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

ADDENDUM
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} $?

In addition:

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

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 <k $ 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.

I have made some progress.

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 $.

A simplification of this inequality if it is true? : For $ t geq 1.22 $: $ | zeta (0.5 + it) | leq 0.5 frac {| Gamma (0.5 + it) |} {| Gamma (-0.5 + it) |} $

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?

Note: The motivation for this question is to look for some prime number distribution connections to the Gaussian distribution.