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I realize this question is a bit long and contains quite a few formulas, but I believe a considerable amount of background and context are needed to fully understand my questions below. Also, I suspect some of the results illustrated below may be new and perhaps of some theoretical interest and/or interest to others as well as myself.


This question is related to the following six pairs of functions where $F_i(s)=s\int\limits_0^\infty f_i(x)\,x^{-s-1}\,dx$. Note $F_6(s)$ evaluated with $a(n)=1$ corresponds to the Riemann Xi function $\xi(s)$. All of the $f_i(x)$ functions defined below are even functions of $x$ and I believe all are examples of non-analytic smooth functions assuming $f_i(0)$ is defined as $f_i(0)=\underset{x\to 0}{lim}\,f_i(x)=0$.


(1) $\quad f_1(x)=\sum\limits_{n=1}^\infty a(n)\,e^{-\frac{n^2}{x^2}}\qquad\qquad\qquad\quad F_1(s)=\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

(2) $\quad f_2(x)=\sum\limits_{n=1}^\infty a(n)\,e^{-\frac{\pi\,n^2}{x^2}}\qquad\qquad\qquad F_2(s)=\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

(3) $\quad f_3(x)=\sum\limits_{n=1}^\infty a(n)\frac{2\,n^2}{x^2}\,e^{-\frac{n^2}{x^2}}\qquad\qquad\quad F_3(s)=s\,\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

(4) $\quad f_4(x)=\sum\limits_{n=1}^\infty a(n)\frac{2\,\pi\,n^2}{x^2}\,e^{-\frac{\pi\,n^2}{x^2}}\qquad\qquad F_4(s)=\pi^{-\frac{s}{2}}s\,\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

(5) $\quad f_5(x)=\sum\limits_{n=1}^\infty a(n)\left(\frac{2\,n^2}{x^2}-1\right)\,e^{-\frac{n^2}{x^2}}\qquad F_5(s)=(s-1)\,\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

(6) $\quad f_6(x)=\sum\limits_{n=1}^\infty a(n)\left(\frac{2\,\pi\,n^2}{x^2}-1\right)\,e^{-\frac{\pi\,n^2}{x^2}}\quad F_6(s)=\pi^{-\frac{s}{2}}(s-1)\,\Gamma\left(\frac{s}{2}+1\right)\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$


I've been investigating all of the functions above with various definitions of the coefficient function $a(n)$ but in this question I'm going to focus on $a(n)=\Lambda(n)$ because the following two formulas for $\frac{\zeta'(s)}{\zeta(s)}$ can be used to derive explicit formulas for all six of the $f_i(x)$ functions defined above when evaluated with $a(n)=\Lambda(n)$. For the remainder of this question $f_{\Lambda i}(x)$ denotes $f_i(x)$ above evaluated with the coefficient function $a(n)=\Lambda(n)$.


(7) $\quad\frac{\zeta'(s)}{\zeta(s)}=\sum\limits_\rho\frac{1}{s-\rho}-\frac{1}{2}\psi^{(0)}\left(\frac{s}{2}+1\right)+\frac{1}{1-s}+\frac{1}{2}\log(\pi)$

(8) $\quad\frac{\zeta'(s)}{\zeta(s)}=s\sum\limits_\rho\frac{1}{\rho\,\left(s-\rho\right)}-\frac{1}{2}H_{\frac{s}{2}}+\frac{s}{1-s}+\log(2\,\pi)$


My previous two questions at the links below are related to the two formulas for $\frac{\zeta'(s)}{\zeta(s)}$ defined in (7) and (8) above, but these earlier questions are a bit tangential to my primary investigation as of late which is more fully illuminated in this question.


What is the convergence of the explicit formula for $\frac{\zeta(s)}{\zeta(s)}$?

What is asymptotic and error bound for $\sum\limits_{k=1}^K\left(\frac{1}{\rho}+\frac{1}{\rho_{-k}}\right)$ as a function of $K$?


