Q:

To what extent can the invention of zero in India as a number be tied to Buddhist philosophy, if at all? 

The Wikipedia entry on zero suggests that the ancient Greeks were unsure about the ontological status of zero.  They asked themselves, 'How can nothing be something?'  whereas in Buddhism, Sunyata or void appears to have positive value (not in an arithmetic sense, but in the sense of beneficial).  Is there any evidence that Buddhist philosophy helped usher in the invention of zero as a number in its own right, other than the fact that it arose in India when Buddhism was vital force there? 
Further, zero is often called the identity, or (very much) less often the neutral element in mathematical circles.  Is that a telling coincidence, rather than simply a curious one? 
Zero is used as a placeholder in the decimal/Arabic number system, but for the purposes of the argument I'm developing here, the more important 'positive' virtue is that it allowed the invention of negative numbers.  Without zero they wouldn't be possible. 
We can then picture two numbers of the same absolute value but of opposite sign put together (i.e. addition) vanishing into the void, and the opposite process of the void separating into two somethings.  So we could answer the Greeks, not only is nothing a something, but somethings (in the plural) can dissolve into nothing as well as arise from it. 

A:

I think there is a connection, but you have the relationship reversed. 
Yes, one of the key terms in Buddhist philosophy is Śūnyatā (emptiness), and yes, this comes directly from the Sanskrit word for zero (śūnya).  However, there is no indication that the Buddhist philosophical concept led to the mathematical concept; on the contrary, it appears that the Buddhists borrowed the existing mathematical concept for their own purposes. 
It should also be remembered that within Buddhist philosophy, the emphasis on Śūnyatā only comes with the advent of Nāgārjuna (which is to say, 1st-2nd century CE).   There are a handful of earlier references; see Luis O. Gomez, "Proto-Madhyamaka in the Pāli Canon", Philosophy East and West, 22(1976):2 pp137-165) for an analysis of these, along with Choong Mun-keat, The Notion of Emptiness in Early Buddhism, 1995, if you are interested. 

A:

Your question heads in the right direction, but you're a little off in the religion itself.  The invention of zero (or rather, its acceptance as more than just a placeholder) had a lot to do with Hinduism, not Buddhism (which, granted, came directly from Hinduism).  
The following explanation is entirely referenced from the excellent book Zero: The Biography of a Dangerous Idea by Charles Seife:

The void had an important place in Hindu religion.  Hinduism had started off as a polytheistic religion... However, over centuries - centuries before Alexander [of Greece] arrived [in the 4th century BC] - the gods began to merge together... at its core Hinduism became monotheistic and introspective.  All the gods became aspects of an all encompassing god, Brahma. 

The book continues to describe the concept of Atman, and how it is "the infinite soul that suffuses the universe, at once everywhere and nowhere at the same time.  It is infinity, and it is nothing." 

So India, as a society that actively explored the void and the infinite, accepted zero.  Indian mathematicians did more than simply accept zero.  They transformed it, changing its role from mere placeholder to number. 

Thus, Buddhism did not have anything to do with the idea of zero, but rather it was  Hinduism that allowed for its acceptance (note "acceptance," and definitely not "invention" - zero existed long before even Hinduism became prominent, as it was used as a placeholder since Babylonian times).  The Indian concept of the void, arising from Hinduism, helped usher in the Arabic numeral system and eventual mathematical applications of zero.  
For further information, I strongly recommend Zero as I mentioned above. 

A:

The (https://en.wikipedia.org/wiki/Bakhshali_manuscript) [Bakhshali manuscript] is one of the oldest(3rd century AD) recorded Indian mathematics.  It was discovered by a local peasant in the Pakhtunkhwa region of Pakistan.  The region was under the control of Buddhism (see (http://whc.unesco.org/en/list/140) [Takhtbai ruins]).  It dates back from 500 years before Brahmagupta.  The zero dot symbol is used in the manuscript instead of a ring symbol.