Are there enhancement solutions for the Riemann Zeta function?
Specifically for 2 actual numbers $a$ and also $b$, I would certainly such as to recognize if there are solutions for $\zeta (a+b)$ and also $\zeta (a-b)$ as a function of $\zeta (a)$ and also $\zeta (b)$.
The closest I can locate online is a paper by Harry Yosh "General Addition Formula for Meromorphic Functions Derived from Residue Theorem" in some unfamiliar journal, yet however I have no accessibility to it and also do not recognize if it would certainly address my inquiry. Possibly this is well - well-known and also I really did not look appropriately ...
Any aid valued, many thanks!
Almost absolutely, the solution is no. Take into consideration $a$ with actual component in the period (1/ 2,1). The Riemann theory mentions that $\zeta(a)\not=0$. If $\zeta(a)$ were a function of $\zeta(2+a)$ and also $\zeta(2)$ after that this would certainly transform the RH right into a declaration concerning the values of $\zeta(s)$ on $5/2<\Re[s]<3$, which would certainly be simply also very easy.
There is an useful formula connecting values of $\zeta$ at $s$ and also $(1-s)$.
If there were solutions connecting values along a continual family members of 1 - dimensional contours, such as $x + y = C$, one would certainly get a differential formula for $\zeta$, or some equally solid restraint. It is recognized that $\zeta$ does not please any kind of ODE is with algebraic features as coefficients. Certainly there can be gamma features or various other extra difficult coefficients yet leads for this sort of added framework in $\zeta$ appear dim. There are no added useful formulas for limited area zetas, as an example.