Multiplicative Number Theory

Frontmatter

1. Primes in Arithmetic Progression

Abstract

Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet’s memoir of 1837 on the existence of primes in a given arithmetic progression.

Harold Davenport

2. Gauss’ Sum

Abstract

In this section we evaluate the sum

$$ G = \sum\limits_{m = 1}^{q - 1} {\left( {\frac{m}{q}} \right)} \,{e_q}(m), $$

where q is a prime other than 2. It is easy to prove that G ² = q if q≡1 (mod 4) and G ² = -q if q ≡ 3 (mod 4), though this does not determine G completely. The computation is as follows. We have

$$ {G^2} = \sum\limits_{{m_1} = 1}^{q - 1} {\,\sum\limits_{{m_2} = 1}^{q - 1} {\left( {\frac{{{m_1}{m_2}}}{q}} \right)\,{e_q}\left( {{m_1} + {m_2}} \right).} } \, $$

.

Harold Davenport

3. Cyclotomy

Abstract

Cyclotomy is concerned with the properties of the roots of unity of a given order, with particular reference to their algebraic character.¹ Our first object must be to establish the result quoted in §1, and this we can do without going very deeply into the theory. Afterward I shall digress briefly from the main theme of these lectures to discuss two topics in cyclotomy which are of general interest.

Harold Davenport

4. Primes in Arithmetic Progression: The General Modulus

Abstract

Dirichlet’s proof of the existence of primes in a given arithmetiω progression, in the general case when the modulus q is not necessarily a prime, is in principle a natural extension of that in the special case. But the proof given in §1 that L_ω(1) ≠ 0 when ω = - 1, which involved separate consideration of the cases q ≡ 1 and q 3 (mod 4), does not extend to give the analogous result that is needed when q is composite.

Harold Davenport

5. Primitive Characters

Abstract

Many results about characters and L functions take a simple form only for the so-called primitive characters, though they may be capable of extension, with complications, to imprimitive characters. We shall now explain the distinction between these two types of character, and afterward investigate in detail the real primitive characters.

Harold Davenport

6. Dirichlet’s Class Number Formula

Abstract

Dirichlet’s class number formula, in its simplest and most striking form, was conjectured by Jacobi¹ in 1832 and (as we said in §1) proved in full by Dirichlet in 1839.

Harold Davenport

7. The Distribution of the Primes

Abstract

Legendre was the first, as far as we know, to make any significant conjecture about the distribution of the primes. Let π(x) denote the number of primes not exceeding x. Then Legendre conjectured, somewhat tentatively, that for large x the number π(x) is given approximately by

$$ \frac{x}{{\log x - 1.08...}} $$

.

Harold Davenport

8. Riemann’s Memoir

Abstract

In his epoch-making memoir of 1860 (his only paper on the theory of numbers) Riemann showed that the key to the deeper investigation of the distribution of the primes lies in the study of ζ(s) as a function of the complex variable s. More than 30 years were to elapse, however, before any of Riemann’s conjectures were proved, or any specific results about primes were established on the lines which he had indicated.

Harold Davenport

9. The Functional Equation of the L Functions

Abstract

The functional equation for Dirichlet’s L functions was first given by Hurwitz in 1882 (Werke I, pp. 72–88), though he confined himself to real characters since he was primarily interested in L functions in relation to quadratic forms. He first obtained the functional equation for the more general ζ function ζ(s, w), which will be given below, and deduced that of the L functions from it. We shall follow the method used by de la Vallée Poussin in 1896, which is an extension of that of Riemann used in the preceding section.

Harold Davenport

10. Properties of the Γ Function

Abstract

We collect some properties of the Γ function for convenience of reference.

Harold Davenport

11. Integral Functions of Order 1

Abstract

The next important progress in the theory of the ζ function, after Riemann’s pioneering paper, was made by Hadamard, who developed the theory of integral functions of finite order in the early 1890’s and applied it to ζ(s) via ξ(s). His results were used in both the proofs of the prime number theorem, given by himself and by de la Vallée Poussin, though later it was found that for the particular purpose of proving the prime number theorem, they could be dispensed with.

