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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900 views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
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1k views

The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...
9
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324 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 ...
9
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242 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}{...
8
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158 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 ...
7
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122 views

How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?

In this question Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$. I fixed $a=2$ and the minus sign and defined: $$ f(s)=\Re \left( \zeta\left(\frac{...
6
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138 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}^{\...
6
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515 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 ...
6
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305 views

implication of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< s<...
6
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356 views

$\zeta(x)$ in terms of $\zeta'(x),\zeta'(1-x),\Gamma,\psi$

By differentiating $\xi$ and solving for $\zeta(1-x)$: $$ \zeta(1-x) = \frac{2(\zeta'(x)\Gamma(x/2)+\Gamma((1-x)/2) \zeta'(1-x)\pi^{x-1/2}) )}{\Gamma((1-x)/2) \pi^{-1/2+x}(2\log\pi -\psi((1-x)/2)-\...
6
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498 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 $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
6
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511 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
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116 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 ...
5
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101 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 ...
5
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187 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 ...
5
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137 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
5
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224 views

Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas: 1) The ...
5
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746 views

Zeta function double product

Is it possible to write the following double product in terms of the zeta function? \begin{align} &\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}} \end{align} Extending the ...
4
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247 views

A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function: $$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$ through substituting $t=nx$, $$\displaystyle \Gamma(s)\frac{1}{n^s} ...
4
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100 views

Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you. This post is related to zeros of partial sum of Taylor series of $e^x-1$. Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
3
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70 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 $$ \...
3
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65 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} \pi^...
3
votes
0answers
255 views

Does the difference of two converging infinite series correctly induce the non-trivial zeros of $\zeta(s)$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ derived here: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+1+ \sum _{n=1}^{\infty } \left( {\frac {s-1-2\,n}{{n}^{s}}...
3
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272 views

Alternating sums of the non-trivial zeros of $\zeta(s)$.

It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the same outcome), ...
3
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281 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 $|f(s)|...
2
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261 views

An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known? $$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
2
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96 views

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 ...
2
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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
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67 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: $$\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$$...
2
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0answers
522 views

The simple zero conjecture for the Riemann zeta function

The simple zero conjecture says that all zeros of the Riemann zeta function are simple. Suppose the conjecture is not true. Namely there is an $s$ in the complex plane such that $\zeta(s)=0$ and the ...
2
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0answers
266 views

Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and this questions. Basically got definite integral that experimentally equals $\zeta(s)$ both numerically and symbolically. Closed form for the indefinite integral is known, but I ...
2
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154 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 $\Re(s) \ne \frac12$: $|g(s)_{+...
2
votes
0answers
298 views

Mellin inverse of the Hadamard product rep. of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta(s)\frac{x^{s}}{...
2
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368 views

Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
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0answers
61 views

The Franca-Leclair approximation does not exactly approximate the Riemann zeta zeros but rather the points where only the real part of zeta is zero

Am I right that the Franca-Leclair approximation is a better approximation to the points on the critical line where the real part of the Riemann zeta function is zero and the imaginary part of log ...
1
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0answers
201 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: $W_{n}(z)...
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0answers
250 views

On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$ with $s_0$ a zero of the Riemann Zeta function in the critical strip. This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
1
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0answers
142 views

Simultaneous vanishing of convolutions of Mertens function with itself

By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$) $$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq x}\prod_{p|n}{\...
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0answers
96 views

Conjectured alternate form for vanishing of $\Re\zeta(1/2+it)$ except at zeros

Heavily based on Agno's question. Define: $$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$ Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, ...
1
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0answers
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 $\...
1
vote
0answers
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
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0answers
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 \...
1
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0answers
193 views

Contour plots of Riemann zeta-function

A glimpse of figures in this preprint seems to suggest that curves $\Re{\zeta(s)}=0$ (or $\Im{\zeta(s)}=0$) do not touch each other in the half-plane $\Re{s}>1$. Question: Is there any conditional(...
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0answers
96 views

Do we know a lower bound for the number of critical zeros of the Riemann zeta-function with irrational imaginary part?

If I'm not mistaken, the imaginary parts of the critical zeros of the Riemann Zeta function are conjectured to be linearly independent over $\mathbb{Q}$, but I think we're very far from proving such a ...
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0answers
77 views

Question about the zeros of the sum/difference of two finite Euler products

The conjecture Are all zeros of $\zeta(0+s) \pm \zeta(0-s)$ except a finite few on the line $\Re(s)=0$? was shown to be unconditionally true. The proof can even be extended towards the domain $\...
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0answers
318 views

Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ was derived here: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } {\frac {s-1-2\,n}{{n}^{s}}} + \...
1
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0answers
250 views

Are these valid expansions of the Riemann $\xi(s)$ function in the Hadamard product?

In this post I derived for $s=a + ti$, that assuming the RH, the following should be true: $$\displaystyle \frac{\xi(\frac12 - a + s)}{\xi(\frac12 - a)} = \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \...
1
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0answers
183 views

Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $ 0 \le \sigma \le \frac12$

Based on limited numerical evidence, I suspect this conjecture. Conjecture: Fix $ 0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\...
1
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0answers
287 views

Zeta sum $\sum_{n=2}^\infty \frac{\zeta(n)}{a^n}$

Probably this is known, but mathworld and wolfram alpha don't recognize this potential identities. Numerical evidence suggests: $$ \sum_{n=2}^\infty \frac{\zeta(n)}{a^n} =? \sum_{n=1}^\infty \frac{1}...
1
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0answers
463 views

How ..did Connes get it (trace formula)

i have been reading or at least trying to understand how Connes get the density (approximate) of states $ N(E)= \frac{E}{2\pi}log \frac{E}{2\pi}- \frac{E}{2\pi}+ \frac{7}{8}+ \frac{1}{\pi}arg \zeta(1/...