Hugh L. Montgomery

Topics in multiplicative number theory

Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value theorems.- Large moduli theorems.- Further results and conjectures concerning mean and large moduli.- Mean moduli of L-functions.- Zero-free regions and the proliferation of zeros.- Distribution of zeros of L-functions.- Least character non-residues and arg L(12+it, x).- The prime number theorems of Hoheisel and Selberg.- The bombieri - Vinogradov theorem.- A lemma in additive prime number theory.- The mean value theorem of Barban.

The large sieve

In this paper, we propose an integer quadratic optimization model to determine the optimal decision for a supplier selection problem. The decision is the optimal product volume that has to be purchased from each supplier so that the total cost is minimum and the constraints are satisfied. The cost function that we used is containing the purchasing cost, transportation cost, penalty cost for product that not satisfy the quality level, penalty cost for product that is late and the holding cost whereas the constraints are consisting of supplier capacity constraint, demand satisfying, supplier assignment, inventory management, and budget constraint. A numerical experiment with generated random data is given to illustrate how the supplier selection problem can be solved by using the proposed mathematical model. From the results, the optimum product volume from each suppliers was determined so that the total cost is minimum.

The analytic principle of the large Sieve

E. Bombieri [12] has written at length concerning applications of the large sieve to number theory. Our intent here is to complement his exposition by devoting our attention to the analytic principle of the large sieve; we describe only briefly how applications to number theory are made. The large sieve was studied intensively during the decade 1965-1975, with the result that the subject has lost its mystery: We now possess a variety of simple ideas which provide very precise results and a host of variants. While the large sieve can no longer be considered deep, it nevertheless gives powerful estimates in many different settings.

Pair Correlation of Zeros and Primes in Short Intervals

In 1943, A. Selberg [15] Deduced From The Riemann Hypothesis (Rh) that

Minimal theta functions

\int\limits_{\rm{1}}^{\rm{X}} {{{\left( {\psi \left( {\left( {{\rm{1 + }}\delta } \right){\rm{x}}} \right){\rm{ - }}\psi \left( {\rm{x}} \right){\rm{ - }}\delta {\rm{x}}} \right)}^2}{{\rm{x}}^{{\rm{ - 2}}}}{\rm{dx}} \ll \delta {{\left( {{\mathop{\rm l}\nolimits} {\rm{ogX}}} \right)}^2}}

Zeros of L-Functions

(1) for X–1 ≤ δ ≤ X–1/4, X ≥ 2. Selberg was concerned with small values of δ and the constraint δ ≤ X–1/4 was imposed more for convenience than out of necessity. For Larger δ we have the following result.

Note on a Diophantine inequality in several variables

Abstract We study the product of two polynomials in many variables, in several norms, and show that under suitable assumptions this product can be bounded from below independently of the number of variables.

Fluctuations in the mean of Euler's phi function

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)