Two explicit formulas for each of the $f_{\Lambda i}(x)$ functions associated with (1) to (6) above are defined below along with an asymptotic function derived from each explicit formula. The sums over $\rho$ in all of the explicit formulas defined below are over the non-trivial zeta zeros in the upper-half plane. As an example of the nomenclature used below, $f_{o\,\Lambda1a}(x)$ and $f_{o\,\Lambda1b}(x)$ denote the explicit formulas for $f_{\Lambda1}(x)$ derived from formulas (7) and (8) above respectively, and $\overset{\text{~}}{f}_{o\,\Lambda1a}(x)$ and $\overset{\text{~}}{f}_{o\,\Lambda1b}(x)$ denote the corresponding asymptotics derived from these two formulas.


The formulas for $f_{\Lambda i}(x)$ associated with (1) to (6) above and their corresponding explicit formulas $f_{o\,\Lambda ia}(x)$ and $f_{o\,\Lambda ib}(x)$ are also illustrated below. Unless otherwise stated the formulas for $f_{\Lambda i}(x)$ associated with (1) to (6) above are illustrated in blue, the first explicit formula $f_{o\,\Lambda ia}(x)$ is illustrated in orange, the second explicit formula $f_{o\,\Lambda ib}(x)$ is illustrated in green, and all series associated with all formulas are evaluated over the first 100 terms. Typically all three formulas evaluate so closely together that the evaluation of the second explicit formula virtually hides the underlying evaluations of the other two formulas in the right-half plane. The explicit formulas for $f_{\Lambda1}(x)$ to $f_{\Lambda4}(x)$ only seem to converge in the right-half plane, whereas the explicit formulas for $f_{\Lambda5}(x)$ and $f_{\Lambda6}(x)$ seem to converge in the left-half plane as well as the right-half plane. All of the explicit formulas also seem to converge for a subset of $x\in\mathbb{C}$.


The explicit formulas and associated asymptotics for $f_{\Lambda1}(x)$ are as follows.


(9) $\quad f_{o\,\Lambda1a}(x)=-\frac{1}{2}\sum\limits_\rho\left(E_{1-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)+E_{\frac{\rho+1}{2}}\left(\frac{1}{x^2}\right)\right)\\$ $\qquad\qquad\qquad\qquad-\frac{1}{2}\sum\limits_{n=1}^\infty\left(E_{n+1}\left(\frac{1}{x^2}\right)-e^{-\frac{1}{x^2}} \log\left(\frac{1}{n}+1\right)\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)-\frac{1}{2} e^{-\frac{1}{x^2}} \log(\pi)$

(10) $\quad f_{o\,\Lambda1b}(x)=\frac{1}{x^2}\sum\limits_\rho\left(\frac{E_{\frac{\rho-1}{2}}\left(\frac{1}{x^2}\right)}{\rho-1}-\frac{E_{-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)}{\rho}\right)+\frac{1}{2\,x^2}\sum\limits_{n=1}^\infty\frac{E_n\left(\frac{1}{x^2}\right)}{n}\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+e^{-\frac{1}{x^2}}-e^{-\frac{1}{x^2}} \log(2\,\pi)$

(11) $\quad\overset{\text{~}}{f}_{o\,\Lambda1a}(x)=\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)-\frac{1}{2} e^{-\frac{1}{x^2}} \log(\pi)$

(12) $\quad\overset{\text{~}}{f}_{o\,\Lambda1b}(x)=\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+e^{-\frac{1}{x^2}}-e^{-\frac{1}{x^2}} \log(2\,\pi)$


The following plot illustrates $f_{\Lambda1}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda1a}(x)$ and $f_{o\,\Lambda1b}(x)$ defined in (9) and (10) above. I'll note that for the asymptotics defined in (11) and (12) above $\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda1a}(x)=\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda1b}(x)=\frac{\sqrt{\pi}}{2}$.


Illustration of formula (1) and associated explicit formulas

Figure (1): Illustration of $f_{\Lambda1}(x)$, $f_{o\,\Lambda1a}(x)$, and $f_{o\,\Lambda1b}(x)$ (blue, orange, and green)


The explicit formulas and associated asymptotics for $f_{\Lambda2}(x)$ are as follows.


