Summatory Function of σ₁(n)

• The summatory function of σ₁(n) is defined as S(x)= Σₙ≤ₓ σ₁(n), aggregating divisor sums to capture essential number-theoretic insights.
• It admits multiple representations including divisor-floor sums, Lambert series, and Eisenstein series, facilitating both theoretical analysis and efficient computation.
• Its precise asymptotic expansion, with key error bounds linked to the Riemann hypothesis, underscores its central role in analytic and computational number theory.

The summatory function of $\sigma_1(n)$, commonly denoted $S(x) = \sum_{n \leq x} \sigma_1(n)$, where $\sigma_1(n)$ is the sum of the positive divisors of $n$, plays a central role in analytic number theory, classical modular forms, and the study of divisor problems. This function admits both explicit finite expressions and highly precise asymptotic expansions, and is tightly linked to deep structures such as the Riemann zeta function and Ramanujan’s $\tau$-function. Its analytic properties encode significant combinatorial and number-theoretic information, and the main term and fluctuation phenomena associated with $S(x)$ provide important benchmarks for research in multiplicative function theory and the spectral analysis of arithmetic functions.

1. Generating Functions and Lambert Series Representations

The generating function for $\sigma_1(n)$ exhibits multiple complementary analytic presentations. A fundamental identity is

$$- \sum_{n=1}^\infty \ln(1 - q^n) = \sum_{n=1}^\infty \frac{\sigma_1(n)}{n} q^n, \quad |q| < 1,$$

from which one deduces, via differentiation,

$$G(q) := \sum_{n=1}^\infty \sigma_1(n) q^n = \sum_{m=1}^\infty \frac{m q^m}{1 - q^m}.$$

This is the classical Lambert series form (Wang, 2011). Alternatively, the generating function is encapsulated as a modular object: $$\sum_{n=1}^\infty \sigma_1(n) q^n = \frac{1 - E_2(q)}{24},$$ where $S(x) = \sum_{n \leq x} \sigma_1(n)$0 is the normalized normalized weight-2 Eisenstein series. This interplay between Lambert series and Eisenstein series connects the analysis of $S(x) = \sum_{n \leq x} \sigma_1(n)$1 to the theory of modular forms.

2. Explicit Finite Expressions for $S(x) = \sum_{n \leq x} \sigma_1(n)$2

The summatory function $S(x) = \sum_{n \leq x} \sigma_1(n)$3 may be expressed exactly via a divisor-floor sum: $S(x) = \sum_{n \leq x} \sigma_1(n)$4 This formula arises directly from interchanging the order of summation after expanding $S(x) = \sum_{n \leq x} \sigma_1(n)$5 (Wang, 2011).

Recent work delivers alternative explicit forms using cyclotomic and Ramanujan expansions: $S(x) = \sum_{n \leq x} \sigma_1(n)$6 where the terms respectively involve the “zeroth” harmonic number, a Ramanujan sum component, and two pieces corresponding to prime power and alternating prime counting contributions. Each is a fully finite sum, with computational complexity dominated by the evaluation of Ramanujan sums and enumeration over relevant primes and their exponents (Schmidt, 2017). This decomposition facilitates efficient, large-scale computational evaluations.

3. Asymptotic Expansions and Error Terms

The classical Dirichlet hyperbola method and Mellin inversion techniques yield the precise leading-order asymptotics: $S(x) = \sum_{n \leq x} \sigma_1(n)$7 with $S(x) = \sum_{n \leq x} \sigma_1(n)$8 the Euler-Mascheroni constant (Wang, 2011, Fel, 2011, Schmidt, 2017, Iwata, 16 Jan 2026). The $S(x) = \sum_{n \leq x} \sigma_1(n)$9-term reflects the state-of-the-art in the error bound for the Dirichlet divisor problem, with the exponent $\sigma_1(n)$0 quoted as “routine,” and the best record approaching $\sigma_1(n)$1.

The scaling and renormalization approach contextualizes $\sigma_1(n)$2 within a universal class for multiplicative arithmetic summatory functions, namely with $\sigma_1(n)$3, $\sigma_1(n)$4, $\sigma_1(n)$5, $\sigma_1(n)$6. Explicitly, in this framework: $\sigma_1(n)$7 The main term constant arises as $\sigma_1(n)$8 (Fel, 2011).

4. Analytic Decomposition and the Riemann Hypothesis

Iwata’s analysis (Iwata, 16 Jan 2026) provides a further decomposition of the error beyond the main term. Defining

$\sigma_1(n)$9

one obtains for $n$0: $n$1 where

$n$2

and

$n$3

Here $n$4 denotes the fractional part. The term $n$5 is considered the arithmetic part of the error, $n$6 the analytic part.

Assuming the Riemann Hypothesis, sharper bounds are established: $n$7 for every $n$8; equivalently, for all $n$9, $\tau$0 (Iwata, 16 Jan 2026). This provides explicit conditional control on the size of the error in $\tau$1 in terms of the zeros of the Riemann zeta function.

5. Connections to Modular Forms and Ramanujan’s $\tau$2-Function

A notable link connects the generating function for $\tau$3 to modular forms, especially via Ramanujan’s discriminant modular form $\tau$4: $\tau$5 This can be rewritten in terms of $\tau$6 as

$\tau$7

Thus, polynomial relations exist between $\tau$8 and the sequence $\tau$9. Deligne’s bound $S(x)$0 imposes nontrivial constraints on mean values of $S(x)$1, but does not yield sharper error exponents for $S(x)$2 than those provided by traditional divisor-problem approaches (Wang, 2011). No sub-$S(x)$3 exponent for the error term has been attained via the theory of modular forms or $S(x)$4 alone.

6. Computational Considerations and Practical Evaluation

The various explicit and finite-sum formulas for $S(x)$5 permit practical computations at very large scales. Both the divisor-floor representation and cyclotomic/Ramanujan sum decompositions are fully explicit and easily parallelizable. Efficient sieving methods permit the evaluation of all required summands—including $S(x)$6 Ramanujan sums and prime power floors—in $S(x)$7 time or faster for large $S(x)$8 (Schmidt, 2017). Practical computations for $S(x)$9 are achievable in seconds with optimized numerical routines. For enhanced efficiency with extremely large $\sigma_1(n)$0, block-wise sieving and memory management may be employed.

7. Summary of Main Representations

The principal exact and asymptotic expressions for the summatory function are:

Formula Type	Expression	Source
Divisor-floor sum	$\sigma_1(n)$1	(Wang, 2011)
Lambert series	$\sigma_1(n)$2	(Wang, 2011)
Eisenstein series (modular form)	$\sigma_1(n)$3	(Wang, 2011)
Cyclotomic-Ramanujan sum	$\sigma_1(n)$4 sum of $\sigma_1(n)$5 (see section 2 above)	(Schmidt, 2017)
Asymptotic expansion	$\sigma_1(n)$6	(Wang, 2011, Fel, 2011, Schmidt, 2017, Iwata, 16 Jan 2026)

All of these representations are fundamentally equivalent and arise from the interplay of divisor sums, modular forms, analytic techniques (Mellin transforms, Dirichlet hyperbola, residue calculus), and explicit algebraic expansions.

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References:

(Wang, 2011, Schmidt, 2017, Iwata, 16 Jan 2026, Fel, 2011)