## Less basic applications of Zeta regularization:

As we all know, zeta regularization is used in quantum field theory and computation regarding the Casimir effect.

Are there less fundamental applications of regularization? By less fundamental, I mean
It appears "naturally" in more than one artificially / purely mathematically ideal constructed scenario.

Thank you!

## nt.number theory – Does this series, linked to the Hasse / Ser series for \$ zeta (s) \$, converge for all \$ s in mathbb {C} \$?

I asked this question at the math stack swap, but it didn't get traction. Always curious to know the answer.

Numerical evidence suggests that:

$$lim_ {N to + infty} sum_ {n = 1} ^ N frac {1} {n} sum_ {k = 0} ^ n (-1) ^ k {n choose k } frac {1} {(k + 1) ^ {s}} = s$$

or equivalent

$$lim_ {N to + infty} H left (N right) + sum _ {n = 1} ^ {N} left ({ frac {1} {n} sum _ {k = 1} ^ {n} { left (-1 right) ^ {k} {n choose k} frac {1 } { left (k + 1 right) ^ {s}}}} right) = s$$

with $$H (N)$$ = the $$N$$-th Harmonic Number.

Convergence is quite slow, but clearly goes faster for negatives $$s$$. In addition, calculations for non-integer values ​​of $$s$$ require high precision parameters (I used Maple, bet / gp and ARB).

However, according to Mathematica, the series diverges according to the "harmonic series test", although $$s$$ as an integer, it agrees on convergence.

Does this series converge for $$s in mathbb {C}$$ ?

Some numerical results below:

``````s=0.5
0.497702121, N = 100
0.499804053, N = 1000
0.499905919, N = 2000

s=-3.1415926535897932385
-3.14160222, N = 100
-3.14159284, N = 1000
-3.14159272, N = 2000

s=2.3-2.1i
2.45310498 - 1.94063637i, N = 100
2.33501943 - 2.09308517i, N = 1000
2.31996958 - 2.09923503i, N = 2000
``````

## referral request – Double sum for \$ zeta (3) \$ and \$ zeta (5) \$

I found the following double sum representations for $$zeta (3)$$ and $$zeta (5)$$

$$zeta (3) = frac {1} {2} sum_ {i, j geq 1} frac { beta (i, j)} {ij}$$

$$zeta (5) = frac {1} {4} sum_ {i, j geq 1} frac {H_ {i} H_ {j} , beta (i, j)} {ij}$$

or $$beta ( cdot, cdot)$$ represents the beta function, and $$H_ {i}$$ represents the $$i$$e harmonic number.

Are these results known in the literature? If yes, please provide some references / evidence for the same.

## Asymptotic of the Hurwitz zeta function

Can anyone please help me with a reference to Hurwitz asymptotics (or just the upper limits) $$zeta (s, z)$$ as $$| t | rightarrow infty$$ with $$Re (z)> 0,$$ $$s = sigma + i t$$ and $$sigma <0$$? I only found limits for the real z

## nt.number theory – Doubt on the proof of the irrationality of \$ zeta (3) \$

I read this article by Henri Cohen on the proof of the irrationality of Apery $$zeta (3)$$ but I don't really have the details of "THEOREM 1".
My first doubt concerns the relationship $$a_n sim A alpha ^ n n ^ {- 3/2}$$.

I know if $$a_n$$ filled the relationship $$a_n-34a_ {n-1} + 1 = 0$$ then as its characteristic polynomial is $$x ^ 2-34x + 1$$ and like $$alpha$$ is one of its roots, if we note by $$bar { alpha}$$ the second root, then we would have $$a_n = A_1 alpha ^ n + A_2 bar { alpha} ^ n$$.

Then, like $$0 < bar { alpha} <1$$ we have that $$a_n / alpha ^ n longrightarrow A_1$$.

However, the relationship for $$a_n$$ East
$$a_n- (34-51n ^ {1} + 27n ^ {- 2} -5n ^ {- 3}) + (n-1) ^ 3n ^ {- 3} a_ {n-2} = 0$$
and I don't know how we can extravagantly deal with additional terms.

Also, how to get the supplement $$n ^ {- 3/2}$$ term?

Second, why does this relationship imply that $$zeta (3) -a_n / b_n = O ( alpha ^ {- 2n})?$$

After that, it remains that it can be shown that from the prime number theorem, we have that $$log d_n sim n$$ or $$d_n = text {lcm} (1,2, cdots, n)$$.

I managed to prove that
$$dfrac { log d_n} {n} leq pi (n) dfrac { log n} {n}$$
but I am unable to prove that $$log d_n / n$$ converges to $$1$$.

Finally, I don't know how this last result is true that for everything $$varepsilon> 0$$ we have
$$zeta (3) – dfrac {p_n} {q_n} = O (q_n ^ {- r- varepsilon})$$

I'm not really good at asymptotic behavior and big-O scoring, so I would really appreciate it if someone could respond with rotting and detailed explanations.

Thank you so much.

## limits and convergence – Does this sum \$ (1- frac {1} {2 ^ s}) ^ {(1- frac {1} {3 ^ s}) ^ {… … ^ {(1- frac {1} {p ^ s})}}} \$ also linked to the Riemann zeta function for \$ Re (s)> 0 \$?

