Area of a disk

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In [[number theory]], the '''Elliott–Halberstam conjecture''' is a [[conjecture]] about the distribution of [[prime number]]s in [[arithmetic progression]]s. It has many applications in [[sieve theory]]. It is named for [[Peter D. T. A. Elliott]] and [[Heini Halberstam]].

To state the conjecture requires some notation. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x''. If ''q'' is a [[negative and positive numbers|positive]] [[integer]] and ''a'' is [[coprime]] to ''q'', we let <math>\pi(x;q,a)</math>, denote the number of primes less than or equal to ''x'' which are equal to ''a'' modulo ''q''. [[Dirichlet's theorem on primes in arithmetic progressions]] then tells us
that

:<math> \pi(x;q,a) \approx \frac{\pi(x)}{\varphi(q)}</math>

where ''a'' is coprime to ''q'' and <math>\varphi</math> is [[Euler's totient function]]. If we then define the error function

:<math> E(x;q) = \max_{(a,q) = 1} \left|\pi(x;q,a) - \frac{\pi(x)}{\varphi(q)}\right|</math>

where the max is taken over all ''a'' coprime to ''q'', then the Elliott–Halberstam conjecture is the assertion that
for every ''θ'' < 1 and ''A'' > 0 there exists a constant ''C'' > 0 such that

:<math> \sum_{1 \leq q \leq x^\theta} E(x;q) \leq \frac{C x}{\log^A x}</math>

for all ''x'' > 2.

This conjecture was proven for all ''θ'' < 1/2 by [[Enrico Bombieri]] and [[A. I. Vinogradov]] (the [[Bombieri–Vinogradov theorem]], sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the [[generalized Riemann hypothesis]]. It is known that the conjecture fails at the endpoint ''θ'' = 1.

The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by [[Dan Goldston]], [[János Pintz]], and [[Cem Yıldırım]],<ref>{{arxiv|math.NT/0508185}}; see also {{arxiv|math.NT/0505300}}, {{arxiv|math.NT/0506067}}.</ref> which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott-Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.

==See also==
*[[Barban–Davenport–Halberstam theorem]]
*[[Barban–Montgomery theorem]]

==Notes==
{{Reflist}}

== References ==
*{{cite journal |first=E. |last=Bombieri |title=On the large sieve |journal=Mathematika |volume=12 |issue= |year=1965 |pages=201–225 |doi= }}
*{{cite journal |first=P. D. T. A. |last=Elliott |first2=H. |last2=Halberstam |title=A conjecture in prime number theory |journal=Symp. Math. |volume=4 |year=1968 |issue= |pages=59–72 }}
*{{cite journal |first=A. I. |last=Vinogradov |title=The density hypothesis for Dirichlet L-series |language=Russian |journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=29 |issue=4 |year=1965 |pages=903–934 |mr=197414 }}
*{{cite journal |first=K. |last=Soundararajan |authorlink=Kannan Soundararajan |title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım |journal=Bull. AMS |volume=44 |year=2007 |issue=1 |pages=1–18 |doi=10.1090/S0273-0979-06-01142-6 }}

{{DEFAULTSORT:Elliott-Halberstam conjecture}}
[[Category:Analytic number theory]]
[[Category:Conjectures about prime numbers]]

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{{About|theoretical attempts to define income|its definition in United States law|Income (United States legal definitions)}}
{{refimprove|date=October 2010}}
'''Income''' is the consumption and savings opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms.<ref name="Barr"/> However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings received... in a given period of time."<ref name="Case & Fair">Case, K. & Fair, R. (2007). ''Principles of Economics''. Upper Saddle River, NJ: Pearson Education. p. 54.</ref>

In the field of [[public economics]], the term may refer to the accumulation of both monetary and non-monetary consumption ability, with the former (monetary) being used as a proxy for total income.<ref name="Barr"/>
==Increase in income==
Income per capita has been increasing steadily in almost every country.<ref>http://www.gapminder.org/world/#$majorMode=chart$is;shi=t;ly=2003;lb=f;il=t;fs=11;al=30;stl=t;st=t;nsl=t;se=t$wst;tts=C$ts;sp=5.59290322580644;ti=2007$zpv;v=0$inc_x;mmid=XCOORDS;iid=0AkBd6lyS3EmpdHo5S0J6ekhVOF9QaVhod05QSGV4T3c;by=ind$inc_y;mmid=YCOORDS;iid=rdCufG2vozTpKw7TBGbyoWw;by=ind$inc_s;uniValue=8.21;iid=phAwcNAVuyj0XOoBL_n5tAQ;by=ind$inc_c;uniValue=255;gid=CATID0;by=grp$map_x;scale=log;dataMin=58;dataMax=108111$map_y;scale=lin;dataMin=26;dataMax=56$map_s;sma=49;smi=2.65$cd;bd=0$inds=</ref> Many factors contribute to people having a higher income such as [[Education]],<ref>http://www.gapminder.org/world/#$majorMode=chart$is;shi=t;ly=2003;lb=f;il=t;fs=11;al=30;stl=t;st=t;nsl=t;se=t$wst;tts=C$ts;sp=5.59290322580644;ti=2007$zpv;v=0$inc_x;mmid=XCOORDS;iid=0AkBd6lyS3EmpdHo5S0J6ekhVOF9QaVhod05QSGV4T3c;by=ind$inc_y;mmid=YCOORDS;iid=pyj6tScZqmEdrsBnj2ROXAg;by=ind$inc_s;uniValue=8.21;iid=phAwcNAVuyj0XOoBL_n5tAQ;by=ind$inc_c;uniValue=255;gid=CATID0;by=grp$map_x;scale=log;dataMin=58;dataMax=108111$map_y;scale=lin;dataMin=8.7;dataMax=100$map_s;sma=49;smi=2.65$cd;bd=0$inds=</ref> [[globalisation]] and favorable political circumstances such as [[Index of Economic Freedom|economic freedom]] and [[peace]]. Increase income also tends to lead to people choosing to work less [[working hours]].
[[Developed countries]] defined as countries with a "developed economy" have higher incomes as opposed to a [[developing countries]] tend to have lower incomes.

