View source for Bombieri-Iwaniec method

Invented in [[#References|[a1]]] and [[#References|[a2]]], where it was used to bound the [[Riemann zeta-function|Riemann zeta-function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301901.png" /> along the mid-line of its critical strip, this is currently (2000) the strongest method around for establishing upper bounds on several wide classes of Weyl sums that have many applications within [[Analytic number theory|analytic number theory]] (cf. also [[Weyl sum|Weyl sum]]). General application to the first such class, exponential sums within the scope of Van der Corput's exponent pairs theory (see [[#References|[a3]]]), was done by M.N. Huxley and N. Watt in [[#References|[a10]]]. Improvements to one aspect of the method (the first spacing problem below) were found by Watt [[#References|[a16]]] and Huxley and G. Kolesnik [[#References|[a8]]], while in [[#References|[a11]]] H. Iwaniec and C.J. Mozzochi made an important adaptation, forming a second branch of the method that yielded new results for the circle and divisor problems. In a series of papers beginning with [[#References|[a4]]] Huxley has generalized and refined this adaptation, applying it to bound lattice point discrepancy (the difference between an area and the number of integer lattice points within the area; cf. also [[Lattice of points|Lattice of points]]; [[Geometry of numbers|Geometry of numbers]]). The original insight of E. Bombieri and Iwaniec into the second spacing problem (see below), which is of crucial importance for both branches of the method, has been significantly augmented by the theory of resonance curves invented and refined by Huxley in [[#References|[a5]]] and [[#References|[a7]]].

See [[#References|[a3]]] for a brief introduction to the method; [[#References|[a6]]] is the most recent book on the method, and covers almost all of its aspects, including applications to, e.g.: exponential sums with a Dirichlet character factor; the mean-square of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301902.png" /> over a short interval; and counting integer lattice points near a curve. It also covers Jutila's independent (but related) method of [[#References|[a12]]] and [[#References|[a13]]], used for the estimation of exponential sums with a Fourier coefficient of a cusp form, or a divisor function, as a factor.

The original branch of the Bombieri–Iwaniec method deals with a sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301903.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301905.png" />, and the derivatives (from first to fourth, say) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301906.png" /> have absolute values ranging within a uniformly bounded factor of a single parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301907.png" /> (sufficiently large) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301908.png" />. The method, which only applies directly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b1301909.png" />, has five main steps:

i) Division of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019010.png" /> into consecutive shorter sums within each of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019011.png" /> is approximated, via [[Taylor series|Taylor series]], by a cubic polynomial with rational quadratic coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019012.png" />.

ii) Use of Poisson summation (cf. also [[Poisson summation method|Poisson summation method]]), evaluation of Gauss sums (cf. also [[Gauss sum|Gauss sum]]) and the stationary phase method (cf. also [[Stationary phase, method of the|Stationary phase, method of the]]) to transform the short sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019013.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019014.png" /> into an even shorter one:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019015.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019018.png" />.

iii) Estimation of the sum over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019020.png" />, in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019021.png" />th power moment, using the [[Hölder inequality|Hölder inequality]] and the double large sieve introduced in [[#References|[a1]]] (cf. also [[Large sieve|Large sieve]]). This leaves two steps to provide data for the sieve.

iv) The first spacing problem: Count pairs of  "neighbours"  amongst vectors of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019022.png" />.

v) Second spacing problem: Count pairs of  "neighbours"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019024.png" /> (coincidences). Steps iv) and v) remain open problems (as of 2000), as does the question of more radical improvement of the whole. E. Fouvry and Iwaniec, using only Van der Corput's method with the double large sieve from Step iii), found useful new bounds for their monomial exponential sums (see [[#References|[a6]]]).

The method's second branch treats

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019025.png" /></td> </tr></table>

where the summation is over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019029.png" /> are as above. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019030.png" /> has been replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019031.png" />, so must the cubic approximation of Step i) be replaced by its derivative. Steps ii)–iv) also change, but the second spacing problem does not. These sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019032.png" /> can arise when a truncated form of the [[Fourier series|Fourier series]] for the sawtooth function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019033.png" /></td> </tr></table>

is used in the estimation of the lattice point discrepancy of an area of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019034.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019038.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019039.png" />-plane. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019040.png" /> gets too large, then one can switch to summing the discrepancies for columns of width <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019041.png" /> to summing those for rows of depth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019042.png" />, the number of which required will be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019043.png" />. Poisson summation makes a similar switch possible in estimating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019044.png" /> alone. In both contexts the critical cases are those around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019046.png" />), so the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019047.png" />, being values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019048.png" />, will satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019049.png" />; something like the situation in applications of the Hardy–Littlewood [[Circle method|circle method]].

Jutila's method [[#References|[a13]]] utilizes a  "twisted"  Wilton summation formula, corresponding to a cusp form of even weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019050.png" /> for the full modular group:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019052.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019053.png" />th Fourier coefficient of the cusp form, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019054.png" /> is a smooth function supported in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019055.png" />, and the linear integral transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019056.png" /> has integrand containing the Bessel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019057.png" />-function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019058.png" /> as a factor (cf. also [[Bessel functions|Bessel functions]]). Step ii) above is essentially the extension of Jutila's formula to the case where the cusp form is replaced by the [[Theta-series|theta-series]] of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019059.png" /> with Fourier coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019060.png" />. M. Jutila has an analogous  "twisted"  Voronoi summation formula for the more delicate case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019061.png" /> is the divisor function (cf. also [[Number of divisors|Number of divisors]]), for which the corresponding modular form is non-holomorphic; this enabled a more elementary proof of a famous theorem of Iwaniec (see [[#References|[a14]]]).

