 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 ...

**8**

votes

**1**answer

638 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 ...

**4**

votes

**0**answers

201 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 ...

**1**

vote

**0**answers

141 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 ...

**2**

votes

**1**answer

271 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}} + ...

**8**

votes

**0**answers

224 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 ...

**1**

vote

**0**answers

88 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 ...

**1**

vote

**0**answers

89 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 ...

**10**

votes

**2**answers

771 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 ...

**6**

votes

**0**answers

273 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 ...

**49**

votes

**4**answers

3k views

### Are there refuted analogues of the Riemann hypothesis?

The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important ...

**1**

vote

**2**answers

133 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

402 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 ...

**47**

votes

**1**answer

3k 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?

**25**

votes

**1**answer

890 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 ...

**3**

votes

**1**answer

155 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) ...

**0**

votes

**1**answer

246 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 ...

**40**

votes

**1**answer

3k 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 ...

**4**

votes

**1**answer

480 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 ...

**30**

votes

**3**answers

3k views

### The Hardy Z-function and failure of the Riemann hypothesis

David Feldman asked whether it would be reasonable for the Riemann hypothesis to be false, but for the Riemann zeta function to only have finitely many zeros off the critical line. I very rashly ...

**8**

votes

**3**answers

1k views

### On Robin's criterion for RH [closed]

\begin{equation}
\sigma(n) < e^\gamma n \log \log n
\end{equation}
In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...

**5**

votes

**2**answers

388 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

424 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

498 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 ...

**46**

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 ...

**7**

votes

**0**answers

1k 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

159 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 ...

**11**

votes

**2**answers

618 views

### Best bounds toward Serre's uniformity conjecture

If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois representations ...

**5**

votes

**2**answers

627 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 ...

**22**

votes

**4**answers

3k views

### Modular forms and the Riemann Hypothesis

Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function ...

**3**

votes

**1**answer

335 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 ...

**13**

votes

**1**answer

1k views

### Exceptional zeros and Liouville's $\lambda$ function

This originated from an textbook exercise (recently posted to math.stackexchange
http://math.stackexchange.com/questions/62883/quadratic-characters-and-liouvilles-function
with no success) but I think ...

**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

488 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

663 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} ...