All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
10
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5
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Riemann–Von Mangoldt formula
Let $$N(T) = \#\{\rho \in \mathbb{C}: \zeta(\rho) = 0,\, \operatorname{Im} \rho \in (0,T]\}$$ denote the number of zeros of $\zeta(s)$, counting multiplicities, with imaginary part lying in the ...
- 4,985
2
votes
1
answer
240
views
On the nearest integer to $\zeta^{(k)}(1-1/B),B \ge 2$
Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$
denote the $k$-th derivative of Riemann zeta function.
For real $x$, let $[x]$ denote the nearest integer to $x$.
Conjecture 1: For all $n \ge 1,[\...
- 25.4k
1
vote
1
answer
138
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On the nearest integer to $\zeta(1-1/B),B \ge 2$
Let $B \ge 2$ be integer and $[x]$ denote the nearest integer
to real $x$.
For $2 \le B \le 10^5$ computations with mpmath suggest:
$$ [\zeta(1-1/B)]=-B+1 \qquad (1)$$
Is (1) true for all $B \ge 2$?
- 25.4k
20
votes
4
answers
1k
views
Bound on $L^2$ norm of $1/\zeta(1+i t)$?
What sort of bounds (explicit of preference) can one give for
$$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$
Some obvious points:
One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \...
- 20.2k
6
votes
1
answer
369
views
Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function
For any fixed $\frac{1}{2} < \sigma < 1$, let
$$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$
It is clear that $\theta > 0$, since we ...
- 81
22
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2
answers
3k
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What are the consequences of an ineffective proof of the Riemann Hypothesis?
Suppose a proof came out (and was verified by credible peer review) of the following statement:
There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$
where $...
- 9,114
8
votes
1
answer
868
views
A question on an equivalence of RH
In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that
$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$
where $\rho$ is ...
- 365
4
votes
0
answers
168
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Is there a relation or formula between the correlations of the nontrivial zeros of the Riemann zeta function and the correlations between high points
Consider an interval of length $(\log T)^{\theta}$ for some fixed $\theta > −1$, around a point $1/2 + i y$ on the critical line where $y\in[T,2T]$ and $T$ is large. How do the correlations between ...
- 219
-1
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1
answer
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Non trivial zeros of Riemann zeta function [closed]
Question Define $f(z)=(s-1)\zeta(s)$ where $s=\frac{1}{1+z^2}$ and $\zeta$ denotes the Riemann zeta function. Prove that if $\rho$ denotes the non trivial zeros of $\zeta(s)$ then, $$\sum_{|\alpha|&...
user292590
1
vote
1
answer
261
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Generalizing closed form representations related to conjectured analytic formulas for $f_a(x)=\sum\limits_{n=1}^x a(n)$
Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$.
$$f_a(x)=\sum\limits_{n=1}^...
- 1,126
28
votes
2
answers
3k
views
What are some consequences of zero free strip of the Riemann zeta function?
A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?
- 3,625
6
votes
0
answers
654
views
Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
- 3,098
6
votes
1
answer
900
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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), ...
- 149
0
votes
0
answers
220
views
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 ...
- 149
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 ...
- 149
0
votes
1
answer
501
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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 $a(n)$.
$$...
- 1,126
0
votes
0
answers
151
views
Abscissa of convergence of transformed Dirichlet series
Let
$$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},$$
where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
- 3,098
0
votes
2
answers
682
views
On integral relating logarithm of absolute value of Zeta function
Sorry for such a direct question:
Consider the following integral:
$$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$
How to find the nature of $I(t)$ as $t\rightarrow\infty$?
- 790
14
votes
1
answer
2k
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Optimality of the Riemann Hypothesis
The Riemann hypothesis is equivalent to the assertion that the prime counting function $\pi(x) := \sum_{p \le x} 1$ deviates from the logarithmic integral $Li(x) = \int_2^x \frac{dx}{\log x}$ in the ...
- 903
5
votes
1
answer
423
views
A generating function for non-trivial zeros of Riemann zeta function
Suppose $0^+_\zeta$ is the set of non-trivial zeros of the Riemann zeta function $\zeta(s)$ which lie on or to the right of the critical line and above the $x$-axis, i.e,
$$0^+_\zeta = \{s \in \mathbb{...
- 51
2
votes
1
answer
561
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}^\...
- 149
2
votes
0
answers
273
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(...