(13) $\quad f_{o\,\Lambda2a}(x)=-\frac{1}{2}\sum\limits_\rho\left(E_{1-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)+E_{\frac{\rho+1}{2}}\left(\frac{\pi}{x^2}\right)\right)\\$ $\qquad\qquad\qquad\qquad-\frac{1}{2}\sum\limits_{n=1}^\infty\left(E_{n+1}\left(\frac{\pi}{x^2}\right)-e^{-\frac{\pi}{x^2}} \log\left(\frac{1}{n}+1\right)\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)-\frac{1}{2} e^{-\frac{\pi}{x^2}} \log(\pi)$

(14) $\quad f_{o\,\Lambda2b}(x)=\frac{\pi}{x^2}\sum\limits_\rho\left(\frac{E_{\frac{\rho-1}{2}}\left(\frac{\pi}{x^2}\right)}{\rho-1}-\frac{E_{-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)}{\rho}\right)+\frac{\pi}{2\,x^2}\sum\limits_{n=1}^\infty\frac{E_n\left(\frac{\pi}{x^2}\right)}{n}\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)+e^{-\frac{\pi}{x^2}}-e^{-\frac{\pi}{x^2}} \log(2\,\pi)$

(15) $\quad\overset{\text{~}}{f}_{o\,\Lambda2a}(x)=\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)-\frac{1}{2} e^{-\frac{\pi}{x^2}} \log(\pi)$

(16) $\quad\overset{\text{~}}{f}_{o\,\Lambda2b}(x)=\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)+e^{-\frac{\pi}{x^2}}-e^{-\frac{\pi}{x^2}} \log(2\,\pi)$


The following plot illustrates $f_{\Lambda2}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda2a}(x)$ and $f_{o\,\Lambda2b}(x)$ defined in (13) and (14) above. I'll note that for the asymptotics defined in (15) and (16) above $\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda2a}(x)=\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda2b}(x)=\frac{1}{2}$.


Illustration of formula (2) and associated explicit formulas

Figure (2): Illustration of $f_{\Lambda2}(x)$, $f_{o\,\Lambda2a}(x)$, and $f_{o\,\Lambda2b}(x)$ (blue, orange, and green)


The explicit formulas and associated asymptotics for $f_{\Lambda3}(x)$ are as follows.


(17) $\quad f_{o\,\Lambda3a}(x)=-\frac{1}{x^2}\sum\limits_\rho\left(E_{-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)+E_{\frac{\rho-1}{2}}\left(\frac{1}{x^2}\right)\right)\\$ $\qquad\qquad\qquad\qquad-\frac{1}{x^2}\sum\limits_{n=1}^\infty\left(E_n\left(\frac{1}{x^2}\right)-e^{-\frac{1}{x^2}} \log\left(\frac{1}{n}+1\right)\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+e^{-\frac{1}{x^2}}-\frac{e^{-\frac{1}{x^2}} \log(\pi)}{x^2}$

(18) $\quad f_{o\,\Lambda3b}(x)=-\frac{1}{x^2}\sum\limits_\rho\left(E_{-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)+E_{\frac{\rho-1}{2}}\left(\frac{1}{x^2}\right)+\frac{2\,e^{-\frac{1}{x^2}}}{\rho-\left(\rho\right){}^2}\right)\\$ $\qquad\qquad\qquad\qquad-\frac{1}{x^2}\sum\limits_{n=1}^\infty\left(E_n\left(\frac{1}{x^2}\right)-\frac{e^{-\frac{1}{x^2}}}{n}\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+\frac{e^{-\frac{1}{x^2}} \left(x^2+2\right)}{x^2}-\frac{2\,e^{-\frac{1}{x^2}} \log(2\,\pi)}{x^2}$

(19) $\quad\overset{\text{~}}{f}_{o\,\Lambda3a}(x)=\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+e^{-\frac{1}{x^2}}-\frac{e^{-\frac{1}{x^2}} \log(\pi)}{x^2}$

(20) $\quad\overset{\text{~}}{f}_{o\,\Lambda3b}(x)=\frac{1}{2} \sqrt{\pi}\,x\,erfc\left(\frac{1}{x}\right)+\frac{e^{-\frac{1}{x^2}} \left(x^2+2\right)}{x^2}-\frac{2\,e^{-\frac{1}{x^2}} \log(2\,\pi)}{x^2}$


The following plot illustrates $f_{\Lambda3}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda3a}(x)$ and $f_{o\,\Lambda3b}(x)$ defined in (17) and (18) above. I'll note that for the asymptotics defined in (19) and (20) above $\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda3a}(x)=\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda3b}(x)=\frac{\sqrt{\pi}}{2}$.