The original treatment by Bombieri and Iwaniec of the second spacing problem rested on the observation that proximity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019063.png" /> (modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019064.png" />) implies that the row vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019066.png" /> will be linked by a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019067.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019068.png" /> being an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019069.png" />-matrix of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019070.png" /> having a  "small"  lower left entry. Huxley's theory of resonance curves showed that the coincident pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019072.png" /> for a given choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019073.png" /> correspond to integer points lying near a certain plane resonance curve, this curve being determined (up to a minor transformation) by the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019075.png" />. This led, in [[#References|[a5]]], to better bounds in the second spacing problem, to improved exponential sum estimates, and (for example) to the result that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019076.png" /></td> </tr></table>

when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019077.png" />, with fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019078.png" /> (an unpublished improvement of this and of resonance curve theory is given in [[#References|[a7]]]). For comparison, Bombieri and Iwaniec's original paper [[#References|[a1]]] had <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019079.png" />, while even a complete resolution of both the first and second spacing problems (alone) could not get <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019080.png" /> below <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019081.png" />.

Integer points near a suitable curve may be counted using exponential sum estimates, so resonance curves invite one to apply the Bombieri–Iwaniec method iteratively. P. Sargos managed to do this in [[#References|[a15]]], using his own simpler theory of (quite different) resonance curves. He obtained results of greatest interest for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019082.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019083.png" />. A third construction of resonance curves (different again) has been given by Huxley and Kolesnik [[#References|[a9]]], and is of interest for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019084.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019085.png" />. They were able to apply this iteratively, but got better results (in most cases) with a single step of an elementary method once employed by H. Swinnerton-Dyer to count lattice points exactly on a curve.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Bombieri,   H. Iwaniec,   "On the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130190/b13019086.png" />"  ''Ann. Scuola Norm. Sup. Pisa Cl. Sci.'' , '''13'''  (1986)  pp. 449–472</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Bombieri,   H. Iwaniec,   "Some mean value theorems for exponential sums"  ''Ann. Scuola Norm. Sup. Pisa Cl. Sci.'' , '''13'''  (1986)  pp. 473–486</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S.W. Graham,   G. Kolesnik,   "Van der Corput's method for exponential sums" , ''London Math. Soc. Lecture Notes'' , '''126''' , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.N. Huxley,   "Exponential sums and lattice points"  ''Proc. London Math. Soc.'' , '''60'''  (1990)  pp. 471–502  (Corrigenda, 66 (1993), 70)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.N. Huxley,   "Exponential sums and the Riemann zeta-function IV"  ''Proc. London Math. Soc.'' , '''66'''  (1993)  pp. 1–40</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.N. Huxley,   "Area, lattice points and exponential sums" , ''London Math. Soc. Monographs'' , '''13''' , Oxford Univ. Press  (1996)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M.N. Huxley,   "Exponential sums and the Riemann zeta-function V"  (unpublished notes)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M.N. Huxley,   G. Kolesnik,   "Exponential sums and the Riemann zeta-function III"  ''Proc. London Math. Soc.'' , '''62'''  (1991)  pp. 449–468  (Corrigenda 66 (1993), 302)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.N. Huxley,   G. Kolesnik,   "Exponential sums with a large second derivative"  M. Jutila (ed.)  T. Metsänkylä (ed.) , ''Number Theory (Proc. Turku Conf. Number Theory in Memory of Kustaa Inkeri)'' , de Gruyter  (2000)  pp. 131–143</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M.N. Huxley,   N. Watt,   "Exponential sums and the Riemann zeta-function"  ''Proc. London Math. Soc.'' , '''57'''  (1988)  pp. 1–24</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  H. Iwaniec,   C.J. Mozzoch,   "On the divisor and circle problems"  ''J. Number Theory'' , '''29'''  (1988)  pp. 60–93</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Jutila,   "On exponential sums involving the divisor function"  ''J. Reine Angew. Math.'' , '''355'''  (1985)  pp. 173–190</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M. Jutila,   "Lectures on a method in the theory of exponential sums" , ''Tata Inst. Fundam. Res. Lect. Math. and Physics'' , '''80''' , Springer  (1987)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  M. Jutila,   "The fourth power moment of the Riemann zeta-function over a short interval" , ''Number Theory I (Budapest, 1987)'' , ''Colloq. Math. Soc. J. Bolyai'' , '''51''' , North-Holland  (1990)  pp. 221–244</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  P. Sargos,   "Points entiers au voisinage d'une courbe, sommes trigonométriques courtes et pairs d'exposants"  ''Proc. London Math. Soc.'' , '''70'''  (1995)  pp. 285–312</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  N. Watt,   "Exponential sums and the Riemann zeta-function II"  ''J. London Math. Soc.'' , '''39'''  (1989)  pp. 385–404</TD></TR></table>