Harold Davenport

12. The Infinite Products for ξ(s) and ξ(s, χ)

Abstract

We apply the conclusions of the preceding section to the integral function

$$ \xi (s) = \frac{1}{2}s(s - 1){\pi ^{ - \frac{1}{2}s}}\Gamma (\frac{1}{2}s)\zeta (s) $$

.

Harold Davenport

13. A Zero-Free Region for ζ(s)

Abstract

It was proved independently by Hadamard and de la Vallée Poussin in 1896 that ζ(s) ≠ 0 on σ = 1. This was a vital step in their proofs of the prime number theorem, and it remained a vital step in all subsequent proofs until the discovery of an elementary proof¹ by Selberg and Erdös in 1948.

Harold Davenport

14. Zero-Free Regions for L(s, χ)

Abstract

There is no difficulty in extending the results of the preceding section to the zeros of L(s, χ) when χ is a fixed character. But this is of limited value ; for many purposes it is important to allow q to vary and to have estimates that are explicit in respect of q. This raises some difficult problems, and the results so far known are better for complex characters than for real characters.

Harold Davenport

15. The Number N(T)

Abstract

In this section we prove the approximate formula for N(T), the number of zeros of ζ(s) in the rectangle 0 > σ > 1, 0 > t > T, which was stated by Riemann and established by von Mangoldt. It was stated as (1) in §8.

Harold Davenport

16. The Number N(T, χ)

Abstract

Let χ be a primitive character to the modulus q, and let N(T, χ) denote the number of zeros of L(s, χ) in the rectangle

$$ 0 < \sigma < 1,{\kern 1pt} \left| t \right| < T $$

.

Harold Davenport

17. The Explicit Formula for Ψ(x)

Abstract

In this section we shall prove von Mangoldt’s formula for Ψ(x), which was stated in §8. We recall that

$$ \psi (x) = \sum\limits_{n \le x} {\Lambda (n) = \sum\limits_{{p^m} \le x} {\log p} } $$

.

Harold Davenport

18. The Prime Number Theorem

Abstract

We shall now deduce, from the results of the last section and those of §13, that

$$\psi (x) = x + 0\left\{ {x\exp {{\left[ {\log x} \right]}^{\frac{1}{2}}}} \right\}$$

(1)

and from this the analogous result for π(x), which includes the prime number theorem. This is by no means the easiest way of proving the prime number theorem, but it is an instructive way. It is also very close to the method used by de la Vallée Poussin, though he worked with the function

$${\psi _1}(x) = \sum\limits_{n \le x} {(x - n)\Lambda (n)}$$

instead of the function φ(x).

Harold Davenport

19. The Explicit Formula for ψ(x, χ)

Abstract

For any character χ to the modulus q, we define

$$ \psi (x,\chi ) = \sum\limits_{n \le x} {\chi (n)\Lambda (n)} $$

(1)

.

Harold Davenport

20. The Prime Number Theorem for Arithmetic Progressions (I)

Abstract

We now apply the last result of the preceding section to obtain approximations to

$$ \psi (x;q,a) = \sum\limits_{{}_{n \equiv a(\bmod q)}n \le x} {\Lambda (n)} $$

(1)

.

Harold Davenport

21. Siegel’s Theorem

Abstract

Siegel’s theorem,¹ in the first of its two forms, states that: For any ε > 0 there exists a positive number C ₁(ε) such that, if χ is a real primitive character to the modulus q, then

$$ L(1,\chi)>{C_1}(\varepsilon){q^{- \varepsilon}} $$

(1)

.

Harold Davenport

22. The Prime Number Theorem for Arithmetic Progressions (II)

Abstract

By appealing to Siegel’s theorem we can obtain a better approximation to ψ(x; q, a) than was possible in §20.

Harold Davenport

23. The Pólya-Vinogradov Inequality

Abstract

Suppose that χ is a nonprincipal character (mod q). Since

$$ \sum\nolimits_{n = 1}^q {\chi \left( n \right)} = 0 $$

, it is clear that

$$ \sum\nolimits_{n = M + 1}^{M + N} {\chi \left( n \right) \ll q} $$

for any M and N. However, a sharper bound is needed to describe the distribution of power residues within the interval 1 ≤ n ≤ q.