Illustration of formula (3) and associated explicit formulas

Figure (3): Illustration of $f_{\Lambda3}(x)$, $f_{o\,\Lambda3a}(x)$, and $f_{o\,\Lambda3b}(x)$ (blue, orange, and green)


The explicit formulas and associated asymptotics for $f_{\Lambda4}(x)$ are as follows.


(21) $\quad f_{o\,\Lambda4a}(x)=-\frac{\pi}{x^2}\sum\limits_\rho\left(E_{-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)+E_{\frac{\rho-1}{2}}\left(\frac{\pi}{x^2}\right)\right)\\$ $\qquad\qquad\qquad\qquad-\frac{\pi}{x^2}\sum\limits_{n=1}^\infty\left(E_n\left(\frac{\pi}{x^2}\right)-e^{-\frac{\pi}{x^2}} \log\left(\frac{1}{n}+1\right)\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)+e^{-\frac{\pi}{x^2}}-\frac{\pi\,e^{-\frac{\pi}{x^2}} \log(\pi)}{x^2}$

(22) $\quad f_{o\,\Lambda4b}(x)=-\frac{\pi}{x^2}\sum\limits_\rho\left(E_{-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)+E_{\frac{\rho-1}{2}}\left(\frac{\pi}{x^2}\right)-\frac{2\,e^{-\frac{\pi}{x^2}}}{\rho\,\left(\rho-1\right)}\right)\\$ $\qquad\qquad\qquad\qquad-\frac{\pi}{x^2}\sum\limits_{n=1}^\infty\left(E_n\left(\frac{\pi}{x^2}\right)-\frac{e^{-\frac{\pi}{x^2}}}{n}\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2} \left(x\,\left(-erf\left(\frac{\sqrt{\pi}}{x}\right)\right)+e^{-\frac{\pi}{x^2}} \left(\frac{4\,\pi}{x^2}+2\right)+x\right)-\frac{2\,\pi\,e^{-\frac{\pi}{x^2}} \log(2\,\pi)}{x^2}$

(23) $\quad\overset{\text{~}}{f}_{o\,\Lambda4a}(x)=\frac{1}{2}\,x\,erfc\left(\frac{\sqrt{\pi}}{x}\right)+e^{-\frac{\pi}{x^2}}-\frac{\pi\,e^{-\frac{\pi}{x^2}} \log(\pi)}{x^2}$

(24) $\quad\overset{\text{~}}{f}_{o\,\Lambda4b}(x)=\frac{1}{2} \left(x\,\left(-erf\left(\frac{\sqrt{\pi}}{x}\right)\right)+e^{-\frac{\pi}{x^2}} \left(\frac{4\,\pi}{x^2}+2\right)+x\right)-\frac{2\,\pi\,e^{-\frac{\pi}{x^2}} \log(2\,\pi)}{x^2}$


The following plot illustrates $f_{\Lambda4}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda4a}(x)$ and $f_{o\,\Lambda4b}(x)$ defined in (21) and (22) above. I'll note that for the asymptotics defined in (23) and (24) above $\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda4a}(x)=\underset{x\to\infty}{lim}\,\frac{\partial }{\partial x}\overset{\text{~}}{f}_{o\,\Lambda4b}(x)=\frac{1}{2}$.


Illustration of formula (4) and associated explicit formulas

Figure (4): Illustration of $f_{\Lambda4}(x)$, $f_{o\,\Lambda4a}(x)$, and $f_{o\,\Lambda4b}(x)$ (blue, orange, and green)


The explicit formulas and associated asymptotics for $f_{\Lambda5}(x)$ are as follows.