==Economic definitions==
In [[economics]], "[[factor income]]" is the return accruing for a person, or a nation, derived from the "factors of production": rental income, wages generated by labor, the interest created by capital, and profits from entrepreneurial ventures.<ref>{{cite web|title=factor income|url=http://www.businessdictionary.com/definition/factor-income.html|work=BusinessDictionary.com|publisher=WebFinance, Inc|accessdate=20 June 2012|author=Staff|year=2012}}</ref>

From labor services, as well as ownership of [[Land (economics)|land]] and [[Capital (economics)|capital]].{{citation needed|date=June 2012}}

In [[consumer theory]] 'income' is another name for the "budget constraint," an amount <math>Y</math> to be spent on different goods x and y in quantities <math>x</math> and <math>y</math> at prices <math>P_x</math> and <math>P_y</math>. The basic equation for this is
:<math>Y=P_x \cdot x + P_y \cdot y</math>
This equation implies two things. First buying one more unit of good x implies buying <math>\frac{P_x}{P_y}</math> less units of good y. So, <math>\frac{P_x}{P_y}</math> is the ''relative'' price of a unit of x as to the number of units given up in y. Second, if the price of x falls for a fixed <math>Y</math>, then its relative price falls. The usual hypothesis is that the quantity demanded of x would increase at the lower price, the [[law of demand]]. The generalization to more than two goods consists of modelling y as a [[composite good]].

The theoretical generalization to more than one period is a multi-period [[wealth (economics)|wealth]] and income constraint. For example the same person can gain more productive skills or acquire more productive income-earning assets to earn a higher income. In the multi-period case, something might also happen to the economy beyond the control of the individual to reduce (or increase) the flow of income. Changing measured income and its relation to consumption over time might be modeled accordingly, such as in the [[permanent income hypothesis]].

===Full and Haig-Simons income===
{{main|Haig-Simons income}}
Full income refers to the accumulation of both, monetary and non-monetary consumption ability of any given entity, such a person or household. According to the what economist [[Nicholas Barr]] describes as the "classical definition of income:" the 1938 Haig-Simons definition, "income may be defined as the... sum of (1) the market value of rights exercised in consumption and (2) the change in the value of the store of property rights..." Since the consumption potential of non-monetary goods, such as leisure, cannot be measured, monetary income may be thought of as a proxy for full income.<ref name="Barr"/> As such, however, it is criticized for being unreliable, ''i.e.'' failing to accurately reflect affluence and that is consumption opportunities of any given agent. It omits the utility a person may derive from non-monetary income and, on a macroeconomic level, fails to accurately chart social welfare. According to Barr, "in practice money income as a proportion of total income varies widely and unsystematically. Non-observability of full-income prevent a complete characterization of the individual opportunity set, forcing us to use the unreliable yardstick of money income." On the macro-economic level, national per-capita income, increases with the consumption of activities that produce harm and omits many variables of societal health.<ref name="Barr">Barr, N. (2004). Problems and definition of measurement. In ''Economics of the welfare state''. New York: Oxford University Press. pp. 121-124</ref>

== Income inequality ==
[[Income inequality]] refers to the extent to which income is distributed in an uneven manner. Within a society can be measured by various methods, including the [[Lorenz curve]] and the [[Gini coefficient]]. Economists generally agree that certain amounts of inequality are necessary and desirable but that excessive inequality leads to efficiency problems and social injustice.<ref name="Barr"/>

National income, measured by statistics such as the [[Net National Income]] (NNI), measures the total income of individuals, corporations, and government in the economy. For more information see [[measures of national income and output]].

== Income in philosophy and ethics ==
Throughout history, many have written about the impact of income on [[morality]] and [[society]]. [[Saint Paul]] wrote 'For the love of money is a root of all kinds of evil:' ([[wikisource:Bible (American Standard)/1 Timothy#6:10|1 Timothy 6:10]] ([[American Standard Version|ASV]])).

Some scholars have come to the conclusion that material progress and prosperity, as manifested in continuous income growth at both individual and national level, provide the indispensable foundation for sustaining any kind of morality. This argument was explicitly given by [[Adam Smith]] in his ''Theory of Moral Sentiments''{{Citation needed|date=April 2009}}, and has more recently been developed by Harvard economist [[Benjamin M. Friedman|Benjamin Friedman]] in his book ''The Moral Consequences of Economic Growth''.{{Citation needed|date=April 2009}}

==Accountancy==
The [[International Accounting Standards Board]] (IASB) uses the following definition: "Income is increases in economic benefits during the accounting period in the form of inflows or enhancements of assets or decreases of liabilities that result in increases in equity, other than those relating to contributions from equity participants." [F.70] (IFRS Framework)

== See also ==
{{Wiktionary|income}}
*[[Basic income]]
*[[Comprehensive income]]
*[[Income tax]]
*[[Unpaid workers]]

==References==
{{reflist}}
*D. Usher (1987). "real income," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 4, pp. 104–05

[[Category:Generally Accepted Accounting Principles]]
[[Category:Income| ]]

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Area of a disk

Icosidodecahedron