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.
743 questions
3 votes
0 answers
120 views
What do we know about $S(t+1)-S(t)$?
Let $S(t)$ denote $\frac{1}{\pi} \arg \zeta\left(\frac{1}{2} + i t\right)$, as usual. Do we have unconditional pointwise (i.e., not average) estimates on $S(t+1)-S(t)$ better than the ones we get from ...
- 22k
0 votes
0 answers
292 views
Apéry series for $\zeta(3)$
I am still now stumped on deriving the series equivalence $$\zeta(3)=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$ Like even I did not get the series for $\frac1{n^2}$ mentioned ...
- 2,135
6 votes
1 answer
715 views
Optimal bound on the growth of Riemann zeta along the critical line
According to the Lindelöf hypothesis, $|\zeta(\frac{1}{2}+it)|=O(t^\epsilon)$ for all $\epsilon>0$ as $t\rightarrow\infty$. I am wondering if there is any expectation for the optimal asymptotic ...
- 315
0 votes
0 answers
146 views
Harmonic analysis on the non-trivial zeros of the Riemann zeta function?
Suppose I have some function $f(x)$ that satisfies constraints roughly as restrictive as those for Fourier series expansions, and I'm interested in alternative ways of expanding it between some bounds ...
- 635
0 votes
0 answers
95 views
Upper bound for maximum of reciprocal of zeta
The following appears in this paper: Lemma Let $H=T^{1/3}$. Then we have $$\min_{T\le t\le T+H}\max_{1/2\le\sigma\le 2}\frac{1}{|\zeta(\sigma+it)|}<\exp(C(\log\log T)^2)$$ where $C$ is an absolute ...
user574181
4 votes
0 answers
242 views
SZC implies explicit formula for Mertens function
To my understanding there is currently no proof of SZC (Simple Zeros Conjecture) implies $$M_0(x)=\lim_{T\to\infty}\sum_{\zeta(\rho)=0\text{ and }\vert\Im(\rho)\vert\leq T}\frac{x^\rho}{\rho\zeta'(\...
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2 votes
1 answer
179 views
Explicit convexity bound for Dirichlet $L$-functions
The following is a well-known convexity bound for Dirichlet $L$-functions. Theorem Let $\chi$ be a primitive character to the modulus $q$. Then, for any fixed $\varepsilon>0$ and any $k\in\mathbb{...
user574181
1 vote
1 answer
353 views
Statistical meaning of zeros of zeta functions
To simplify the discussion, I consider the Riemann zeta function but my question should make sense for most zeta functions (Selberg, algebraic varieties over finite fields, ...). Let $\zeta(s) := \...
- 3,108
3 votes
0 answers
355 views
A new generalization of Euler product formula?
I show below a formula that I've derived recently from the well-known Euler product formula, which could be considered as a generalization of it. Let's start with a definition. For any non-empty set ...
- 153
0 votes
1 answer
187 views
Is phase $S(T)$ of Riemann Zeta function jumping maximum by one for small increase of $T$?
The number of non-trivial zeros of the $\zeta$ function is strongly coupled to the hypothetical number of zeros outside of the critical line that are counter-examples for the Riemann Hypothesis. Hence,...
2 votes
1 answer
172 views
Zero density estimates for zeta translated to Dirichlet $L$-functions
In this recent paper of Bellotti the author gives a new zero-density estimate for $\zeta(s)$ close to the boundary of the Vinogradov-Korobov region. It is natural to ask if this generalises to ...
user574181
3 votes
1 answer
341 views
Barriers to a fixed-width zero-free region for zeta
The classical zero-free region of the Riemann zeta function $\zeta(s)$ says there is a constant $A>0$ such that there are no zeta zeros $$\rho=\sigma+iT$$ with $\sigma>1-\frac{A}{\log T}$. ...
user574181
0 votes
0 answers
101 views
Why is it hard to get a zero-free half-plane $\Re s>\tfrac12$ for $\zeta(s)$ via Rouché using the Maclaurin (Euler–Maclaurin) formula?
I start from the classical “Maclaurin” form of Euler–Maclaurin for every integer $n\ge2$: $$ \zeta(s) =\frac{1}{s-1}+\frac12+\sum_{k=2}^{n} B_k\,\frac{s(s+1)\cdots(s+k-2)}{k!} -\frac{s(s+1)\cdots(s+n-...
- 471
8 votes
1 answer
558 views
Sign of Laurent coefficients of $-\zeta'(s)/\zeta(s)$ at $s=1$
Let $\zeta(s)$ be the Riemann zeta function. Write $-\zeta'(s)/\zeta(s)-1/(s-1) = \sum_{n=0}^\infty a_n (s-1)^n$. It would seem that this is an alternating sum: $$a_0<0, \;\;\;a_1>0, \;\;\;a_2&...
- 22k
0 votes
1 answer
213 views
Average of $\Lambda(n)^2$
Let $\Lambda$ be the von Mangoldt function. I am interested in understanding the average $$\sum_{n=1}^x \Lambda(n)^2.$$ By partial summation and the prime number theorem one can prove that this is $$ ...
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