(25) $\quad f_{o\,\Lambda5a}(x)=\frac{1}{2}\sum\limits_\rho\left(\left(1-\rho\right)\,E_{1-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)+\rho\,E_{\frac{\rho+1}{2}}\left(\frac{1}{x^2}\right)-4\,e^{-\frac{1}{x^2}}\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2}\sum\limits_{n=1}^\infty\left((2\,n+1)\,E_{n+1}\left(\frac{1}{x^2}\right)-\frac{e^{-\frac{1}{x^2}} \left(\left(x^2-2\right) \log\left(\frac{1}{n}+1\right)+2\,x^2\right)}{x^2}\right)\\$ $\qquad\qquad\qquad\qquad+e^{-\frac{1}{x^2}}+\frac{\left(x^2-2\right) e^{-\frac{1}{x^2}} \log(\pi)}{2\,x^2}$

(26) $\quad f_{o\,\Lambda5b}(x)=-\frac{1}{x^2}\sum\limits_\rho\frac{\left(\rho-1\right){}^2\,e_{-\frac{\rho}{2}}\left(\frac{1}{x^2}\right)+\left(\rho\right){}^2\,e_{\frac{\rho-1}{2}}\left(\frac{1}{x^2}\right)-2\,e^{-\frac{1}{x^2}}}{\rho\,\left(\rho-1\right)}\\$ $\qquad\qquad\qquad\qquad-\frac{1}{2\,x^2}\sum\limits_{n=1}^\infty\frac{(2\,n+1)\,E_n\left(\frac{1}{x^2}\right)-2\,e^{-\frac{1}{x^2}}}{n}+\frac{2\,e^{-\frac{1}{x^2}}}{x^2}+\frac{\left(x^2-2\right) e^{-\frac{1}{x^2}} \log(2\,\pi)}{x^2}$

(27) $\quad\overset{\text{~}}{f}_{o\,\Lambda5a}(x)=e^{-\frac{1}{x^2}}+\frac{\left(x^2-2\right) e^{-\frac{1}{x^2}} \log(\pi)}{2\,x^2}$

(28) $\quad\overset{\text{~}}{f}_{o\,\Lambda5b}(x)=\frac{2\,e^{-\frac{1}{x^2}}}{x^2}+\frac{\left(x^2-2\right) e^{-\frac{1}{x^2}} \log(2\,\pi)}{x^2}$


The following plot illustrates $f_{\Lambda5}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda5a}(x)$ and $f_{o\,\Lambda5b}(x)$ defined in (25) and (26) above. I'll note that for the asymptotics defined in (27) and (28) above $\underset{x\to\infty}{lim}\,\overset{\text{~}}{f}_{o\,\Lambda5a}(x)=1+\frac{\log (\pi)}{2}$ and $\underset{x\to\infty}{lim}\,\overset{\text{~}}{f}_{o\,\Lambda5b}(x)=\log(2\,\pi)$ the latter of which corresponds to the dashed-gray horizontal reference line in the plot below.


Illustration of formula (5) and associated explicit formulas

Figure (5): Illustration of $f_{\Lambda5}(x)$, $f_{o\,\Lambda5a}(x)$, and $f_{o\,\Lambda5b}(x)$ (blue, orange, and green)


The explicit formulas and associated asymptotics for $f_{\Lambda6}(x)$ are as follows.


(29) $\quad f_{o\,\Lambda6a}(x)=\frac{1}{2}\sum\limits_\rho\left(\left(1-\rho\right)\,E_{1-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)+\rho\,E_{\frac{\rho+1}{2}}\left(\frac{\pi}{x^2}\right)-4\,e^{-\frac{\pi}{x^2}}\right)\\$ $\qquad\qquad\qquad\qquad+\frac{1}{2}\sum\limits_{n=1}^\infty\left((2\,n+1)\,E_{n+1}\left(\frac{\pi}{x^2}\right)-\frac{e^{-\frac{\pi}{x^2}} \left(\left(x^2-2\,\pi\right) \log\left(\frac{1}{n}+1\right)+2\,x^2\right)}{x^2}\right)\\$ $\qquad\qquad\qquad\qquad+e^{-\frac{\pi}{x^2}}+\frac{e^{-\frac{\pi}{x^2}} \left(x^2-2\,\pi\right) \log(\pi)}{2\,x^2}$

