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.

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What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?

Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
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Mertens function via Perron's formula without assuming the simplicity of the Riemann zeros

Let $\mu$ denote the Möbius function, and define the the Mertens function $M(x) = \sum_{n \leq x} \mu(n)$. By Person's formula, one can express $M(x)$ as a sum over the nontrivial zeros of the ...
user257465's user avatar
8 votes
2 answers
2k views

Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis?

Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple. I have often heard of the statement that the SZC is stronger than the Riemann ...
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0 votes
1 answer
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Questions on analytic representations of the Kronecker delta function $\delta(x-1)$ and the Moebius function $\mu(n)$

This question is related to analytic formulas for $a(n)$ where $f_a(x)$ and $F_a(s)$ defined in formulas (1) and (2) below are the summatory function and Dirichlet series associated with $a(n)$. $$...
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Are there any papers about this observation of the distribution of the zeros of the zeta function?

Choose some $x > 1$. Then $$ \lim_{T\to\infty} \sum_{\Im(\rho)<T}\cos(\ln(x)\Im(\rho))=-\infty $$ where $\rho$ ranges over all zeros of the zeta function iff $x$ is prime or the power of some ...
Mr.Mustache Man's user avatar
7 votes
0 answers
243 views

Computability assertions for Riemann zeta zeros

While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this ...
Pace Nielsen's user avatar
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0 answers
143 views

Abscissa of convergence of transformed Dirichlet series

Let $$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$ where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
Vincent Granville's user avatar
2 votes
1 answer
612 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)+\...
Vincent Granville's user avatar
2 votes
0 answers
171 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 ...
Uzu Lim's user avatar
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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 $w=1$. ...
Vincent Granville's user avatar
-4 votes
1 answer
390 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 $...
Vincent Granville's user avatar
2 votes
1 answer
435 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,...
Vincent Granville's user avatar
0 votes
2 answers
479 views

On integral relating logarithm 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 $t\rightarrow\infty$?
TPC's user avatar
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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 ...
Uzu Lim's user avatar
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5 votes
1 answer
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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{...
00...'s user avatar
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4 votes
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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<...
Shree's user avatar
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2 votes
1 answer
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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}^\...
Zaza's user avatar
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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 ...
Wane's user avatar
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2 votes
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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(...
davidlowryduda's user avatar
6 votes
2 answers
1k views

On modified Euler product

Consider the modified Euler product as follows: $$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$ Here $c$ is a constant My questions are Is there a compact representation for this ...
Zaza's user avatar
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9 votes
3 answers
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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 ...
H A Helfgott's user avatar
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6 votes
2 answers
300 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 ...
H A Helfgott's user avatar
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1 vote
1 answer
286 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)^{...
Agno's user avatar
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2 votes
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135 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^\...
H A Helfgott's user avatar
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5 votes
1 answer
642 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 ...
geocalc33's user avatar
2 votes
0 answers
152 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: ...
Max Muller's user avatar
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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 ...
Vincent Granville's user avatar
1 vote
1 answer
458 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{...
Agno's user avatar
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3 votes
0 answers
289 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 ...
Vincent Granville's user avatar
4 votes
1 answer
160 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_{...
Agno's user avatar
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0 votes
1 answer
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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 ...
Vincent Granville's user avatar
1 vote
1 answer
1k 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\...
Vincent Granville's user avatar
29 votes
1 answer
2k 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 ...
Mustafa Said's user avatar
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23 votes
1 answer
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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 ...
Vincent Granville's user avatar
2 votes
0 answers
209 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 :...
mohammad-83's user avatar
3 votes
1 answer
428 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 ...
Turbo's user avatar
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8 votes
0 answers
271 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 ...
user avatar
2 votes
0 answers
289 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 $s_{0}$ ...
Juu's user avatar
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6 votes
0 answers
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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 ...
Rachid Atmai's user avatar
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2 votes
4 answers
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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\lim_{n \rightarrow \infty} \left( \left[ 1- \left( \sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{...
Mats Granvik's user avatar
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1 vote
0 answers
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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 ...
Cristian Dumitrescu's user avatar
7 votes
0 answers
225 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 $\...
Agno's user avatar
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2 votes
0 answers
230 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 ($\...
Agno's user avatar
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10 votes
1 answer
706 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 &...
Max Muller's user avatar
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4 votes
0 answers
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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, ...
Max Muller's user avatar
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17 votes
2 answers
2k 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 $\mathbb{C}$?...
Richard Stanley's user avatar
1 vote
1 answer
159 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} \...
Max Muller's user avatar
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4 votes
1 answer
278 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 $...
Farzad Aryan's user avatar
9 votes
2 answers
890 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 ...
Vincent Granville's user avatar
0 votes
1 answer
575 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}(...
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