Questions tagged [riemann-zeta-function]

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 relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.

Filter by
Sorted by
Tagged with
2
votes
1answer
274 views

Analytic continuation and convergence of a Riemann zeta related function

The functions in question are $$L(s)=\sum_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^*(s)=\frac{1}{2}\sum_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\...
2
votes
0answers
116 views

Special zeta value and zeroes

Are there known relationships linking special values of the Riemann zeta function or MZV (multiple zeta values, i.e. $\zeta(n_1, \cdots n_k)$ with $n_i \in \mathbb Z^+$) to the nontrivial zeroes of ...
1
vote
0answers
161 views

Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
-4
votes
1answer
305 views

Scaled Riemann zeta function with no zero in the critical strip

Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues. Prime numbers are denoted as $...
2
votes
1answer
137 views

Truncated Euler products, Dirichlet eta function, and convergence issues

Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as $$W(\sigma,...
0
votes
2answers
106 views

On integral relating logarithmic of absolute value of Zeta function:

Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?
9
votes
1answer
1k views

Optimality of the Riemann Hypothesis

The Riemann hypothesis is equivalent to the assertion that the prime counting function $\pi(x) := \sum_{p \le x} 1$ deviates from the logarithmic integral $Li(x) = \int_2^x \frac{dx}{\log x}$ in the ...
5
votes
1answer
299 views

A generating function for non-trivial zeros of Riemann zeta function

Suppose $0^+_\zeta$ is the set of non-trivial zeros of the Riemann zeta function $\zeta(s)$ which lie on or to the right of the critical line and above the $x$-axis, i.e, $$0^+_\zeta = \{s \in \mathbb{...
1
vote
0answers
120 views

$\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $

Consider,$$I=\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $$ where $\zeta$ is the Riemann Zeta function. Since , Hardy (...
4
votes
0answers
189 views

Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$

Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function. The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<...
3
votes
1answer
313 views

On infinite sum containing logarithmic derivative of Zeta function and Möbius function:

Consider the following function: $$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$ Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
9
votes
0answers
215 views

Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag*{(4)}$$ with $\operatorname{Ri}$ denoting the Riemann function ...
2
votes
0answers
105 views

Applications of Jensen's Formula to entire functions of finite order

I am trying to understand a frequently omitted technical detail in applications of Jensen's Formula to bound the number of zeros of entire functions of finite order. We say that an entire function $f(...
8
votes
2answers
860 views

On modified Euler product

I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it. Consider the modified Euler product as ...
6
votes
2answers
301 views

Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?

Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
7
votes
2answers
229 views

Functional equation and/or growth estimates for a shifted L function

Consider the $L$-series defined by $$L_{\alpha,\chi}(s) = \sum_{n\geq 1} \frac{e^{2\pi i \alpha \Omega(n)} \chi(n)}{n^s} = \prod_p \left(1 - \frac{e^{2\pi i \alpha} \chi(p)}{p^s}\right)^{-1}.$$ It ...
1
vote
1answer
147 views

Continuing an analytic continuation of the Dirichlet $\eta$-function?

The Dirichlet $\eta$-function is defined as: $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$ and has the full analytical continuation: $$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...
0
votes
0answers
598 views

Trying to evaluate an integral relating to $\zeta (3)$

So similarly to my search for $\zeta (3)$ over at the mathematics stack exchange, I have continued to attempt to work towards a closed-form for it. The following integral is related to a search of ...
2
votes
0answers
117 views

Mean values of $\zeta(s)$ for $\Re(s)=1/2$ vs $\Re(s)\ne 1/2$

Say I have a good estimate for the $L^2$ mean of the Riemann zeta function $\zeta(s)$ for $\Re s = 1/2$, $|t|\leq T$: $$\int_0^T |\zeta(1/2+i t)|^2 = T \log T - T (1 + \log 2 \pi - 2\gamma) + O(T^\...
4
votes
0answers
341 views

Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$

About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$ What's the maximal analytic continuation of $\varphi(s)?$ Doing this will help me better understand how ...
0
votes
0answers
124 views

A *natural* polynomial expansion of the Riemann $\xi(s)$ function

This is something that I've known for some time. Its an expansion of Riemann's $\xi$ function based on the traditional representation $$ \xi(s)=\frac{1}{2}\left(1-s(1-s)\int_{1}^{\infty}\frac{\psi(x)}{...
2
votes
0answers
137 views

How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?

It has been discovered long ago that $$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: ...
0
votes
0answers
307 views

Incredibly accurate recursions for the Riemann Zeta function

Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here. During some ...
1
vote
1answer
263 views

Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers

When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used: $$n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...
3
votes
0answers
137 views

Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$

Let $s=\sigma + it$, with $0.6 < \sigma < 1$ and $\sigma=\Re(s)$. I am trying to get good enough approximations for $\eta(s)$, hoping something useful might come out of it. I stumbled upon a ...
3
votes
1answer
110 views

A Hadamard product representation for Keiper's $\tau$-function?