- 1,570
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 ...
- 149
11
votes
4
answers
707
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 ...
- 20.2k
6
votes
2
answers
315
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 ...
- 20.2k
2
votes
0
answers
154
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: ...
- 4,829
1
vote
0
answers
360
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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 ...
- 3,098
3
votes
0
answers
315
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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 ...
- 3,098
1
vote
1
answer
2k
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 $\frac{1}{2}<\Re(s) < 1$.
$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\...
- 3,098
23
votes
1
answer
3k
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More mysteries about the zeros of the Riemann zeta function
Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...
- 3,098
2
votes
0
answers
210
views
Binomial transform of Dirchlet series (2)
Referring to this MO question, i managed to do the following :
We denote by $J(k+1,z)$ the sum :
$$J(n+1,z)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$
and by $S(k+1,z)$ the sum :...
- 181
3
votes
1
answer
436
views
Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
Robin's inequality
$$\sigma_1(n)<e^\gamma n\log\log n$$
at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
- 13.9k
2
votes
0
answers
313
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Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
1
vote
0
answers
253
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Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?
Does anyone know who formulated this conjecture related to Riemann's zeta function?
Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$.
In ...
10
votes
1
answer
731
views
What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?
A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik &...
- 4,829
4
votes
0
answers
170
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Can $ x \sum_{k=1}^{\infty} \frac{1}{k} \Big{(}- \gamma - \psi \big{(}1-\frac{x}{k} \big{)} \Big{)} $ be simplified?
I'm interested in sums of the form $$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{)} .$$
(That is, ...
- 4,829
1
vote
1
answer
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Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?
On p. 286 of Borwein's paper entitled "Computational Strategies for the Riemann zeta function", the author mentions a formula due to Ramaswami: $$(1-2^{1-s})\zeta(s) = \sum_{n=1}^{\infty} \...
- 4,829
4
votes
1
answer
291
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 $...
- 499
0
votes
1
answer
607
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On Soundararajan's explicit formula
I'm reading Soundararajan's https://arxiv.org/pdf/0705.0723.pdf, and on page 5, one has
$$\sum_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(...
- 1,019
5
votes
0
answers
343
views
Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
5
votes
0
answers
161
views
On the asymptotics of some sum involving the Mertens function
Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
- 1,019
3
votes
3
answers
493
views
Show that the ratio of limits converges to the nearest Riemann zeta zero except when the ratio is a singularity
Let $h(s,n)$ be:
$$h(s,n)=\lim_{c\to 1} \, \frac{(-1)^{n-2}}{(n-2)!}\zeta (c)^{n-2} \sum _{k=1}^{n-1} \frac{(-1)^{k-1} \binom{n-2}{k-1}}{\zeta ((c-1) (k-1)+s)}$$
and let $g(s,n)$ be:
$$g(s,n)=\lim_{c\...
- 1,183
7
votes
2
answers
788
views
Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$
Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
- 1,019
6
votes
0
answers
177
views
Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions
Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
- 1,243
7
votes
1
answer
811
views
Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]
There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
4
votes
1
answer
925
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A question on the use of fractional derivatives in Riemann Hypothesis
We already know that Riemann-zeta function on the critical band is defined as follows:
$$(1-2^{1-\alpha})\zeta(\alpha) = \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha},\quad \Re(\alpha) \in ]0, 1[ $$
Is ...
- 71
3
votes
0
answers
219
views
What is known about products of zeta values?
A couple of years ago, I asked this MSE question on the evaluation of the product of even zeta values: $$ \prod_{n=1}^\infty \zeta(2n) \approx 1.82 \quad .$$ While it can be shown that the product ...
- 4,829
4
votes
1
answer
247
views
Are there any extensive treatments on rational zeta series?
I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
- 4,829
3
votes
1
answer
242
views
Efficient boxing for a mean value in the Bombieri Iwaniec method
One of the nice applications of decoupling is Bourgain’s record towards Lindelöf:
https://arxiv.org/pdf/1408.5794.pdf
Wooley has developed some techniques known as efficient congruencing which allow ...
user156885
-5
votes
1
answer
561
views
On the Absolute Value of the Riemann Zeta Function on the Critical Line [closed]
Is there any sort of (closed form preferably, though if not, it's fine) function for $|\zeta(\frac12+it)|$ where $\zeta$ is the Riemann zeta function? Anything is welcome, so I can take it from there. ...
- 265