Q: my new iphone 6S stopped synchronizing with iTunes in my PC. It tells me that I have to update my iTunes version but it is up to date

What version of iTunes are you running?

On what version of OSx?

The most current version of iTunes is 12.4.1.

iOS9 on a mobile device requires iTunes 12.3 or higher which in turn requires a computer running OSX 10.8.5 or higher.  System Update only checks for updates for the current system version you are running but that itself may be out of date.  It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync this phone to your present computer. If it can run 10.8.5 or higher you can either buy a download code for OSX 10.8 Mountain Lion from Apple online, or you can try installing the free El Capitan OSX 10.11.  El Capitan may run slower on older machines and require additional RAM purchase.  Doing a big jump in system versions is also more likely to affect old software.

Snow leopard to el capitan, will it cause my macbook to be slow?  - https://discussions.apple.com/thread/7412959

Mountain Lion 10.8 purchase link U.S.A. - http://www.apple.com/shop/product/D6377Z/A/os-x-mountain-lion

Mountain Lion 10.8 purchase link U.K. - http://www.apple.com/uk/shop/product/D6377ZM/A/os-x-mountain-lion

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/