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Upsilon function

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File:Upsilon-function.png
Modular Surface of The Upsilon function

The Upsilon function, Υ(s), is a function of a complex variable s that analytically continues the sum of the infinite series

for

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

Definition[edit]

The original Upsilon function is defined as

For all complex numbers .

The Upsilon function Υ(s) is a meromorphic function of a complex variable , which has isolated singularities at points where is a non-zero integer. At point the function has a removable singularity; one can readily remove this singular point by assigning the value ( see Basel problem ).

The Upsilon function is actually Generalized Upsilon function of order π when the generalized Upsilon function of order is defined as

Singularities[edit]

Anyway, we know that at the function has a removable singularity and we can remove it as mentioned above.

Roots and Upsilon conjecture[edit]

It was conjectured by Emami and Jamali that All roots of the Upsilon function have the form of for all satisfying the equation .

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[edit]

Known as the Jamali's identity, first appeared in a paper by Alireza Jamali and Mohamad Emami.
  • See Basel problem.
  • where φ is the Golden Ratio (Jamali's series).


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