In this paper J.B. Keiper defined the following function: $$\tau_k = \sum_{j=1}^k (-1)^j\,{k-1 \choose j-1} \sigma_{j+1} \qquad k \ge 1, k \in \mathbb{N} \tag{1}$$ where $\displaystyle\sigma_r = \sum_{...
0
votes
0answers
102 views

Some properties of special Dirichlet series, connection to Riemann Hypothesis

The series in question is $$\phi(\sigma,\mu,t; \alpha,\beta,\gamma) = \sum_{n=2}^\infty (-1)^{n+1} \cdot \frac{W\big(\gamma +\alpha t+\beta t\lambda(n)\big)}{n^\sigma (\log n)^\mu}$$ The wave $W$ is a ...
1
vote
1answer
213 views

On some property of the zeros of $\zeta(s)$ in the complex plane

This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
1
vote
1answer
274 views

About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$

The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$. $$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\...
27
votes
1answer
1k views

Riemann's attempts to prove RH

I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I ...
16
votes
1answer
1k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
0
votes
0answers
79 views

Analytic continuation of the Euler product of odd primes to $s=1$?

From this question, the following expression emerges that is valid for $n \in \mathbb{N}, n \gt 1$: $$\zeta(n) = \frac{\sigma^*_n - \sigma_n + 1}{1-2^{-n}} \tag{1}$$ or equivalently with $p$ = a prime:...
2
votes
0answers
201 views

Binomial transform of Dirchlet series (2)

Referring to this MO question, i managed to do the following : We denote by $J(k+1,z)$ the sum : $$J(n+1,z)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$ and by $S(k+1,z)$ the sum :...
2
votes
1answer
305 views

Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

Robin's inequality $$\sigma_1(n)<e^\gamma n\log\log n$$ at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
7
votes
0answers
240 views

Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note: This question has been brought here from MSE. I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had ...
2
votes
0answers
232 views

Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem

Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$? If so: Let $s_{0}$ ...
6
votes
0answers
508 views

Value of $\zeta(3/2)$?

Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be ...
2
votes
2answers
611 views

Prove that the real part of this limit converges to $\frac{1}{2}$

Let: $s= 1/3 + 14i$ Prove that the real part of this limit converges to $\frac{1}{2}$: $$\Re\left(\lim_{n \rightarrow \infty}\left(\frac{1}{1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\...
1
vote
0answers
231 views

Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?

Does anyone know who formulated this conjecture related to Riemann's zeta function? Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$. In ...
0
votes
0answers
77 views

Is the set of subsequential limits of the normalized gaps between critical zeroes of zeta stable under $s\mapsto 1/s$?

Let $g(n):=\frac{(\gamma_{n+1}-\gamma_{n})\log n}{2\pi}$ the normalized gap between consecutive critical zeroes of the Riemann zeta function written as $1/2+i\gamma_{n}$. Denoting by $\mu_{0}:=\lim\...
7
votes
0answers
203 views

Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?

Lehmer 1988 and Keiper 1992 made major progress on evaluating the series: $$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$ where $\...
2
votes
0answers
212 views

Could analytically deriving the next non-trivial zero of $\zeta(s)$ be made rigorous up to a fixed accuracy?

In this question., a very inefficient, yet rigorous analytic approach for finding the next prime was established. I wondered whether a similar approach could exist to find the next non-trivial zero ($\...
8
votes
1answer
402 views

What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?

A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$ Many more identities can be found in articles by e.g. Borwein and Adamchik &...
4
votes
0answers
154 views

Can $ x \sum_{k=1}^{\infty} \frac{1}{k} \Big{(}- \gamma - \psi \big{(}1-\frac{x}{k} \big{)} \Big{)} $ be simplified?

I'm interested in sums of the form $$f_{p} (x) = \sum_{k=2}^{\infty} \zeta(k)^{p} x^{k} .$$ For $p=1$, the following result is known: $$f_{1} (x) = -x \big{(}\psi(1-x) + \gamma \big{)} .$$ (That is, ...
17
votes
2answers
1k views

Algebraic independence of shifts of the Riemann zeta function

Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{ \zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\, \colon\, z\in\mathbb{C}\}$, algebraically independent over $\mathbb{C}$?...
1
vote
1answer
132 views

Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?

On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...
3
votes
1answer
191 views

Generalization of the The Liouville Lambda function

Let $n=p^{\alpha_1}_1 \cdots p^{\alpha_m}_m,$ and define $$\lambda_k(n)= (-1)^{ [\frac{\Omega(n)}{k} ]},$$ where $\Omega(n)= \alpha_1 + \cdots + \alpha_k,$ and $[\cdot]$ is the floor function. For $...
8
votes
2answers
688 views

Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula

I was trying to get some interesting result for $\zeta(3)$, exploring the following function: $$W(a) = \sum_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim_{a\rightarrow 0} W(a) = \zeta(3).$$ Let ...
-1
votes
1answer
440 views

On Soundararajan's explicit formula

I'm reading Soundararajan's https://arxiv.org/pdf/0705.0723.pdf, and on page 5, one has $$\sum_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(...
4
votes
0answers
196 views

Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?

In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...

1
2 3 4 5
10