## plotting – Problem with a finite sum involving HurwitzZeta Function

I’m trying to reproduce some plots from the analytical expression:

$$f(xi)=left(frac{2}{beta^2}-1right)+left(frac{theta,e^{-beta}}{2}-frac{1}{2}right)xi+sum_{n=1}^{60}left(frac{(-beta)^n}{n!}xi^{1+n/2}zeta_Hleft(-frac{n}{2},1+frac{1}{xi}right)right)$$.

I need to plot $$f(-4x)$$ for several values of $$beta$$ and with $$theta=0$$ and $$theta=1$$. In the attached picture, the left panel is what I have to get (ignore the blue curves), but Mathematica gives me a lot of noise.

My code is:

``````f((Beta)_, (Xi)_,M_, (Theta)_) := ((2/(Beta)^2 -1) + (((Theta) Exp(-(Beta)))/2 - 1/2) (Xi) + Sum(((-(Beta))^n (Xi)^(1 + n/2))/n! HurwitzZeta(-(n/2), 1 + 1/(Xi)), {n, 1, M}));

Plot({f(0.5, -4 x, 60, 1), f(0.5, 4 x, 60, 0),
f(0.8, -4 x, 60, 1), f(0.8, 4 x, 60, 0), f(1.1, -4 x, 60, 1),
f(1.1, 4 x, 60, 0)}, {x, -6, 6}, AxesOrigin -> {0, 0},
PlotStyle -> {Green, Green, Red, Red, Black, Black},
PlotRange -> {0, 8})
``````

Do you have an idea of what is going on?

Thanks!

## plotting – Can this 2D-Venn diagram code–involving Disk–be “upgraded” to a 3D one, involving Sphere?

This is only a start, but is this within your capabilities/understanding?

``````a={1,0,-1/Sqrt(2)};b={-1,0,-1/Sqrt(2)};c={0,1,1/Sqrt(2)};d={0,-1,1/Sqrt(2)};
Graphics3D({Opacity(1/2),
Sphere(a,3/2),Sphere(b,3/2),Sphere(c,3/2),Sphere(d,3/2),
Text("A",a),Text("B",b),Text("C",c),Text("D",d),
Text("AB",(a+b)/2),Text("AC",(a+c)/2),
Text("BD",(b+d)/2),Text("CD",(c+d)/2),
Text("ABC",(a+b+c)/3),Text("ABD",(a+b+d)/3),
Text("ACD",(a+c+d)/3),Text("BCD",(b+c+d)/3),
Text("ABCD",(a+b+c+d)/4)
})
``````

Can you see how that was done? Can you see how each part might work?
Can you adjust the Opacity and size and Text size to make that better?
Can anyone else suggest why dragging the box with the mouse doesn’t
seem to correctly show the Text sometimes?

There are a LOT of labels to be able to distinguish in one diagram, but perhaps you can learn a bit from this and make some progress and be better prepared to do other things in the future.

## calculus and analysis – How to get the output for the following code involving differentiation and integration?

I need to evaluate the following Mathematica code to get a numerical value as my output. But during compilation I get the error message

NumericalMath`FixedPrecisionEvaluate::precbd: Requested precision (Infinity) is not a machine-sized real number between $$MinPrecision and$$MaxPrecision.

This is my coding

``````ee = Integrate((x - y)^(-((Alpha) - 1) -
1) ((y^(3))*(Sum(((Exp(I*1))*(1 -
Exp(y *I*((Lambda))^s) ))/(Lambda)^(s*(2 - d)), {s, -20,
20}))), {y, 0, x});
ew = D(ee / Gamma(-((Alpha) - 1)), {x, 1}) /. x -> 1 /. (Lambda) ->
3 /. d -> 1.5 /. (Alpha) -> 0.6
``````

## nt.number theory – A conjecture involving \$P_n=prod_{k=1}^np_k\$

For each positive integer $$n$$ let $$P_n=prod_{k=1}^n p_k$$, where $$p_k$$ is the $$k$$th prime.

Question. Is my following conjecture true?

Conjecture. For any integer $$n>1$$, there are $$k,min{1,ldots,n-1}$$ such that
$$P_nequiv P_kpmod n text{and} P_nequiv -P_mpmod n.$$

For example, $$P_{32}equiv P_{23}pmod{32}$$ and $$P_{32}equiv -P_8pmod{32}$$.

I have verified the conjecture for all $$n=2,3,ldots,70000$$. When $$n$$ is squarefree, the conjecture holds trivially. I’m unable to prove the conjecture fully.

For the motivation of the conjecture, one may look at Conjecture 1.5 and Remark 1.7 in my paper available from http://dx.doi.org/10.1016/j.jnt.2013.02.003.

## combinatorial – Need help formulating a problem for planning events involving multiple clients and class types

I'm having trouble understanding what should be a simple problem!