I asked this question a month ago in SE but I don't have an answer. I want help for MO researchers

I'm interesting for the amount of repeated exemption from the form $$(z_1) ^ {z_2) ^ {… ^ {z_k}}}$$ such as $$z_1$$ and $$z_2$$, $$z_k$$ are different real exponents, this type of sum has been studied by many authors such as: Barraw ((exponential infinite, monthly 28 (1921) pp 141-143), For the Euler product which is linked to the Riemann zeta function, we have this well-known identity:$$prod_ {p in mathbb {P}} (1-p ^ {- s}) = frac {1} { zeta (s)}$$ , for $$s = 2$$ This product gives $$frac {6} { Pi ^ 2}$$ Now I am thinking of making this product as sum of exposure as follows:$$S_p = (1- frac {1} {2 ^ s}) ^ {(1- frac {1} {3 ^ s}) ^ {… ^ {(1- frac {1} { p ^ s})}}}$$ , I want to know if this iterated exponential sum linked to Riemann's zeta function as an Euler product for a sufficiently large prime number

I have a connection log function on this sum and increases exponentially I have the following identity:

$$S_p = exp ((1-2 ^ {- s}) sum_ {p geq 3} (1-p ^ {- s}))$$ , from this identity, I don't know the relationship between the sum of the exhibitor on $$p geq3$$ and Riemann's zeta function, part of the observation that I have has is this sum for $$s = 2$$ and $$s = 1$$ is this sum delimited by $$frac {1} { zeta (s)}$$ for $$p to infty$$ as for $$s = 1$$ ,$$S_p leq frac {1} { zeta (2)}$$ and for $$s = 2$$ We have $$S_p leq frac {1} { zeta ^ 2 (2)}$$ .

## nt.number theory – Can this quantity be expressed in \$ x cdot zeta (k) + y, x, y in mathbb {Q} \$?

For each natural number $$a$$ consider the sequence $$l (a): = left ( frac { gcd (a, b)} {a + b} right) _ {b in mathbb {N}}$$.

Then I calculated it for $$k ge 2, k in mathbb {R}$$ and $$p$$ first, we have:

$$| l (1) | _k ^ k = zeta (k) -1$$

$$| l (p) | _k ^ k = frac {2 p ^ k-1} {p ^ k} zeta (k) – left (1+ sum_ {j = 1} ^ {p-1} frac {1} {j ^ k} right)$$

I also calculated for $$n = 4$$ this:

$$| l (4) | _k ^ k = zeta (k) left (3- frac {1} {4 ^ k} – frac {2} {2 ^ k} + frac {1} {2 ^ {2k}} droite ) -3- frac {1} {3 ^ k}$$

My question is, if $$| l (a) | _k ^ k = x zeta (k) + y$$ or $$x, y in mathbb {Q}$$?

$$langle l (1), l (2) rangle = sum_ {k = 1} ^ infty frac {3k + 1} {2k (k + 1) (2k + 1)}$$

Is this last quantity equal to $$log (2)$$?

## riemann zeta function – A counterexample to the conjecture below?

This question is the clarification of my recent closed question, an example satisfying this conjecture $$n = 180$$ is mentioned here and the smallest integer $$k$$ satisfying this guess is for $$n = 3$$ Which one is $$60,480$$ look in this example.

Conjecture: Is$$k$$ and$$alpha$$ to be prime positive integers and$$alpha such as:
if $$4 ^ {- n} zeta (2n) = frac { pi ^ {2n} alpha} {k}$$ then $$k$$ always divisible by the first integers of $$1$$ at $$9$$ for each $$n> 2$$.

Now my question here: a counterexample for this conjecture? And if it is true how can I prove it?

## 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 $$x$$.
When I tried to set limits $$zeta (0.5 + it)$$ using certain transformations on the Gamma function using the function $$f (x) = exp (-n x)$$ all over the beach $$(0, + infty)$$ , For $$Re (s) = frac12$$ and $$t> 0$$ I'm coming to the final limits for $$zeta (0.5 + it)$$ which is represented by the following formal: For $$t geq 1.22$$: $$| zeta (0.5 + it) | leq 0,5 frac {| Gamma (0.5 + it) |} {| Gamma (-0,5 + it) |} tag {1}$$, For the limits of $$Gamma (s)$$ we find that the monotonic increasing function for $$| t | geq 5/4$$ with respect for the real part of $$s$$ and it was wrong with $$| t | leq 1$$ in this article called On the horizontal monotony of $$| Gamma (s) |$$ by Gopala Krishna Srinivasan and P. Zvengrowski, |$$Gamma (s)$$| is given in the introduction to this document for $$s = sigma + i t$$ by this formula:
$$| Gamma ( sigma + it) | = lambda frac { Gamma (1+ sigma)} { sqrt { sigma ^ 2 + t ^ 2}} sqrt { frac {2 pi t} { exp ( pi t) – exp (- pi t)}} tag {2}$$, it seems that the right side of this formal linked to the hyperbolic function cos, now when I tried to plug this formal into the RHS of my limits, it gives me a complicated form such as no simple formula for simplification, my question here how I can simplify RHS OF $$1$$ if this is true?