Analytic Number Theory

Prerequisite path Preparation & skip guidance · Elementary Number Theory · Complex Analysis & Several Complex Variables

Studying Analytic Number Theory

Analytic number theory combines arithmetic structure with estimates, so manipulating a Dirichlet series is only the beginning of an argument. Kedlaya organizes a route through multiplicative functions, Euler products and complex methods for prime distribution. Richert develops sieve techniques as a focused branch. Choose the first route while learning which sums and products may be rearranged; choose the sieve continuation after asymptotic estimates have become dependable.

What to work on

Justify an Euler product in a half-plane of absolute convergence.
Use Dirichlet convolution and Mobius inversion.
Identify the Dirichlet series of divisor and von Mangoldt functions.
Distinguish the analytic input for prime-number and arithmetic-progression theorems.

A starting text

Notes on Analytic Number Theory — Kiran S. Kedlaya. Begin with 2.1-2.3 on Dirichlet series, multiplicativity and Euler products. Then read 1.2-1.4 on zeta and the Tauberian route to the prime number theorem; 3.1-3.4 and 4.1-4.3 introduce characters, L-functions and primes in progressions.

Choosing what to read

Prove an arithmetic-function identity by counting divisor pairs, then justify the corresponding Dirichlet-series identity in a region of absolute convergence. Write error terms with the variable and constants they depend on. Use the hyperbola method on a divisor sum to see how a geometric decomposition produces an estimate. Before reading a prime-number theorem proof, separate its analytic input from its arithmetic conclusion. For sieve arguments, identify the set being sifted and the local exclusions before handling weights.

Notes on Analytic Number Theory

Kiran S. Kedlaya

Learn to justify Euler products, convolutions and the first asymptotic arguments.

How it reads: Course notes linking arithmetic functions with Dirichlet series and prime distribution.
What it expects: Elementary number theory and complex analysis.
What to keep in mind: The opening selection is preparation for deeper prime-distribution theorems, not evidence of mastering their proofs.
Practice: Practice page for this work at https://kskedlaya.org/ant/part-1-3.html: Public exercises 2.4.1-2.4.10 verified, including a hint to 2.4.7. Public worked solutions were not verified. Section 2.4: use exercises 3-4 for convolution and arithmetic functions, 6 for von Mangoldt and the logarithmic derivative, and 9 for the hyperbola method.

Complete public HTML text.

Edition, reading range and source

Begin with 2.1-2.3 on Dirichlet series, multiplicativity and Euler products. Then read 1.2-1.4 on zeta and the Tauberian route to the prime number theorem; 3.1-3.4 and 4.1-4.3 introduce characters, L-functions and primes in progressions.

Source · Checked 2026-10-16

Lectures on Sieve Methods

Hans-Egon Richert

Study a separate method for bounding sifted sets after foundational analytic reading.

How it reads: Sustained specialized exposition of sieve techniques.
What it expects: Analytic number theory and asymptotic estimates.
What to keep in mind: Historical notation may differ from current treatments; this is a specialization rather than a general first course.
Practice: Public worked solutions were not verified; work a small sifted-set example as an editorial task alongside the text.

Complete TIFR archive PDF; individual-use access only, with no redistribution.

Edition, reading range and source

Follow the archive contents into the sieve-method development after reviewing the prerequisites; exact section ranges were not checked.

Source · Checked 2026-10-11

Practice

Dirichlet-series and arithmetic-function exercises. Section 2.4: use exercises 3-4 for convolution and arithmetic functions, 6 for von Mangoldt and the logarithmic derivative, and 9 for the hyperbola method.. Public exercises 2.4.1-2.4.10 verified, including a hint to 2.4.7. Public worked solutions were not verified.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

For complex s with real part greater than 1, prove that the Dirichlet series of the divisor-counting function d(n) equals zeta(s) squared. Justify rearranging the double sum.
A hint
Each positive divisor corresponds to an ordered pair ab=n. Bound the absolute double sum at the real part of s.
Work through the solution
Let $s\in\mathbb C$ , $\sigma=\operatorname{Re}s>1$ , and define $n^{-s}=e^{-s\log n}$ with the real logarithm of positive $n$ . The positive series $\sum n^{-\sigma}$ converges by the integral test. Therefore

$\sum_{a,b\ge1}|a^{-s}b^{-s}|=\left(\sum_{a\ge1}a^{-\sigma}\right)^2=\zeta(\sigma)^2<\infty.$

Absolute convergence justifies multiplying the two series for $\zeta(s)$ and regrouping their terms by $n=ab$ . For every positive integer $n$ , there are exactly $d(n)$ ordered pairs $(a,b)$ of positive integers with product $n$ , since $a$ determines $b=n/a$ . Also $a^{-s}b^{-s}=(ab)^{-s}$ . Consequently

$\zeta(s)^2=\sum_{a,b\ge1}(ab)^{-s}=\sum_{n\ge1}d(n)n^{-s}.$

This argument proves absolute convergence of the final series as well; it does not assert the identity outside $\operatorname{Re}s>1$ .

If this was the sticking point:
- Real Analysis — Review absolute convergence and rearrangements of infinite sums.
- Elementary Number Theory — Review the divisor function and divisor-pair counting.

Already comfortable with this? Skip arithmetic-function review if you can derive both identities and control convergence. Continue with the zeta and L-function readings if analytic continuation, nonvanishing and Tauberian hypotheses are still gaps; a formal Euler-product expansion alone does not justify the prime number theorem.

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Where this can lead

Complex Analysis & Several Complex Variables — Review analytic continuation, zeros and contour estimates before complex prime-distribution methods.
Elementary Number Theory — Repair multiplicativity and congruence calculations before advanced estimates.

Reading-source dates

Notes on Analytic Number Theory · 2026-10-16
Dirichlet-series and arithmetic-function exercises · 2026-10-16

Reading choices revised 2026-10-17.

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Notes on Analytic Number Theory

Online text

Kiran S. Kedlaya · Course notes · Advanced

A graduate course on primes, analytic methods and sieve ideas. The freely readable HTML format supports selective study; complex analysis is an essential preparation.

Preparation: Complex analysis; elementary number theory.

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Additional reading

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Graduate reading

Lectures on the Riemann Zeta-Function

K. Chandrasekharan · Long lecture notes · Graduate · English · PDF

Develops analytic properties of the zeta function. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.