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[[File:Upsilon-function.png|400px|thumb|right|Modular Surface of The Upsilon function]]

The '''Upsilon function''', {{math|''Υ''(''s'')}}, is a [[function (mathematics)|function]] of a [[complex variable]] ''s'' that [[analytic continuation|analytically continues]] the sum of the [[infinite series]] 
:<math>\Upsilon(s) =\sum_{n=1}^\infty \frac 1 {n^2+s^2}</math>
for <math>s\neq ki, k\in\mathbb{Z}-\{0\}</math>

It was suggested in 2016 by Mohamad Emami and Alireza Jamali, analogous to the [[Riemann zeta function]].

==Definition==
The original Upsilon function is defined as
:<math>\Upsilon(s) =\sum_{n=1}^\infty \frac 1 {n^2+s^2} = \frac \pi {2s} \coth(\pi s) - \frac 1 {2s^2}</math>
For all complex numbers <math>s\in\mathbb{C}-\{ki \in\mathbb{Z}\mid k\neq 0\} </math>.

The Upsilon function ''Υ''(''s'') is a [[meromorphic]] function of a complex variable <math>s</math>, which has isolated singularities at points <math>ki</math> where <math>k</math> is a non-zero integer.
At point <math>s=0</math> the function has a [[removable singularity]]; one can readily remove this singular point by assigning the value <math> \zeta(2)=\frac{\pi^2}{6}</math> ( see [[Basel problem]] ).

The Upsilon function is actually a Generalized Upsilon function of order π when the generalized Upsilon function of order <math>\nu</math> is defined as

: <math>\Upsilon_\nu(s)=\sum_{n=1}^\infty \frac 1 {s^2 + (\frac{n\pi}\nu)^2} = \frac \nu {2s}\coth(\nu s)-\frac{1}{2s^2} </math>

==Singularities==

{{cquote| One can easily see that for <math> s=ki, k\in\mathbb{Z} </math> the function has singularities and the sum diverges, ...  }}

Anyway, we know that at <math>s=0</math> the function has a [[removable singularity]] and we can remove it as mentioned above.

==Roots and Upsilon conjecture== 
It was conjectured by Emami and Jamali that '''All roots of the Upsilon function have the form of <math>s=ix</math> for all <math>x\in\mathbb{R}-\{0\}</math> satisfying the equation <math>\pi x=\tan\pi x </math> '''.

So far, it is not known whether all roots are of the above form or not, leaving it as an open problem '''(Upsilon conjecture)'''.

==Specific values==
*<math>\Upsilon_1(\log\sqrt{x})=\sum_{n=1}^\infty \frac 1 {(\log\sqrt{x})^2 +(n\pi)^2} = \frac 1 {2\log x} \left(\frac{x+1}{x-1} - \frac 1 {\log x}\right)</math>
::Known as the '''''Jamali's identity''''', first appeared in a paper by Alireza Jamali and Mohamad Emami.

*<math>\Upsilon_\pi(0)=\zeta(2)=\frac{\pi^2}6. </math> See [[Basel problem]].
*<math>\Upsilon_1(\log\varphi) = \sum_{n=1}^\infty \frac 1 {(\log\varphi)^2 + (n\pi)^2} = \frac 1 {2\log\varphi} \left(\sqrt{5}-\frac 1 {\log\varphi}\right), </math> where ''φ'' is the [[Golden Ratio]] ('''''Jamali's series''''').

[[Category:Functions and mappings]]
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