**1**

vote

**0**answers

66 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 ...

**5**

votes

**0**answers

203 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

112 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 $n$, ...

**-3**

votes

**1**answer

338 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 ...

**23**

votes

**1**answer

708 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 ...

**2**

votes

**1**answer

104 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) ...

**1**

vote

**0**answers

163 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}} + ...

**0**

votes

**1**answer

234 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 ...

**33**

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

873 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

305 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

393 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

383 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 ...

**5**

votes

**1**answer

215 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 ...

**6**

votes

**0**answers

796 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 ...

**0**

votes

**0**answers

138 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

568 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 ...

**0**

votes

**0**answers

80 views

### Closed forms for paired factors of a 4-factor infinite product. Is their shape fixed?

As a follow up on this question, I have now composed the following model of closed forms for infinite products of pairs of factors. At the heart is a 4-factor infinite product shown in red at the ...

**3**

votes

**1**answer

310 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 ...

**12**

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

470 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 ...

**6**

votes

**1**answer

576 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} ...

**4**

votes

**1**answer

716 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 ...

**39**

votes

**1**answer

2k 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

915 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 ...

**40**

votes

**5**answers

4k 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

363 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

229 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 ...

**9**

votes

**2**answers

676 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 ...

**9**

votes

**2**answers

473 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 ...

**46**

votes

**4**answers

2k 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 ...

**4**

votes

**2**answers

327 views

### Mertens function limits using $\phi+2$n

As we can see in the plot below, Mertens function: $M(x)\equiv \sum_{n=1}^{x}\mu(n)$ has wild swings from positive to negative and back again.
When we use: $$x=\frac{1}{2+\frac{1}{\phi+2}}\text{, ...

**18**

votes

**4**answers

1k 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 "expectional zeros" of course first ...

**4**

votes

**1**answer

422 views

### Is there a connection between the closed forms of these two infinite products?

Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$$\displaystyle ...

**5**

votes

**2**answers

293 views

### Equivalence of two well-known forms of (RH): reference-request.

This is a reference-request about a very simple statement.
The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to
$$(2)\ \ ...

**2**

votes

**2**answers

238 views

### On the location of zeros of L functions from modular forms

I understand that the Mellin transform of a modular form is expected to satisfy RH when it is an eigenform of all Hecke operators, in which case it has an Euler product. Now about when the form is not ...

**4**

votes

**3**answers

558 views

### A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?

Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- ...

**14**

votes

**1**answer

891 views

### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ?

Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ...

**4**

votes

**4**answers

442 views

### What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are “scaled” linearly?

I found that the following infinite product with $\mu = a +n b i$ and a,b real, $s \in \mathbb{C}$:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \frac{s}{1-\mu} ...

**1**

vote

**1**answer

300 views

### $\zeta(2k+1)$ expressed in a product of two infinite products of non-trivial zeros.

Take the Hadamard product for $\zeta(s)$:
$$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} ...

**1**

vote

**0**answers

118 views

### The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$

I was exploring the formula:
$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$
and found that for all $\Re(s) \ne \frac12$:
...

**14**

votes

**1**answer

818 views

### A question about Speiser's 1934 result on the Riemann hypothesis

A number of sources concerning Speiser's 1934 result state that the Riemann Hypothesis (RH) implies $\zeta'(s)\neq 0$ for all $0<\text{Re}(s)<1/2$. But I have seen some (possibly less reliable) ...

**0**

votes

**0**answers

217 views

### Does there exist a Weierstrass/Hadamard factorization for $\chi(s)-1$ ?

Would like to build once more on this question.
Take $s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1$.
Let's assume it is proven that:
$$\zeta(1-s) - \zeta(s)$$
has all its zeros on the ...

**0**

votes

**1**answer

649 views

### Interplay between Riemann and Swinnerton-Dyer

Hello everyone,
After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ...

**2**

votes

**1**answer

209 views

### Deriving the Riemann non-trivial zeros from $\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$

The Hurwitz zeta function:
$$\zeta_{H}(s,a)$$
reduces to $\zeta(s)$ when $a=1$ and to $(2^s-1)\zeta(s)$ when $a=\frac12$.
However, I stumbled upon a peculiar third connection:
$$\zeta_{H}(s,a) + ...

**8**

votes

**1**answer

1k views

### What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?

Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?
I've heard Freeman Dyson say that ...

**3**

votes

**1**answer

838 views

### What happens when infinite values of $\zeta_{H}(s,z)$ approach $\zeta(s)$ ?

Take the following Hurwitz zeta:
$$\zeta_{H}(s,z)$$
with $s=\sigma \pm ti$ and $\displaystyle z=1 \pm \frac{i}{a}$ and $t,a \in \mathbb{R}$.
In the critical strip $0 \lt \sigma \lt 1$, this Hurwitz ...

**3**

votes

**0**answers

266 views

### Definite integral of $\zeta(s)$ over the critical strip

Take the following definite integral:
$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$
with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$.
The graph of ...

**3**

votes

**2**answers

411 views

### Are all zeros of ζ^{k}(s)±ζ^{k}(1−s) on the critical line (k=k-th derivative)?

The non-trivial zeros of $\zeta^{k}(s)$, with $k=k^{th}$ derivative, do not lie on a line and seem to be distributed randomly in the region $\sigma > \frac12$. However the non-real zeros in the ...

**1**

vote

**1**answer

462 views

### Zeros of the function $\zeta(s) \pm \zeta(\overline s)$

Building on this question: Zeros of $\zeta(s) \pm \zeta(1-s)$, I experimented further with:
$$\zeta(s) \pm \zeta(\overline s)$$
Assuming $s=\sigma + ti$, I observed that this function also has many ...