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Godfrey Harold Hardy was a mathematician whose work joined analysis with number theory, including Fourier series, the Riemann zeta function and the distribution of prime numbers. His long collaboration with John Edensor Littlewood and his work with Srinivasa Ramanujan produced influential methods and results in analytic number theory.^[ref] The Hardy-Littlewood circle method exemplifies this connection: it studies arithmetic counting problems through an integral of products of exponential sums, separating regions near rational points from the remaining contribution. His teaching also helped establish rigorous accounts of limits, functions and convergence in English undergraduate analysis.

Srinivasa Ramanujan developed results in number theory, including the study of highly composite numbers: positive integers having more divisors than every smaller positive integer.^[ref]

His analysis connects record values of the divisor function with prime factorization. Prime exponents must be arranged in nonincreasing order as the primes increase: swapping an inverted pair would produce a smaller integer with the same divisor count, contradicting the record property. For example, $12=2^2\cdot 3$ has six positive divisors. The subject turns irregular arithmetic data into a constrained optimization problem.^[ref]