Harold G. Diamond

University of Illinois at Urbana-Champaign

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- An elementary proof of the prime number theorem with a remainder term (with John Steinig), Inventiones Mathematicae 11 (1970), 199-258.
" Elementary" methods in prime number theory involve the use only of real variable techniques; methods of complex function theory or harmonic analysis or use of the properties of the Riemann zeta function in

**C**are not allowed. This article contains an elementary proof of the prime number theorem in the formpi(x) - li(x) = O{x exp(-c log^alpha x)}.

The proof uses a generalization of the Selberg formula and a tauberian argument.

- Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. N. S. 7 (1982), 553-589.
This is a survey article on elementary methods.

- Chebyshev estimates for Beurling generalized numbers, Proc. Amer. Math. Soc. 39 (1973), 503-508.
Beurling's numbers are a generalization of the integers in which only multiplicative structure is preserved. This article is one of a series that I have written on this theme. Under a weak hypothesis upon the distribution of the Beurling integers, an analogue of Chebyshev's O bound for pi(x)/(x/log x) is established.

- Beurling primes with large oscillation, Math. Annalen
(2006) (with Hugh L. Montgomery and Ulrike M. A. Vorhauer).
Methods of E. Landau from a hundred years ago show that Beurling generalized numbers having an integer counting function N(x) = cx + O(x^a) with c > 0 and 1/2 < a < 1, satisfy the prime number theorem with the error bound of de la Vallee Poussin. Here we show by a probabilistic construction that this error bound is in fact optimal for a discrete Beurling number system.

- One-sided Tauberian theorems for kernels which changes sign (with Matts Essen), Proc. London Math. Soc. (3) 36 (1978), 273-284.
A tauberian theorem is a kind of " unaveraging" result. For technical reasons the " kernel" function which creates the average is usually assumed to be positive. Here it shown that one can convert a class of kernels which change sign into positive ones for which tauberian methods succeed.

- Gauss sums and Fourier analysis on multiplicative subgroups of Z_q (with Frank Gerth and Jeffrey Vaaler), Trans. Amer. Math. Soc. 277 (1983), 711-726.
For f a periodic function mod q, let f^ denote the finite Fourier transform of f. A criterion is given for deciding when f can be recovered from the restriction of f^ to the integers relatively prime to q. Applications are given when this representation applies.

- Combinatorial sieves of dimension exceeding one, II (with H.
Halberstam and H.-E. Richert), 265 - 308, in Analytic Number Theory:
Proceedings, B. C. Berndt, et al. eds., Series PM 138, Birkhauser,
Boston, 1996.
In sieve theory estimates are made of the number of integers remaining in a given collection when those falling into certain residue classes are removed. The authors have devised a new sieve whose estimates are expressed in terms of a pair of functions defined by a coupled system of difference differential equations. This article is the conclusion of a series in which the authors establish the existence of the required functions.