# 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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This is a reference question: Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that $$\int_2^x (\psi(y)... 1 vote 1 answer 119 views ### Best possible unconditional partial sum estimate of \sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}: Consider the following partial sum:$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$Here p runs through primes and n is constant What is the best possible unconditional( using best known version ... 5 votes 0 answers 251 views ### What is the winding behavior of the Riemann zeta function around zero along the line s=1+it? Let \phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C be defined by$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$(the nonvavishing of the denominator being a bit weaker than the prime number ... 3 votes 1 answer 210 views ### What is the behavior of the argument of Riemann’s Zeta function on other verticals inside the critical strip, apart from the critical line? What is known about the behavior of the argument of Riemann’s Zeta function on other verticals inside the critical strip apart from the critical line ? Are there any omega type theorems in this case, ... 4 votes 2 answers 607 views ### How can one deduce an approximation for the density function of prime numbers from this Euler's theorem? The author of Riemann's Zeta Function, H.M.Edwards, says: According to Euler, \sum_{p<x}\frac{1}{p}\sim \log(\log(x)) when x\longrightarrow\infty. \log(\log(x))=\int_{1}^{\log(x)} \frac{du}{... -4 votes 1 answer 343 views ### What is the proof for any non trivial zero? [closed] There are many known nontrivial zeros of the Riemann Zeta function, but I have never seen proof that any of them actually resolve to zero. The trivial zeros make sense because there is a more ... 0 votes 0 answers 98 views ### Prime races in two competing arithmetic progressions - error bound I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: (p_{1,n}) and (p_{3,n}... 4 votes 1 answer 262 views ### Zeros of Dirichlet function L(s,\chi_4) I am wondering if there are some know results for the non-trivial roots at {\rm Re}(s) = \frac{1}{2}, even maybe a table of the first few roots with t>0. This sister function$$ L_4^* (s,\chi_4)...
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 $I$ ...