Harold Davenport

24. Further Prime Number Sums

Abstract

When f is monotonic we can use the prime number theorem and partial summation to estimate Σ_p≤N f(p). For certain multiplicative functions, namely those of the form f(n) = χ(n)n ^-s, we can estimate Σ_p≤N f(p) by using the zero-free region of L(s, χ). In 1937 Vinogradov introduced a method for estimating sums Σ_p≤N f(p) which f is oscillatory but not multiplicative.¹ His starting point was a simple sieve idea. Let

$$ P = {\Pi _{p < {N^{{\textstyle{1 \over 2}}}}p}} $$

. For n in the range 1 ≤ n ≤ N the sieve of Eratosthenes asserts that (n, P) = 1 if and only if n = 1 or n is a prime number in the interval N < < n ≤ N. Hence

$$ f\left( 1 \right) + \sum\limits_{{N^{\frac{1}{2}}} < p \le N} {f\left( p \right) = \sum\limits_{\mathop {n \le N}\limits_{\left( {n.P} \right) = 1} } {f\left( n \right) = \sum\limits_{\mathop {t|P}\limits_{t \le N} } {\mu \left( t \right)\sum\limits_{r \le N/t} {f\left( {rt} \right)} } } } $$

.

Harold Davenport

25. An Exponential Sum Formed with Primes

Abstract

Vinogradov first used his method to estimate the important sum

$$ S(\alpha ) = \sum\limits_{n \le N} { \wedge (n)e(n\alpha )} s$$

.

Harold Davenport

26. Sums of Three Primes

Abstract

Hardy and Littlewood¹ showed, assuming the generalized Riemann hypothesis, that every sufficiently large odd number is a sum of three primes. In their argument, the hypothesis was required to provide estimates corresponding to our estimates of S(α) in §25. In 1937 Vinogradov² used his new estimates to treat sums of three primes unconditionally.

Harold Davenport

27. The Large Sieve

Abstract

The large sieve was first proposed by Linnik¹ in a short but important paper of 1941. In a subsequent series of papers, Rényi developed the method by adopting a probabilistic attitude. His estimates were not optimal, and in 1965 Roth² substantially modified Rényi’s approach to obtain an essentially optimal result. Bombieri³ further refined the large sieve, and used it to describe the distribution of primes in arithmetic progressions; this we shall discuss in the following section.

Harold Davenport

28. Bombieri’s Theorem

Abstract

Rényi used the large sieve to show that prime numbers are well distributed in arithmetic progressions (mod q) for most q; his rather complicated result allowed him to show that every large even number is representable in the form

$$p + {p_1}{p_2} \ldots {p_r},$$

where r is bounded by some absolute constant. The subsequent refinements of Bombieri¹ and A. I. Vinogradov² enable one to take r = 3, and recently Chen³ has added an ingenious new idea to obtain r = 2.

Harold Davenport

29. An Average Result

Abstract

We now consider the mean square error in the prime number theorem for arithmetic progressions. Work in this direction was initiated by Barban¹, and by Davenport and Halberstam². Their results were sharpened by Gallagher³, who showed that

$$ {\sum\limits_{q \le Q} {\sum\limits_{\scriptstyle a = 1 \hfill \atop\scriptstyle (a,q = 1) \hfill}^q {\left( {\psi \left( {x;q,a} \right) - \frac{x}{{{\o}\left( q \right)}}} \right)} } ^2} \ll xQ\log x $$

(1)

for x(log x) ^-A ≤ Q ≤ x; here A > 0 is fixed. This estimate is best possible, for Montgomery⁴ has shown that the left-hand side is ~ Qx log x for Q in the stated range. Moreover, Hooley⁵ has shown that (1) can be combined with some of Montgomery’s ideas to give, in a simple way, a very precise asymptotic estimate.

Harold Davenport

30. References to other Work

Abstract

The principal omission in these lectures has been the lack of any account of work on irregularities of distributions, both of the primes as a whole and of the primes in the various progressions to the same modulus q.

Harold Davenport

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