I run training courses. A class "B", a class "C" and an advanced class "Adv".

Classes B take place every Saturday, last 3 hours and have a maximum of 4 clients.
Classes C follow classes B and last 2 hours. There are 3, sequentially, one per week (although this frequency may be increased). Max 4 customers.
Advanced courses follow courses C and take place every 6 weeks, take 2 hours. There are 8, 1 course every 6 weeks. Max 4 customers.

The time available on Saturdays for lessons is 8 hours. Could go up to 10 hours maximum, but ideally no more than 8 hours.
Classes C take place ideally on Saturday, but could also take place on Monday (maximum time allocated 4 hours).
Advanced courses can take place on Mondays, Fridays (maximum duration 4 hours) and Saturdays.

I'm trying to find an optimal use of time that gets the most customers from B to C to Adv.

Any help to at least formulate the above as a specific problem would be greatly appreciated!
Thank you
David

## ruby – How should I approach creating a custom form involving different template attributes?

I'm pretty new to Ruby, so I need advice on how to approach this. I know how to create very simple forms, but I am thinking of implementing a custom form that can generate pay for a business on a specific start and end date. I already have business and payroll templates, but I don't know how to implement this type of custom form and how to route it properly to make it fully functional. Are there any resources or tips I can get to help me on this?

## Do I need a Grad filter with ND for long exposures involving the sky?

If I was on a beach taking a photo of the sunset and trying to get a long exposure but the sky was too bright.

If I put an ND filter, it would darken the sky and the foreground, which would mean that I would have to bring the foreground to post-processing to illuminate it?

Do I need a Grad ND filter for long exposure when the sky is in to equalize the exposure between the sky and the foreground?

I have seen a lot of people using ND diplomas for long exposures and the foreground is not dark and they have not used any ND grads.

## calculability – decidability of equality of expressions involving exponentiation

Let's have expressions which are trees of finite size, with elements of $$mathbb N$$ like leaf nodes and operations {$$+, times, -, /$$, ^} with their usual semantics as internal nodes, with the special note that we allow arbitrary expressions like the right side of the exponentiation operation. Is equality between such nodes (or, equivalently, a comparison with zero) decidable? Is the closure of these operations a subset of algebraic numbers or not? (It turns out that it is not: https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem, which prohibits an easy reduction to the solutions concerning algebraic numbers.)

This question is similar to this one: decidability of the equality of the radical expressions but with the difference that here the operator of exponentiation is symmetric in the type of the base and the & # 39; 39; exhibitor. This means that we could have exhibitors such as $$3 ^ sqrt 2$$. It is not clear to me, whether allowing exponentiation with irrationals keeps the algebraic closure.

This question is also similar to the computability of zero equality for simple language, but the answers to this question focused on the transcendental properties of combinations of $$pi$$ and $$e$$, which I consider out of reach here.

## analytical number theory – Contour integration involving the Zeta function

I am trying to calculate the integral of the contour
$$frac {1} {2 pi i} int_ {c – i infty} ^ {c + i infty} zeta ^ 2 ( omega) frac {8 ^ omega} { omega} omega$$
or $$c> 1$$, $$zeta (s)$$ is Riemann's zeta function.

Use Perron's formula and define $$D (x) = sum_ {k leq x} sigma_0 (n)$$, or $$sigma_0$$ is the usual function of counting the divisors, we can show that
$$D (x) = frac {1} {2 pi i} int_ {c – i infty} ^ {c + i infty} zeta ^ 2 ( omega) frac {x ^ omega } { omega} d omega.$$
So for this purpose, we can just calculate $$D (8)$$ and call it a day. However, for my own needs, I want to redefine $$D (x)$$ by the full above instead. Therefore, why I state the problem for a specific case $$x = 8$$, for example.

Considering a modified Bromwich contour which avoids branch cutting and $$z = 0$$, let's call it $$mathcal {B}$$, we can apply the Cauchy residue theorem:
$$oint _ { mathcal {B}} zeta ^ 2 ( omega) frac {8 ^ omega} { omega} d omega = 2 pi i operatorname * {Res} ( zeta ^ 2 ( omega) frac {8 ^ { omega}} { omega}; 1) = 8 (-1 + 2 gamma + ln 8)$$
or $$gamma$$ is the Euler-Mascheroni constant. I got this by extending $$zeta ^ 2 ( omega) frac {8 ^ omega} { omega}$$ in his Laurent series. To obtain the desired integral, it would then be necessary to subtract from this value the parts of the contour which are not the vertical line $$c – iR$$ at $$c + iR$$, subtract them from the residual value obtained, then take the limit as $$R to infty$$ and $$r to 0$$ or $$C_r$$ is the circle of radius $$r$$ where the $$mathcal {B}$$ dodges the origin.
Feel free to modify this outline in any shape or form, or consider a positive integer value different from $$x$$.