(30) $\quad f_{o\,\Lambda6b}(x)=-\frac{\pi}{x^2}\sum\limits_\rho\frac{\left(\rho-1\right){}^2\,e_{-\frac{\rho}{2}}\left(\frac{\pi}{x^2}\right)+\left(\rho\right){}^2\,e_{\frac{\rho-1}{2}}\left(\frac{\pi}{x^2}\right)-2\,e^{-\frac{\pi}{x^2}}}{\rho\,\left(\rho-1\right)}\\$ $\qquad\qquad\qquad\qquad-\frac{\pi}{2\,x^2}\sum\limits_{n=1}^\infty\frac{(2\,n+1)\,E_n\left(\frac{\pi}{x^2}\right)-2\,e^{-\frac{\pi}{x^2}}}{n}\\$ $\qquad\qquad\qquad\qquad+\frac{2\,\pi\,e^{-\frac{\pi}{x^2}}}{x^2}+\frac{e^{-\frac{\pi}{x^2}} \left(x^2-2\,\pi\right) \log(2\,\pi)}{x^2}$

(31) $\quad\overset{\text{~}}{f}_{o\,\Lambda6a}(x)=e^{-\frac{\pi}{x^2}}+\frac{e^{-\frac{\pi}{x^2}} \left(x^2-2\,\pi\right) \log(\pi)}{2\,x^2}$

(32) $\quad\overset{\text{~}}{f}_{o\,\Lambda6b}(x)=\frac{2\,\pi\,e^{-\frac{\pi}{x^2}}}{x^2}+\frac{e^{-\frac{\pi}{x^2}} \left(x^2-2\,\pi\right) \log(2\,\pi)}{x^2}$


The following plot illustrates $f_{\Lambda6}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda6a}(x)$ and $f_{o\,\Lambda6b}(x)$ defined in (29) and (30) above. I'll note that for the asymptotics defined in (31) and (32) above $\underset{x\to\infty}{lim}\,\overset{\text{~}}{f}_{o\,\Lambda6a}(x)=1+\frac{\log (\pi )}{2}$ and $\underset{x\to\infty}{lim}\,\overset{\text{~}}{f}_{o\,\Lambda6b}(x)=\log(2\,\pi)$ the latter of which corresponds to the dashed-gray horizontal reference line in the plot below.


Illustration of formula (6) and associated explicit formulas

Figure (6): Illustration of $f_{\Lambda6}(x)$, $f_{o\,\Lambda6a}(x)$, and $f_{o\,\Lambda6b}(x)$ (blue, orange, and green)


The following plot illustrates $f_{\Lambda5}(x)$ and $f_{\Lambda6}(x)$ in blue and orange respectively where the dashed-gray horizontal reference line corresponds to $y=\log(2\,\pi)$. Both formulas are evaluated over the first $1000$ terms.


Illustration of formulas (5) and (6)

Figure (7): Illustration of $f_{\Lambda5}(x)$ (blue) and $f_{\Lambda6}(x)$ (orange)


The following two plots illustrate the first four $\rho$ terms for $f_{o\,\Lambda6a}(x)$ and $f_{o\,\Lambda6b}(x)$ defined in (29) and (30) above where each successive $\rho$ term decreases in magnitude. Note the first $\rho$ term illustrated in blue exhibits a noticeable oscillation for both formulas. I believe all of the $\rho$ terms exhibit an analogous oscillation where the frequency of this oscillation increases with each successive $\rho$ term but the magnitude of this oscillation decreases with each successive $\rho$ term. I also believe the magnitudes of these $\rho$ term oscillations increase as $x\to\infty$.


Illustration of the first four zeta zero terms in formula (29)

Figure (8): Illustration of first four $\rho$ terms in $f_{o\,\Lambda6a}(x)$


Illustration of the first four zeta zero terms in formula (30)

Figure (9): Illustration of first four $\rho$ terms in $f_{o\,\Lambda6b}(x)$


The following plot illustrates the first $\rho$ term in $f_{o\,\Lambda6a}(x)$ and the first $\rho$ term in $f_{o\,\Lambda6b}(x)$ in blue and orange respectively evaluated over an extended range in the right-half plane.


Illustration of the first zeta zero term in formulas (29) and (30)

Figure (10): Illustration of first $\rho$ term in $f_{o\,\Lambda6a}(x)$ (blue) and $f_{o\,\Lambda6b}(x)$ (orange)


The following plot illustrates the difference between $f_{\Lambda6}(x)$ and the asymptotic $\overset{\text{~}}{f}_{o\,\Lambda6b}(x)$ defined in (32) above where the series associated with $f_{\Lambda6}(x)$ is evaluated over the first 1,000 terms. This plot illustrates the evaluation of $f_{\Lambda6}(x)$ produces oscillations similar to those of the $\rho$ terms illustrated in figures (8), (9), and (10) above.


