Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

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Analytic Number Theory

The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves.

This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

Exponential sums and lattice points III

Computing the Continuous Discretely

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

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The Riemann zeta function

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds - called Moore bounds - for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem 'from above', remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem 'from below'. This survey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on the other hand.

Moore Graphs and Beyond: A survey of the Degree/Diameter Problem

Introduction Part I Elementary Methods 1 The rational line 2 Polygons and area 3 Integer points close to a curve 4 Rational points close to a curve Part II The Bombieri-Iwaniec Method 5 Analytic methods 7 The simple exponential sum 8 Exponential sums with a difference 9 Exponential sums with a difference 10 Exponential sums with modular form coefficients Part III The First Spacing Problem: Integer Vectors 11 The ruled surface method 12 The Hardy Littlewood method 13 The first spacing problem for the double sum Part IV The Second Spacing Problem: Rational vectors 14 The first and second conditions 15 Consecutive minor arcs Part V Results and Applications 17 Exponential sum theorems 18 Lattice points and area 19 Further results 20 Sums with modular form coefficients m 21 Applications to the Riemann zeta function 22 An application to number theory: prime integer points Part IV Related Work and Further Ideas 23 Related work 24 Further ideas References

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Area, Lattice Points, and Exponential Sums

A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T < 1$. There are different bounds for $S$ in different ranges for $\alpha $. In the middle range where $\alpha $ is near ${1\over 2}$, $S = O(\sqrt{M} T^{\theta + \epsilon })$. This $\theta $ bounds the exponent of growth of the Riemann zeta function on its critical line ${\rm Re} s = {1\over 2}$. Van der Corput used an iteration which changed $\alpha$ at each step. The Bombieri?Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of $y = f'(x)$ coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1?40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ?resonance curve?, one of a family of curves in some dual space, now calculated accurately in the paper ?Resonance curves in the Bombieri?Iwaniec method?, which is to appear in Funct. Approx. Comment. Math.

We turn the whole Bombieri?Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$. Ends and cusps of resonance curves are treated separately. Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve $\theta $ from 0.156140... in the previous paper and monograph to 0.156098.... In fact $(32/205 + \epsilon , 269/410 + \epsilon)$ is an exponent pair for every $\epsilon > 0$.

https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/plms/s3-66.2.302-s

Exponential sums and the Riemann zeta function v

On the difference between consecutive primes

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Martin Neil Huxley

Papers

Exponential sums and lattice points III

Area, Lattice Points, and Exponential Sums

Exponential sums and the Riemann zeta function v

On the difference between consecutive primes

Large values of Dirichlet polynomials, III