# 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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### Spiralling cycles surrounding zeros

The following came up, as a vague idea, in dialogue with a bright, female, 20 year old student of mine. It is a bit vague, but it seems that conjecture 1 is not present in the literature, which seems ...
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### Prime number theorem via the explicit formula

Can the prime number theorem be obtained from the explicit formula, $\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$? Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
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### Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?

The well-known integral expression for the entire function: $$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$ ...
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### On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:

Consider the analytic function $g(x)$ Now define $f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$ Such that $|f(x+it)|=o(e^{2πt})$ uniformly for every $x$...
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### On the upper bound for $|\zeta(s)|$ near the zeta zeros

Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho_T| \approx T$ and let $\varepsilon_T \approx \frac{\log \log T}{\log T}$. Is it ...
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### Zeros of the derivative of $\xi$

In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
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### Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.$$ Is there a general formula that ...
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### Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?

Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
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### On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
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### What heuristic arguments support Montgomery's conjecture for primes in short intervals?

I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
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### proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}.$$ When this ...
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### Asymptotics of the Liouville sum at the primes

Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
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### Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?

There are two proofs of $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$ which I'm aware of. I'll call the first one the Sieve proof and the second one ...
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### Derivative of the Riemann zeta function at $z=-2$

I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
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### Reference for explicit formula used by Ramanujan

In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
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### Derivative of zeta at positive even integers

Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
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### Large values of $\zeta(1/2+it)$ from sums of short moments

In a now classical paper, Iwaniec proved the following theorem. Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
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### Can we tweak the Möbius function sum to better converge on the critical line and maybe also to the left of it?

Let the constant $c = -3/4$ and let the usual divisibility matrix $B(n,k)=1$ if $k\mid n$ else $B(n,k)=0$ for all integers $n \geq 1$ and $k \geq 1$ and let the matrix $A$ be: $$A=B-I(1+c)$$ where $I$ ...
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