Illustration of formula (1) minus asymptotic formula (32)

Figure (11): Illustration of $f_{\Lambda6}(x)$ minus $\overset{\text{~}}{f}_{o\,\Lambda6b}(x)$


Figures (1) to (4) above illustrate the explicit formulas for $f_{\Lambda i}(x)$ associated with (1) to (4) above seem to converge for $x\ge 0$, whereas the explicit formulas for $f_{\Lambda i}(x)$ associated with (5) and (6) above seem to converge for $x\in\mathbb{R}$. I've also noticed the explicit formulas for $f_{\Lambda i}(x)$ associated with (1) to (4) above seem to converge for a subset of $x\in\mathbb{C}\land\Re(x)>0$, whereas the explicit formulas for $f_{\Lambda i}(x)$ associated with (5) and (6) above seem to converge for a subset of $x\in\mathbb{C}$.


The following plot illustrates the absolute parts of $f_{\Lambda6}(x)$ and the corresponding explicit formulas $f_{o\,\Lambda6a}(x)$ and $f_{o\,\Lambda6b}(x)$ defined in (29) and (30) above where all formulas are evaluated along the line $x=t+\frac{1}{2}i\,t$. Note evaluation along this line amplifies the oscillation illustrated in figures (8) to (11) above.


Illustration of absolute values of formula (6) and associated explicit formulas evaluated along the line z=t+1/2it

Figure (12): Illustration of absolute part of $f_{\Lambda6}(x)$ and associated explicit formulas $f_{o\,\Lambda6a}(x)$ and $f_{o\,\Lambda6b}(x)$ evaluated along the line $x=t+\frac{1}{2}i\,t$


The following plot illustrates the absolute parts of $f_{\Lambda6}(x)$ and the asymptotic $\overset{\text{~}}{f}_{o\,\Lambda6b}(x)$ defined in (32) above in blue and orange respectively where both formulas are evaluated along the line $x=t+\frac{1}{2}i\,t$ and the series associated with $f_{\Lambda6}(x)$ is evaluated over the first 1,000 terms. The dashed-gray horizontal reference line corresponds to $y=\log(2\,\pi)$.


Illustration of absolute values of formula (6) and asymptotic (32) evaluated along the line z=t+1/2it

Figure (13): Illustration of absolute part of $f_{\Lambda6}(x)$ (blue) and asymptotic $\overset{\text{~}}{f}_{o\,\Lambda6b}(x)$ (orange) evaluated along the line $x=t+\frac{1}{2}i\,t$


The density plots in figures (14) to (19) below for $\left|f_{\Lambda i}(x)\right|$ associated with (1) to (6) above perhaps provide some insight into the convergence of $f_{\Lambda i}(x)$ for $x\in\mathbb{C}$. In the density plots below the horizontal axis corresponds to $\Re(x)$ and the vertical axis corresponds to $\Im(x)$, and all series associated with formulas (1) to (6) above are evaluated over the first $1000$ terms. I'm fascinated by the patterns that emerge at the sharp boundary in figures (14) to (19) below and wonder how they're related to the prime numbers and non-trivial zeta zeros.


Density plot of formula (1) absolute value for complex input

Figure (14): Density Plot of $\left|f_{\Lambda1}(x)\right|$ for $x\in\mathbb{C}$


Density plot of formula (2) absolute value for complex input

Figure (15): Density Plot of $\left|f_{\Lambda2}(x)\right|$ for $x\in\mathbb{C}$


Density plot of formula (3) absolute value for complex input

Figure (16): Density Plot of $\left|f_{\Lambda3}(x)\right|$ for $x\in\mathbb{C}$


Density plot of formula (4) absolute value for complex input

Figure (17): Density Plot of $\left|f_{\Lambda4}(x)\right|$ for $x\in\mathbb{C}$


Density plot of formula (5) absolute value for complex input

Figure (18):Density Plot of $\left|f_{\Lambda5}(x)\right|$ for $x\in\mathbb{C}$


Density plot of formula (6) absolute value for complex input

Figure (19):Density Plot of $\left|f_{\Lambda6}(x)\right|$ for $x\in\mathbb{C}$


I believe the absolute value of the exponential associated with $f_{\Lambda i}(x)$ also provides some insight into the convergence of $f_{\Lambda i}(x)$ for $x\in\mathbb{C}$.

