The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation ...

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235 views

Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...
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0answers
19 views

The derivatives of Riemann xi function [migrated]

What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the ...
3
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0answers
55 views

Square integral of finite Euler product

Consider the finite Euler product $$ P(t) = \prod_{r=1}^R \left(1 + p_r^{i t} \right). $$ (Here $p_1, p_2, \dots$ are of course the primes.) Question: What is a good asymptotic upper bound for $$ ...
8
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1answer
267 views

Riemann zeta function: pair correlations vs. neighbor spacings

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
33
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1answer
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The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
3
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1answer
474 views

What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question) I think I understood the concept of fractional derivatives applied to ...
41
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9answers
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Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
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88 views

For which integer values of $k$ can we find one solution to the equation $\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$ by iteration? [closed]

I am trying to find solutions to the well known equation: $$\sum\limits_{n=1}^{n=k} \frac{1}{n^s}=0$$ Now with this program below I have found that for certain values of the integer $k$ one can find ...
5
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1answer
332 views

Can $\zeta(s)$ for $\Re(s)>1$ be split into two factors that each can be analytically continued?

Assuming the RH and $s \in \mathbb{C}, \rho_n =\frac12 \pm i\gamma_n$, the following (altered) Hadamard product: $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n i ...
5
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0answers
393 views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where ...
5
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2answers
518 views

Bounds for $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$

Let $$ f(x) = \sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$$ Are there lower bounds, upper bounds or (unlikely) simpler closed form for $f(x)$? The bounds for Bernoulli numbers I ...
8
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3answers
781 views

Upper bounds on the difference of consecutive zeta zeros

There are many results on the spacing of the gaps between nontrivial zeros of the $\zeta$ function, from trivial (average value is $\frac{2\pi}{\log\gamma_n}$) to difficult (bounds on max and min ...
27
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3answers
2k views

Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation $$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$ "proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = ...
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0answers
38 views

estimates of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le c_k Y^k(t)$

By the Vinogradov-Korobov estimate we have $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le c_k Y^k(t)$ where $Y(t)= \ln^{2/3} (|t|+3) (\ln\ln (|t|+3))^{1//3}$ for some $c_k$. I search the value ...
21
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4answers
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 ...
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0answers
161 views

Is the difference of these two real-rooted functions real-rooted?

During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1). Our final goal is to prove that: Proposition 1: ...
7
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2answers
189 views

How to formalize the *loci of equal arg($\zeta(s)$)* (“isogones”) in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
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0answers
85 views

An apparent closed form for a slightly tweaked Dirichlet L-function. Could it be proven? [closed]

I made a small tweak to the well-known Dirichlet L-function ($p$=prime): $$L(s, \chi_4) :=\prod_p \bigg(\frac {p^s}{p^s-\chi_4(p)} \bigg)=\prod_p \bigg(\frac {p^s}{p^s-\sin\left(\frac{p ...
8
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150 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
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1answer
825 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 ...
5
votes
1answer
175 views

Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ implies $\sigma$ continuous?

Let $\sigma$ be a field automorphism of $\mathbb{C}$ that commutes with the Riemann Zeta function. Can we use Voronin's universality theorem to prove that $\sigma$ is necessarily continuous? Thanks in ...
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0answers
123 views

What is the connection between the Riemann Xi-function and n-sphere? [closed]

Riemann's Xi-function is defined as $$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$ At the same time we have the following formulas for n-sphere's area and volume: ...
1
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1answer
158 views

Values of the completed Riemann $\xi(1+it)$ for small t?

I'm editing this question heavily for clarity: I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function $$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ along the line ...
3
votes
2answers
125 views

Deduce average order of $\phi(n)/n$ from probability that two integers are coprime

I've seen proofs of the fact that the probability of two random integers being coprime is $\frac{6}{\pi^2}$ (all of them leading to a use of the Riemann Zeta function and the Basel problem). In ...
5
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0answers
114 views

Moments of completed L-functions?

This is a follow up question to this one. It seems that results on moments of L-functions, that is, estimates for integrals of the form $$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$ are typically for the ...
7
votes
1answer
193 views

Behaviour of $\zeta(1-it)/\zeta(1+it)$?

I am trying to understand the behaviour of $$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$ where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...
6
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1answer
134 views

Zeta zeros standard normal distribution about $\vartheta (\gamma_n)$

Asked at MSE here without response. I realise that this resembles Odlyzko's famous nearest neighbours plot, and was wondering whether this is simply a manifestation of the same phenomenon. That ...
35
votes
3answers
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Is this integral representation of $\zeta(2n+1)$ known?

Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
28
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4answers
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What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
5
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0answers
95 views

On existence of rapid Arithmetic geometric procedure?

We know that $\pi$ can be computed by Arithmetic Geometric mean using Gauss-Legendre procedure which does provide fastest convergence rate as well with a guarantee of $2^n$ bits of $\pi$ at $n$th ...
17
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1answer
529 views

stable homotopy groups and zeta function

I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...
42
votes
4answers
3k views

If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. ...
35
votes
1answer
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Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
3
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2answers
311 views

Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
25
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6answers
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Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
6
votes
2answers
330 views

Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function $$ \begin{align} \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a ...
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votes
1answer
243 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, ...
3
votes
2answers
258 views

Approximations to the Mertens function

The Mertens function $M(x)$ is the summatory Möbius function i.e. $$M(x) = \sum_{k=1}^{x} \mu (k)$$ The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be ...
11
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0answers
460 views

One-to-one correspondance between zeta zeros and the prime powers? [closed]

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here. I have noticed an interesting property related to the ...
3
votes
1answer
295 views

Is this differential equation for zeta on the critical line? One can compute it from its derivative and simpler functions

Looks like on the critical line one can compute $\zeta(1/2+it)$ from $\zeta^{'}(1/2+it)$ and simpler functions. Let $$ \begin{aligned} f(t)= & 2\, \left( {\frac { \left( \left| \zeta^{'} ...
5
votes
0answers
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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 ...
3
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0answers
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Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
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0answers
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Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?

veThe balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as: $$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$ Note $\Psi(s)$ is the digamma ...
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0answers
57 views

Function series involving a suite of imprimitive Dirichlet characters and a zero of $L(\chi,s)$

I have difficulties to understand the behavior of following suite of functions near zero : $$F_P(x)= \sum\limits_{n=P}^{\infty} \chi_P(n) f(nx)$$ With $f(x) = x^{-s_0} e^{-x}$ where $s_0$ is a non ...
6
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494 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} \left[ {\rm li}(t^{\frac zn-s}) \right]^{x}_2 \tag{7} ...
5
votes
1answer
265 views

Are all complex zeros of $\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$ on the critical line $\Re(s)=\frac12$?

Numerical evidence suggests that all complex zeros (real ones exist as well) of: $$\frac{\zeta'}{\zeta}(s) \pm \frac{\zeta'}{\zeta}(1-s)$$ reside on the critical line with $\Re(s)=\frac12$. I made ...
1
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1answer
160 views

Counting prime powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is: $$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...
3
votes
0answers
64 views

how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...
8
votes
1answer
412 views

Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...
5
votes
1answer
252 views

The horizontal distribution of zeros of $\zeta^\prime(s)$

I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998. Throughout I will ...