With respect to $f_{\Lambda 1}(x)$, $f_{\Lambda 3}(x)$, and $f_{\Lambda 5}(x)$, $\left|e^{-\frac{n^2}{x^2}}\right|=e^{-n^2\,\Re\left(\frac{1}{x^2}\right)}=e^{-n^2\frac{(\Re(x))^2-(\Im(x))^2}{\left((\Re(x))^2+(\Im(x))^2\right)^2}}$.

With respect to $f_{\Lambda 2}(x)$, $f_{\Lambda 4}(x)$, and $f_{\Lambda 6}(x)$, $\left|e^{-\frac{\pi\,n^2}{x^2}}\right|=e^{-\pi\,n^2\,\Re\left(\frac{1}{x^2}\right)}=e^{-\pi\,n^2\frac{(\Re(x))^2-(\Im(x))^2}{\left((\Re(x))^2+(\Im(x))^2\right)^2}}$.

So both exponentials are decaying exponentials only for $|\Im(x)|<|\Re(x)|$.


Question (1): What is the convergence of $f_{\Lambda i}(x)$ associated with (1) to (6) above and the corresponding explicit formulas defined in (9) to (32) above for $x\in\mathbb{C}$? Is it true they all converge for $|\Im(x)|<|\Re(x)|$?


The Prime Number Theorem and Riemann Hypothesis both predict constraints on the behavior of the second Chebyshev function $\psi(x)=\sum\limits_{n=1}^x\Lambda(n)$.


Question (2): What do the Prime Number Theorem and Riemann Hypothesis predict for constraints on the behavior of $f_{\Lambda i}(x)$ associated with (1) to (6) above and the corresponding explicit formulas defined in (9) to (32) above? Do the Prime Number Theorem and Riemann Hypothesis constrain the behavior of $f_{\Lambda i}(x)$ and the associated explicit formulas for $x\in\mathbb{C}$ as well as $x\in\mathbb{R}$?


The term of $f_6(x)$ defined in (6) above evaluated as a function of $n$ with $x$ held constant has a fairly simple Maclaurin series illustrated in (33) below. This Maclaurin series can also be used to evaluate the term as a function of $x$ with constant $n$ for $x>0$ and seems to converge fairly rapidly except near the origin.


(33) $\quad \left(\frac{2\,\pi\,n^2}{x^2}-1\right)\,e^{-\frac{\pi\,n^2}{x^2}}=-\sum\limits_{k=0}^\infty\frac{(-\pi)^k\,(2\,k+1)\,n^{2\,k}}{k!\,x^{2\,k}}$


The relationship illustrated in (6) above and the Maclaurin series illustrated in (33) above can be used to derive formulas for $F_6(s)$ such as the one illustrated in (34) below. For $a(n)=1$, $F_6(s)$ corresponds to the Riemann Xi function $\xi(s)$, and in this particular case formula (34) is valid for $\Re(s)>1$.


(34) $\quad F(s)=\pi^{-\frac{s}{2}}\left(\frac{1}{e}+(s-1)\,E_{-\frac{s}{2}}(1)-s\sum\limits_{k=0}^\infty\frac{(-1)^k\,(2\,k+1)}{k!\,(2\,k+s)}\right) \sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$


$\endgroup$
  • $\begingroup$ $\sum_{n=1}^\infty \Lambda(n) n^k e^{-n^l x}= \mathcal{M}^{-1}[\Gamma(s) \frac{-\zeta'}{\zeta}(ls-k)]= \sum_\beta Res[\Gamma(s) \frac{-\zeta'}{\zeta}(ls-k)x^{-s-1}, \beta]$ where $\beta$ are the poles of $\Gamma(s) \frac{-\zeta'}{\zeta}(ls-k)$. The obtained series converges for $arg(x) \in (-\pi/(2l),\pi/(2l))$. This is the residue theorem, applying it to things related to $\zeta'/\zeta$ is more or less what does the proof of the prime number theorem. $\endgroup$ – reuns Nov 7 '18 at 20:12

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